Temperaments. Tunings of the scale in which some or all of the concords are made slightly
impure in order that few or none will be left distastefully so. Equal temperament, in which the
octave is divided into 12 uniform semitones, is the standard Western temperament today except
among specialists in early music. This article traces the history of temperaments in performing
practice and in relation to the main lines of development in the history of harmony; for additional
technical and historical details, see TUNING, PYTHAGOREAN INTONATION, JUST INTONATION,
MEANTONE, WELL-TEMPERED CLAVIER, EQUAL TEMPERAMENT and INTERVAL.
1. Introduction. Since the 15th century, tempered tuning has characterized keyboard music and
(in Western culture) the art music of fretted instruments such as the lute. Its prevalence is due
mainly to the fact that the concords of triadic music – octaves, 5ths and 3rds – are in many cases
incommensurate in their pure forms. Three pure major 3rds (e.g. A -C-E-G ) fall short of a pure
octave by approximately one fifth of a whole tone (lesser diesis); four pure minor 3rds (G -B-DF-A ) exceed an octave by half as much again (greater diesis); the circle of twelve 5ths, if the
5ths are pure, does not quite cumulate in a perfect unison; and, most important of all in the context of Renaissance and Baroque music, the whole tone produced by subtracting a pure minor
3rd from a pure 4th (C-F-D) is about 11% smaller than the whole tone produced by subtracting
a pure 4th from a pure 5th (C-G-D). These discrepancies are summarized in Table 1.
Table 1
Not all timbres, however, are equally conducive to temperament. In general, it is only when the
component overtones of the timbre (together with the fundamental tone of the note) form a virtually pure harmonic series that consonant intervals will sound sufficiently different in quality
from dissonant ones for the need for tempering the concords to arise. A pronounced degree of
inharmonicity in the timbre, as in a set of chimes or a xylophone, eliminates the qualitative difference, except in the case of a unison or octave, between the sound of a pure concord and that
of a slightly impure or tempered one. (An artful exception to this general rule is mentioned in
the last paragraph of §6 below.)
Moreover, theorists are obliged to contrive specific mathematical schemes for tempering the
scale only when the medium of performance allows little or no flexibility of intonation, for only
then does the incommensurability of pure concords have to be dealt with systematically by the
tuner rather than ad hoc by the performer. A singer or violinist will inflect certain intervals depending on their immediate context; but this does not produce thereby a specific temperament
because at different moments the performer will represent each note of the scale by different
shades of pitch within a fairly narrow band (about half as wide again, perhaps, as the vibrato).
On most keyboard instruments and the harp, however, the complete lack of such flexibility obliges the tuner to impose a specific temperament, while on fretted instruments, the clavichord,
and many wind instruments, the tuner or maker establishes a certain model of intonation, albeit
one that an ingenious player can modify significantly during performance.
For present purposes four main types of temperament may be distinguished: equal temperament,
meantone temperament, irregular temperaments and quasi-Pythagorean temperament. This essay
will describe primarily the history of their use, but some plain and easy arithmetic is necessary
here to distinguish one from another.
For a series of twelve 5ths or 4ths to produce cumulatively a perfect unison and so comprise the
‘circle of 5ths’, each must be tempered by an average of 1/12 of the Pythagorean comma (hence
about two cents) – 4ths larger than pure, 5ths smaller. If three major 3rds are to reach a full octave they must average about 14 cents (1/3 of the lesser diesis) larger than pure, while four minor
3rds, in order not to exceed a pure octave, must average about 16 cents (1/4 of the greater diesis)
smaller than pure. In equal temperament, which is modeled on these averages, the major 3rds are
thus tempered seven times as much as the 5ths, and the minor 3rds and major 6ths eight times as
much. Historically this is nearly as important an aspect of equal temperament as the well-known
fact that it divides the octave into 12 equal semitones. As a harmonic interval the major 3rd f-a
at modern concert pitch, for instance, beats seven times per second – too fast for the ear to trace,
even subliminally, the rise and fall of each beat, and fast enough to cause an intermittence of
tone that is likely to strike an unaccustomed ear as unpleasant in many contexts.
To avoid such heavily tempered 3rds and 6ths, Renaissance and Baroque keyboard musicians
tempered their 5ths two or even three times as much as in equal temperament. This choice entailed ‘breaking’ the circle of 5ths, a fact reflected in traditional pitch notation and nomenclature
with their enharmonic distinctions between flats and sharps. If all the good 5ths in such a tuning
are diminished the same amount, the result may be called a regular meantone temperament, although some writers restrict the term ‘meantone’ to that scheme in which 5ths are tempered so
as to produce exactly pure major 3rds. In any meantone temperament the diatonic semitones are
larger than the chromatic (for instance, D is lower in pitch than E ).
Certain irregular keyboard temperaments, in which different 5ths are tuned differently but none
rendered unserviceable, were favoured during the late 17th century and the 18th because they
enabled the more frequently used 3rds to be tempered less than those used infrequently, and because the various keys thereby gained a diversity of intonational shading that was highly valued
by connoisseurs and formed a prominent aspect of 18th-century musical thought. In 17th-century
France an irregular scheme midway between a regular meantone and a typical 18th-century irregular temperament seems to have played a role in the contemporaneous development of the
French keyboard style. Yet another, very different kind of temperament – ‘quasi-Pythagorean’ –
can be derived from the fact that the sum of two major 3rds each of which is larger than pure by
a syntonic comma (for example, A -C-E when the 5th and 4ths A -E -B -F-C-G-D-A-E are all
pure) falls short of an octave by almost a pure major 3rd (in this case E-G ).
The word ‘temper’ and its derivatives were originally applied to music in a broader context than
they now denote. Carter’s Dictionary of Middle English Musical Terms (1961) indicates that the
adjective ‘temperate’ was once used in a general sense to mean ‘musically controlled’, as in a
reference (1398) by John of Trevisa to a ‘swete voys an temperate sowne’. But the verb ‘temperen’ and its past participle ‘tempered’ referred in Middle English unequivocally to tuning,
albeit pure tuning as well as tempered tuning in the modern sense; and this meaning was still
encountered in 1593: ‘Whereupon M. Barleycap tempered up his fiddle, and began’ (‘Temper’,
OED). In the writings of Zarlino, the eminent 16th-century Italian theorist and musician, ‘temperamento’ was closely associated with the term ‘participatio’ by which, according to Gaffurius
in 1496, Renaissance organists referred to their use of 5ths diminished ‘by a very small and hidden and somewhat uncertain quantity’.
2. Quasi-Pythagorean temperament. Late 14th- and early 15th-century keyboard instruments
that were fully chromatic, that is, with 12 notes in the octave, were normally tuned with 11 pure
5ths among the naturals and hence automatically with one sour 5th, a comma smaller than pure,
which was, by most accounts between B and F (the evidence for this comes mainly from Italy
and southern France: Dijon, Liège, Padua, Mantua, Parma, Ferrara, Milan, Florence, Urbino).
Such a tuning happens to yield virtually pure 3rds between the naturals and sharps (D-F -A-C E-G -B-D ) and much less nearly pure 3rds, a comma larger or smaller than pure, among the
naturals (D-F-A-C-E-G-B-D) and between the naturals and flats (D -F-A -C-E -G-B -D). The
apparent consequences in some extant early to mid-15th-century keyboard music are discussed in
PYTHAGOREAN INTONATION. The kind of triadic harmony which was involved – wherein, among
the major triads, those on D, A and E are treated as distinctly more euphonious and stable than
those on B , F, C and G – can be found also in non-keyboard compositions by Matteo da Perugia, Andrea dei Servi and, in his earliest songs, Guillaume Du Fay (Lindley and Boone).
Singers can hardly be expected to have maintained the distinction between pure and impure triads; if they could manage a fairly pure triad at one place in the music, they would most likely be
unwilling to produce on purpose a distinctly impure one somewhere else. However, the notion
that a chain of ‘perfect’ concords (5ths, 4ths) yields ‘imperfect’ ones (3rds, 6ths) which are so
euphonious that they serve as stable sonorities in the harmony, as in Exx.1 and 2, is a hallmark
of temperament. And the fact that even in these vocal works, only triads containing a sharp were
treated as stable (see Ex.19 and the discussion in §11) suggests that some kind of keyboard instrument, perhaps the clavichord if not the organ, was used as a help in composing them. The
theoretical amounts of tempering – less than 1/4 of a cent for each 5th – are so small that no-one
at the time noticed; and indeed Helmholtz in the 19th century tempered in this way the intervals
on his ‘justly intoned harmonium’ with 24 different pitches in each octave (see JUST INTONATION, §2).
Ex.1. Du Fay, Mon chier amy (GB-Ob Can.misc.213, fasc.9, p.314), end of the penultimate
section
Ex.2. Matteo da Perugia, Pres du soloeil (I-MOe".M.5.24, f.16), excerpt
3. Regular meantone temperaments to 1600. In the last chapter of his Musica practica (Bologna 1482) Bartolomeo Ramis de Pareia indicated that meantone temperament was in common
use on keyboard instruments of his day. Earlier in the book he had presented a new, mathematically simple scheme which he said even young singers could use for a monochord to provide a
model scale for plainchant. In that scheme the wolf 5th was ostensibly between G and D. But his
elaborate and comprehensive rules, in the last chapter, for avoiding ‘bad’ intervals on fully chromatic instruments (‘instrumenta perfecta’) leave no doubt that in general practice G–D was perfectly serviceable and the wolf 5th was between C and A . This is corroborated by his lists of
‘good’ and ‘bad’ semitones, whole tones, and major and minor 3rds, and by the pattern of his
nomenclature for the sharps and flats. Ramis expressed particular concern that only ‘good’ major
6ths and 3rds be used in cadential progressions. There is confirmation that ‘good’ did not mean
‘Pythagorean’ in a letter (published in 1518) from his disciple Giovanni Spataro to Franchinus
Gaffurius. Criticizing Gaffurius for saying that the syntonic comma was an insignificant intervallic quantity, Spataro argued, on behalf of Ramis, that for practical music the harsh Pythagorean
monochord had to be reduced and smoothed for the ear (‘el duro monochordo pythagorico …
riducto in molle al senso de lo audito’). Gaffurius, although a strict Pythagorean in his mathematical calculations of intervals, acknowledged in 1496 that organists tempered their 5ths, while
his chromatic monochord of 1518 (reprinted in 1520) indicates that G and D differed from A
and E . Ramis’s discussion of ‘good’ cadential 6ths and 3rds mentions the device, which later
became widespread in Italy on keyboard instruments designed for meantone temperament (see
Barnes 1971 and Dupont 1935), of doubling the accidentals A and E to provide their enharmonic neighbors G and D ; and the 1480 contract for the cathedral organ at Lucca appears to
refer to such a device. The use of regular meantone temperaments had probably evolved from
ad hoc alterations of certain earlier 15th-century untempered schemes in which, as in Ramis’s
own pedagogical monochord, the wolf 5th was placed ostensibly among the naturals (Lindley,
1975-76). Perhaps the oldest known keyboard composer whose music requires a meantone temperament for its proper effect was Conrad Paumann. His use of triads, as in Ex.3, suggests that
he regarded them as solid vertical sonorities, more so than they would in fact be in Pythagorean
intonation. The use of triads in contemporary vocal music (e.g. Ockeghem, Busnoys and late
Du Fay) is similar in this respect.
Ex.3. Paumann (D-Mbs Cim 325b, f.104v)
No particular shade of meantone temperament on keyboard instruments – such as 2/7-comma,
1/4-comma or 1/5-comma (see MEANTONE) – was favored exclusively at any time during the
Renaissance. The earliest mathematically specific formula was Zarlino’s scheme of 1558 for
2/7-comma meantone, in which major 3rds are very slightly smaller than pure. But Lanfranco’s
keyboard tuning instructions of 1533 are unequivocally for some form of meantone, such as 1/5comma or 1/6-comma, with major 3rds slightly larger than pure (and with the wolf 5th lying between G and E instead of between C and A as Ramis suggests). The same can be inferred
from other writers, and not just those who cited Lanfranco. Schlick had implied in 1511 that
some organists would prefer such a tuning to his own irregular scheme (see §4 below). The
clavichord tuning instructions published by Tomás de Santa María in 1565, although clearly
for some form of meantone, do not specify the exact quality of the major 3rds; but when Pietro
Cerone copied these instructions in 1613 he added a few words to specify major 3rds slightly
larger than pure and declared that this was the method most used by master organ builders (‘es
manera mas usada de los Maestros de hazer Organos’). An approximation of 2/9-comma meantone (with major 3rds very slightly larger than pure) was described in 1590 by Cyriacus Schneegass, who may have confused it with 1/4-comma meantone.
Meanwhile Zarlino in 1571 described 1/4-comma meantone, which has pure major 3rds, in mathematically clear terms. He said it was new (‘un novo Temperamento & … una nova Participatione’); but Francisco de Salinas implied in 1577 that he had been using it in the 1530s. Many
scholars attribute 1/4-comma meantone to Pietro Aaron in his harpsichord tuning instructions of
1523, but an equally legitimate interpretation of Aaron’s text suggests that while he would not
have faulted regular ¼-comma meantone, neither did he specify it. Salinas’s 1/3-comma meantone (1577), with its pure minor 3rds at the expense of major 3rds distinctly smaller than pure,
does not seem to have been used much, even though it is very easy to tune. Zarlino described it
in 1571 as less sonorous than 1/4- or 2/7-comma meantone, and added in 1588 that ‘it seems to
me a bit more languid’ (‘anzi al mio parere è un poco più languido’). Its intervals are virtually
identical, however, with those produced by an equal division of the octave into 19 parts, and in
that guise it may have been familiar to Guillaume Costeley (Levy 1955), as well as to Salinas
himself and, later, Jehan Titelouze.
Evidently, then, all the forms of meantone temperament for which the essential mathematics
were worked out by an anonymous early 17th-century Dutch colleague or disciple of Simon
Stevin (Lindley 1984) – namely 1/3-, 2/7-, 1/4- and 1/5-comma meantone – had been in use,
although 1/3-comma probably least and 2/7-comma, which is difficult to tune precisely, perhaps
less than 1/4-comma or some shade of meantone with major 3rds slightly larger than pure. No
doubt most tuners, rather than trying to exemplify any particular mathematical model, merely
sought to achieve sonorous 3rds and 6ths without making any 5th or 4th (other than the wolf)
beat obstreperously. The instructions of G.B. Benedetti (1585), E.N. Ammerbach (1571) and
G.P. Cima (1606), as well as those of Aaron and Santa María, reflect a certain freedom in this
respect. Yet some of the best tuners may have sought a fairly precise regularity in the tempering
of their intervals, if only for the sake of craftsmanship.
Musical evidence cannot show that any specific shade of regular meantone was exclusively
favored by any composer, but certain compositions benefit particularly from certain shades.
The sprightliness of the sharps and B♮ in Ex.4, the first strain of an alman by John Bull, is better
served by 1/5-comma than by 1/4- or 2/7-comma meantone. Ex.5, the opening of a toccata by
Giovanni Gabrieli, gains a certain warmth and dignity in 2/7-comma meantone, while the relatively untuneful style of the entire toccata minimizes the dull melodic effect of the rather large
diatonic semitones of 2/7- or 1/4-comma meantone. In 1/4-comma meantone the contrast between pure major 3rds and tempered minor 3rds prevents that effect of banality which in equal
Temperament would afflict the middle bars of Ex.6, the opening of an organ verso by Antonio
Valente; and in a more elaborate way the same kind of contrast – but involving tempered 5ths as
well – is a positive delight when Ex.7, the opening of a Cabezón tiento, is played in 1/4-comma
meantone.
Ex.4. Bull (GB-Och Mus. 1113, p.227)
Ex.5. G. Gabrieli (I-Tn Giordano II, f.28v)
Ex.6. Valente (Libro secondo, Naples 1580, p.27)
Ex.7. Cabezón (Obras de música, Madrid 1578, f.62v)
The possible use of regular meantone on 16th-century fretted instruments is discussed in §9
below.
4. Irregular keyboard temperaments to 1680. Perhaps the earliest suggestion of an irregular
keyboard temperament is an anonymous English prescription for organ pipes of about 1373
(‘Incipit mensura’, transcribed in K.-J. Sachs 1970) instructing the builder that the correct pipe
length for a chromatic note will halve the difference between its diatonic neighbors that form
a Pythagorean whole tone (‘Ubicumque vis habere semitonium, semper fistulam inferiorem
et superiorem in duas divides’). In the wake of the Renaissance revival of Euclidian geometry,
Henricus Grammateus published in his arithmetic book of 1518 a simple diagram (Fig.1) showing how the geometrical mean between two pipe lengths in the ratio 9:8 could be determined by
drawing a semicircle on the sum of the lengths 9 and 8 taken as diameter, and then measuring the
perpendicular from the juncture of the two lengths. By this means the Pythagorean whole tone
would be, in theory, divided into two musically equal semitones. Grammateus referred to these
as ‘minor semitones’, and other theorists later used that term to designate equal semitones on
the lute (see §9 below); but the context of the prescription leaves no doubt that he intended his
‘amusing reckoning’ (‘kurtzweyllig rechnung’) to be applied to organ pipes. Such a temperament
would make both B-F and B -F sour, however, and even Barbour (1951), although ideologically
attracted to it, described it as an organ tuning which ‘may have been used in practice but hardly
by anyone who was accustomed, like Schlick, to tune by ear’. Nor does any extant Renaissance
organ music avoid B -F as well as B–F .
Figure 1. Grammateus’s method for determining the geometrical mean between two pipe lengths
in the ratio 9:8
The irregularities of a number of other schemes similarly deserve scant attention from the student
of keyboard performing practice because they are only trivially different from regular schemes
(e.g. Aaron 1523), or quite incompetent (Reinhard 1604) or of such limited historical and geographical scope (Douwes 1699) as to be no more important for the history of musical style than
the survival, to this day, of Pythagorean intonation as a provincial keyboard tuning method. Also
relatively inconsequential, insofar as normal keyboard instruments are concerned, were expert
and elaborate irregular schemes for the arcicembalo such as those of Vicentino (1555) and Trasuntino (instrument of 1606). However, irregular temperaments of musical consequence were
published by Arnolt Schlick (1511) and Marin Mersenne (1635-6 and 1636-7).
Schlick’s temperament was an artful variant of regular meantone with major 3rds slightly larger
than pure. The ten 5ths forming a chain from E to C were tempered more or less alike yet not all
quite the same, for the major 3rds among the naturals were to be tempered less than those involving an accidental. Implicitly, B-D was not obliged to be serviceable at all, nor was the 5th from
C to G . But to gain a G that could be used (if camouflaged by ornamentation or treated warily
as in Ex.8) to cadence on A, Schlick advised tempering the 5th A -E larger than pure. The result
was to render E-G more or less Pythagorean and A -C (as in Ex.9) more or less as in equal temperament (Husmann 1967, and Lindley 1974). Schlick remarked that some patrons would prefer
to have an unequivocally serviceable G and no A at all, but he considered that arrangement
an impoverishment of the harmonic resources of the organ; and he disapproved also of splitting
accidentals to gain both enharmonic forms in a euphonious intonation. It is not known how widespread the use of Schlick’s tuning ever became, but his intended readership encompassed the
Holy Roman Empire under Maximilian I.
Ex.8. Schlick (Tabulaturen etlicher Lobgesang, Mainz 1512, p.44)
Ex.9. Schlick (op. cit., p.51)
The context of Mersenne’s step-by-step tuning procedure of 1635-6, and his subsequent acknowledgments of errors in the organ instructions, indicate that he had intended merely to prescribe
regular 1/4-comma meantone (for the organ on pp.364-65 and for the spinet on pp.108-09) where
an ambiguity would allow an equivalent error to be inferred by a hasty reader. This error, concerning the 5ths down from F to B and thence to E , was evidently due to the fact that Mersenne had not tuned a keyboard instrument himself but was transmitting instructions from someone else (perhaps Jean Denis): hence he did not adequately clarify the fact that when one tempers an ascending 5th, for instance by tuning a G to a C, one makes the note G lower than pure
in order to make the interval in C-G smaller than pure, but when one is tempering B to F one
must tune the note B higher than pure in order to produce the same kind of interval, namely a
5th tempered smaller than pure. Virtually every set of step-by-step instructions for meantone
temperament includes an awkward sentence or two about this point (Praetorius devoted nearly
half a page to it). In Mersenne’s treatise the awkwardness was aggravated into succinct confusion. At one point he did specify that ‘il faut tenir la note de dessouz [B or E ] un peu forte’; but
elsewhere he gave a contrary impression, and in the instructions for tuning the spinet he said ambiguously: ‘Cette quinte doit estre augmentée au lieu que les précédentes ont esté diminuées’.
The likelyhood that some 17th-century readers – particularly musicians using the instructions
without reading all the accompanying theory – interpreted this passage as a prescription for irregular temperament is shown by Chaumont’s ‘Méthode d’accorder le clavessin’ (in his Pièces
d’orgue, 1695) which indicates explicitly that E and B were treated, by various tuners, as ‘foible ou forte’. It happens that Mersenne’s inadvertent novelty was musically opportune because
the cumulative lowering of E would more than achieve, on behalf of D, that which Schlick’s
temperament had achieved on behalf of G , albeit at the cost of rendering E -G slightly ‘darker’
than the A -C of Schlick’s temperament. It may further be argued that Louis Couperin’s use of E
and even B (as in Ex.10) differs from that of his teacher Chambonnières (Ex.11), the older man’s
music requiring a regular meantone temperament while the younger found an opportunity for
chiaroscuro in the kind of tuning inadvertently implied by Mersenne. Thus Couperin’s long Gminor Passacaille, with its recurring bass line G-F-E -D, may gain a dimension of structure as
well as expressiveness when G-B beats about eight times per second but ‘resolves’ to a pure or
nearly pure F-A, and then E -G, beating about ten times per second, likewise yields to a more
euphonious D-F ; and this would represent a significant stylistic departure from Frescobaldi’s
music in the same genre, which was known to Couperin. But when inferences are thus drawn
beyond what can be proved by documents, a careful assessment – in this case requiring the use of
accurately reconstructed 17th-century instruments – must be supplied by impartial judges listening.
Ex.10. L. Couperin (F-Pn Mus.Rés. Vm7 675, f.74r)
Ex.11. Chambonnières (Les pièces de clavessin, 1670, p.40)
In 1697 Andreas Werckmeister suggested that organists unwilling to adopt his circulating temperament (see §7 below) or to install a split key with separate pipes for E and D might compromise by tuning E down to serve for both notes. Composers like Samuel Scheidt (1587-1654) and
his contemporaries used D enough to tempt any organist without the split key at his disposal to
consider such a makeshift adjustment of the temperament.
5. Equal temperament to 1735. Although no Renaissance keyboard musician is known to have
advocated or adopted equal temperament, it appears to have been used on fretted instruments
such as the lute and viol at least since the early 16th century (see §9 below). A line of theorists
from Vicentino (1555) to Mersenne and beyond took for granted that fretted and keyboard instruments were incompatible because of their distinct styles of intonation. The development of methods for calculating or approximating the intervals of equal temperament has been traced in J.M.
Barbour’s dissertation (1932) and summarized in Tuning and Temperament (1951), where he
wrote, ‘The simplest way … is to choose a correct ratio for the semitone and then apply it twelve
times’. The ratio 18:17, familiar to theorists from well before the Renaissance and recommended
by Vincenzo Galilei in 1581 for equal temperament on the lute, corresponds mathematically to
a semitone of 99 cents, virtually indistinguishable from the 100-cent semitone of equal temperament. Therefore the theoretical refinements of a Zarlino (1588) or a Stevin (ca.1600) may have
had less bearing than one might imagine on the historical status of equal temperament in musical
practice: an essentially equal temperament was feasible in practice before those theoretical refinements were achieved, and such compositions as Adrian Willaert’s famous Quid non ebrietas
(published in 1530) or Francesco Orso’s setting (1567) of Petrarch’s Il cantar nuovo indicate that
some 16th-century composers appreciated the enharmonic advantages of equal temperament (see
§11 below) which its advocates have always emphasized. In 1603 G.M. Artusi attributed to the
unnamed opponent whom he criticized for certain modern tendencies (Monteverdi) a mathematical theory of intervals approximating to equal temperament and justifying the use of diminished
4ths and 7ths in vocal music (Lindley 1982). The practical history of equal Temperament, then,
is largely a matter of its refinement in various respects and its gradual acceptance by keyboard
musicians from the late 1630s, when Frescobaldi endorsed it, to the 1870s, by which time even
the conservative English cathedrals were won over.
Zarlino in 1588 attributed the following prophetic arguments to Girolamo Roselli, abbot of S
Martino delle Scale in Sicily:
This way of dividing the diapason or octave into 12 equal parts … could alleviate all the difficulties of
singers, players and composers by enabling them generally … to sing or play … DO–RE–MI–FA–SOL–
LA upon whichever of the 12 notes they wish, touring through all the notes, making, as he [Roselli] says,
a circular music; hence all the instruments will be able to keep their tuning and be in unison, and organs,
as he says, will be neither too high nor too low in pitch.
About 50 years later an ‘old man in rags’, who had spent most of his life in Sicily and Calabria
and knew ‘nothing except how to play the harpsichord’, retired to Rome and made a stir by advocating equal temperament on the harpsichord and even inducing Frescobaldi, with the aid of
‘frequent and gratuitous beverages’, to recommend it for the organ in Bernini’s new apse at S
Lorenzo in Damaso. The malicious details of this story are due to G.B. Doni (in his book of 1647
and in a letter to Mersenne of February 1640), who prevailed upon Cardinal Francesco Barberini
to ignore Frescobaldi’s advice. The renovated apse, with ‘doi bellissimi organi’ in meantone,
was inaugurated in August 1640 (and witnessed, towards the end of the century, performances by
Corelli under the patronage of Cardinal Ottoboni). But already in April 1638 Doni had written to
Mersenne that Frescobaldi failed to grasp the difference between major and minor semitones (‘ne
scait pas ce que c’est semiton majeur ou mineur’) – an unlikely accusation to make of a former
pupil of Luzzaschi and yet perhaps a fair indication of Frescobaldi’s attitude towards niceties
of intonation. Whether his magnificent Cento partite sopra passacagli (1637) were intended for
equal temperament or for a harpsichord with split accidentals for D /C as well as G /A and
D /E , the influence of Frescobaldi’s acceptance of equal temperament is apparent in the later
keyboard music of Froberger, who was his pupil in Rome when the old Sicilian harpsichordist
was also there.
Froberger was not the first north European to use equal temperament on keyboard instruments.
In 1637 Mersenne said that the engineer Jean Gallé had ‘accommodated this tuning to the organ
and spinet’. By 1645 Gallé or someone else ‘very learned in mathematics’ had persuaded some
‘honest men’ in Paris to use equal temperament, and the instrument maker Jean Denis had been
told, to his dismay, that perhaps he would not disapprove of it on the harpsichord so vehemently
if he were accustomed to the sound.
It was not mere conservatism that prompted most Baroque keyboard musicians to resist or ignore
the suggestions of a Stevin or a Mersenne that they consider adopting equal temperament, but
rather their appreciation of the virtues of the feasible alternatives: regular meantone in the early
17th century, certain irregular temperaments thereafter. The same interest in timbre and sonority
that nourished the Baroque refining of wind and string instruments must also have nourished
an interest in the sensuous qualities of relatively subtle tunings. Some representative early 18thcentury views are summarized in the following passage from Neidhardt (1732):
Most people do not find in this tuning that which they seek. It lacks, they say, variety in the beating of
its major 3rds and, consequently, a heightening of emotion. In a triad everything sounds bad enough;
but if the major 3rds alone, or the minor 3rds alone, are played, the former sound much too high, the latter much too low…. Yet if oboes, flutes & the like, and also violins, lutes, gambas & the rest, were all
arranged in this same [tuning], then the inevitable church- and chamber-pitch would blend together
throughout in the purest [way].… Thus equal temperament brings with it its comfort and discomfort,
like blessed matrimony.
In the late 17th century and early 18th, however, a circle of German theorists became interested
in Equal temperament, including Werckmeister, Meckenheuser, Neidhardt and Mattheson. The
wave of interest that they represent never thereafter lost momentum, although most of the best
German organ builders are said to have resisted until after the generation of Gottfried Silbermann
and Wender (Mattheson 1722-25). In England the builder Renatus Harris, wishing to discredit
the use of split accidentals by his competitor ‘Father’ Smith, induced John Wallis to publish in
the Philosophical Transactions of 1698 an article ‘On the Imperfections of the Organ’ (Williams
1968). Though admittedly ‘little acquainted’ with the instrument, Wallis said that Equal temperament had been ‘found necessary (if I do not mistake the practice)’ on organs. This exaggerated
claim appears to indicate that equal temperament in any case was no longer unthinkable for organs. And when Brossard in 1703 remarked on the existence of two schools of opinion regarding
temperament in practice, he did not, as so many had done before him, associate Equal temperament with fretted instruments and its alternatives with keyboard instruments.
Special comment is warranted by the fact that equal temperament is often particularly associated
with the name of Werckmeister, or with John Bull’s long Ut-re-mi-fa-sol-la in the Fitzwilliam
Virginal Book (No.51). Bull’s piece does use all five sharps in some sections and all five flats in
others, but never both a sharp and a flat in the same section; it was most probably conceived for
arcicembalo. Werckmeister’s attitude at the end of his life is summarized in a book published
posthumously (1707) where in effect he apologized for not having included equal temperament
among the various circulating temperaments for which he had earlier published a monochord
diagram. (The incompetent engraver, he said, had complained about dividing the narrow space
of a comma into 12 parts.) He quite approved of equal temperament, yet was willing ‘to have the
diatonic 3rds left somewhat purer than the other, less often used ones’, a procedure for which he
had expressed clear preference in his earlier writings (see §7 below).
6. Regular meantone temperaments from 1600. The late-Renaissance system of church modes
or tones was suited to the harmonic dimensions of a 12-note meantone palette (from E to G in
the chain of 5ths) as long as V-I cadences were not required in the Phrygian mode (E minor)
and as long as the Dorian, with its often flattened sixth degee, was not transposed more than one
place down in the chain of 5ths to G minor. Cadences to E could be accommodated by the use of
a split accidental on the keyboard to provide D as well as E ; and a split accidental for A and
G was also often provided (see Ortgies 2003, in regard to organs, as well as Barnes 1971 in
regard to harpsichords). But the growing popularity, from the late 17th century, of more expansive modulations rendered regular meantone temperaments musically problematical well before
the end of the Baroque era, although the old tunings remained familiar on many organs.
Michael Praetorius in his De organographia (1618) prescribed exclusively 1/4-comma meantone
for keyboard instruments, a tuning later referred to by German theorists as the ‘Praetorianische
Temperatur’. In France Mersenne, relying heavily upon Salinas’s De musica, also neglected to
mention any shade with major 3rds larger than pure; he described 1/4-comma meantone as ‘la
manière d’accorder parfaictment les Orgues ordinaires’. The pre-eminence of these authorities
and Zarlino caused 1/4-comma meantone to be regarded by many subsequent writers as the exemplary ideal of meantone temperament and even of Renaissance and Baroque keyboard tunings
in general. An indication of its use on 17th-century German organs lies in Werckmeister’s assertion (1697) that an instrument would be converted to his circulating temperament, in which the
5ths C-G-D-A are tempered 1/4 comma, by resetting only some of the pipes. In the late 17th
century and the 18th the increasing popularity of tierce ranks in mixture stops promoted the
brilliance of 1/4-comma meantone, with its pure major 3rds, on the organ.
But meantone temperament with major 3rds larger than pure also flourished. Cerone attributed
it in 1613 to master organ builders. Lemme Rossi (1666) described 1/5-comma and mentioned
2/9-comma meantone. The instructions of Jean Denis (1643), a professional harpsichord tuner,
specify no particular shade of meantone, prescribing only ‘good’ 3rds together with 5ths tempered by ‘un poinct’; but in 1707 and 1711 the acoustician Joseph Sauveur, using an elaborate
monochord to compare 1/4-, 1/5- and 1/6-comma meantone, reported that organ and harpsichord
makers adhered more closely to 1/5-comma meantone than to the other two forms, even though
‘les musiciens ordinaires’ used 1/6-comma meantone. W.C. Printz in 1696 described the same
three shadings and also 2/9-comma meantone, which he said was ‘even earlier’ (‘noch eher’)
than 1/5-comma. Loulié in 1698 said that 1/5-comma meantone was ‘better and more in use’
than any other temperament. Sorge in 1748 attributed the use of 1/6-comma meantone to Gottfried Silbermann (and a posthumous book of Sorge (1773) confirmed the presence on two
‘especially good’ Silbermann organs of a ‘horribly large beating 5th, G -D , along with four
unbearably barbaric major 3rds’). J.-B. Romieu in 1758 expressed preference for 1/6-comma
meantone because he liked the relative amount of tempering that it allots to 5ths and 3rds. The
same criterion led Giordano Riccati in 1762 to give preference to his own 3/14-comma system
(among regular temperaments).
Meantone with major 3rds smaller than pure seems to have disappeared from normal practice,
although Cerone reproduced Zarlino’s instructions of 1558, and Cima’s instructions (1606) confused it (even more plainly than Aaron’s of 1523) with 1/4-comma meantone. Lemme Rossi,
Mersenne and Printz described it, and in the 18th century Henfling (1710) and Smith (1749)
advocated it as an innovation. More recently Kornerup sought to champion it in the 1930s, and
in 1948 A.R. McClure published a good account of its sound. According to Mersenne, Titelouze
had a harpsichord with 19 notes per octave tuned in equal microtones – the equivalent, as it happens, of 1/3comma meantone. Salinas had published tuning instructions for 1/3-comma meantone applicable to such an instrument, but it is not certain whether the 19-note harpsichord built
by Pesarese for Zarlino or the one built by Elsasz for Luython and described by Praetorius were
tuned thus. Costeley published in 1570 a chanson spirituelle, Seigneur Dieu, ta pitié, which he
had composed as an exercise in the use of the 19-note scale (Levy 1955). In the 20th century
interest in the 19-note equal division was developed by, among others, Yasser (1932), Handschin (1927) and Mandelbaum (1961).
For various shades of meantone the corresponding schemes of dividing the octave into equal
microtones were worked out, with the aid of logarithms, during the 17th century. The correspondence between 1/4-comma meantone and the 31-part division was described clearly in
publications by Lemme Rossi (1666), Joseph Zaragoza (1674) and Christiaan Huygens (1691).
Zaragoza also equated the 19-part division with 1/3-comma meantone. Joseph Sauveur (1701)
matched the 43-part division (which Zaragoza had mentioned) with 1/5-comma meantone, and
the 55-part division (illustrated in a circular diagram by António Fernandez in 1626) with 1/6comma meantone. This kind of theoretical speculation continued in the 18th century. By 1637
Gallé had properly reckoned the intervals of just intonation in terms of the 53-part division,
similarly favored by Isaac Newton in 1665 and Nicolas Mercator in the 1670s.
When an 18th-century writer advocated one of these schemes he is much more likely to have
been referring to a general theoretical model – a shade of meantone (the French term was ton
moyen) regarded as an ideal of intonation – than to an experimental keyboard instrument. In this
sense the 55 division, corresponding to 1/6-comma meantone, was particularly eminent. Not only
did Sauveur attribute it to ‘ordinary musicians’, but Mattheson (1722-25) was familiar with it
and Sorge (1748) labelled it ‘Telemann’s system’. In this scheme the whole-tone comprises nine
‘commas’; but not every reference to that feature necessarily implies the 55 division, as the same
feature characterizes the nearly Pythagorean 53 division, which happens to contain virtually pure
3rds as well as 5ths and which had been referred to in a Pythagorean context by Italian theorists
since the 15th century (e.g. Anselmi, 1434). Neither Praetorius (1618) nor Nassarre (1723-4) had
in mind 1/6-comma meantone when referring to the whole-tone as made up of nine commas, and
one may entertain doubts in the case of other writers, like Brossard (1703) or Tosi (1723), who
were not specialists in the theory of microtonal scales. The status of 1/6-comma meantone in
performing practice might better be confirmed by systematic research into 18th-century woodwind instruments that survive unaltered. Certainly the expressive semitones and harmonic 3rds
and 6ths in Ex.12, the opening of Lully’s overture to Bellérophon in a 17th-century keyboard
transcription, are ill-served by 1/4-comma meantone.
Ex.12. Anon: contemporary transcription of the overture to Lully’s Bellérophon (1679)
Some scholars have assumed that Baroque harpsichordists habitually retuned wholesale the accidentals on their instruments in order to produce various dispositions of regular meantone for
music in various keys (Barbour and Kuttner, 1958; Rayner, 1969). But no Baroque musician or
theorist other than Cima (1606) is known to have described this procedure; and evidently the
standard 17th-century 12-note disposition (E –G ) was used by some late 17th-century and 18thcentury musicians as if it were a circulating temperament. Apparent references to this practice
can be found in the writings of Christiaan Huygens (?1670s), Werckmeister (1700), Romieu
(1758) and Bédos de Celles (1766-78), in the fourth edition of the Encyclopaedia Britannica
(1810) and elsewhere. As late as 1847, I.F. Holton in New York wrote: ‘In Unequal Temperament some of the chords are very good, while the aggravated dissonance of others, called by
tuners the WOLF, imparts a peculiarity to the keys in which it occurs, much admired by certain
musicians … These predilections … must be chiefly attributed to fancy and prejudice. Some
inequality of temperament may be preferable, but no key ought to he made so bad as to give it
a character for harshness.’ Similar opinions had been expressed throughout the 18th century.
Extant specimens of Baroque carillon are said to be tuned to some form of meantone; in view of
their inharmonic timbre this is a remarkable indication of the precision in tuning and control of
timbre achieved by the Hemonys and other master bell-founders.
7. Irregular temperaments from 1680. The most characteristic type of 18th-century keyboard
tuning was an irregular temperament with no wolf 5th but with the 3rds in the C major scale tempered lightly, more or less as in some form of meantone temperament; most of the 3rds in ‘modern’ keys (such as B major, A major or F minor) were thereby rendered distinctly more impure
than in equal temperament. As a modulation of triadic harmonies moved about the circle of 5ths
the amount of tempering in the 3rds would thus change, in a fairly unabrupt fashion, according to
whether one was closer to the front or back of the circle as shown in Fig.2. The degree and exact
pattern of the differences might vary to suit taste and circumstance without forfeiting this principle; in fact different instruments require different quantities to achieve equivalent results. But
there is occasional evidence of a tendency for the E major triad to be tuned approximately as in
equal temperament. Concomitant to variously tempered 3rds were diverse sizes of semitone, the
largest being E-F and B-C and the smallest C-D and perhaps F-G (or E -F ). Hence the major
keys with few sharps or flats had the most resonant and limpid triads but the least keenly inflected leading notes. In minor keys the effects were more intricate, E minor for instance having
a sharper leading note but a less harsh tonic triad than F minor, a key often remarked upon (e.g.
in M.-A. Charpentier ca.1692, Mattheson 1713, Rousseau 1768, and Gervasoni 1800) for its dark
qualities.
Figure 2. Circle of 5ths indicating the relatice amount of tempering of 3rds in characteristic
18th-century keyboard tuning
Routine sequences as in Ex.13 (from a sonata of G.B. Platti) gain savour in this kind of tuning,
and more elaborate sequential manipulations are enhanced. In lieu of an extensive discussion of
key character (see Steblin, 1983), Ex.14 (the opening of an organ prelude of J.C.F. Fischer) may
represent the fact that in the key of A major, D is often a particularly tender note by virtue of its
low intonation and its affinity in part-writing to C (which is inflected high in relation to A and
D ). Shadings of this sort lend an unimaginable dimension of beauty to such lavishly flat-laden
pieces as J.S. Bach’s chorale prelude O Mensch, bewein’ or the Adagio of Beethoven’s ‘Pathétique’ Sonata. Qualities of nuance in various keys created by this kind of tuning contributed
much to the beauty of the Baroque French harpsichord and its music by such masters as d’Anglebert and François Couperin, who according to J.-J. Rousseau ‘proposed and abandoned’ equal
temperament (‘Tempérament’, in the Encyclopédie, 1768). Handel’s harmonically daring recitatives are also brilliantly enhanced, and the static tendency of the harmony in some of his arias
(compared with Bach’s) is rendered more logical. William Croft (1700) was evidently the first
English keyboard composer to exploit the resources of this type of intonation (Meffen 1978).
Ex.13. Platti, op.4 no.3 (Nuremberg ca.1746)
Ex.14. Fischer (Ariadne musica neo-organoedum, 1702)
Except for Werckmeister’s harpsichord tuning instructions of 1698 (summarized by F.T. Arnold,
1931), no Baroque prescription in German is known to have called for any 5ths larger than pure
or, consequently, any 3rds less pure than Pythagorean 3rds. In German tunings the 5ths among
the naturals averaged a heavier degree of tempering than those among the accidentals, the latter
tending in fact to be pure. Some German writers characterized as ‘good’ any temperament well
suited to harmony involving all 24 major and minor keys, thereby including equal temperament;
but others regarded equal temperament as distinctly less ‘good’ than irregular temperaments of
the type described here (see WELL-TEMPERED CLAVIER).
French tuning instructions characteristically required two or three 5ths at the back of the circle
of 5ths to be tempered slightly larger than pure (most probably A -E -B -F; see §4 above), thus
producing a more pronounced difference in size and quality between the 3rds D -F-A -C and the
3rds among the seven diatonic notes. This kind of tuning was often referred to by 18th-century
French musicians as the ‘ordinary’ or ‘common’ temperament, although some occasionally confused it with regular meantone (e.g. Bédos de Celles 1766-78) or with some tuning similar to the
irregular temperament described in §4 above (e.g. Corrette 1752). Hence it is difficult to be certain exactly what shade of temperament Ozanam had in mind when he wrote in 1691:
Whatever precaution we might take in tuning our instruments to render all the chords equal, there is always left therein some inequality that causes us to notice a je-ne-sais-quoi of sadness or gaiety, of the
melodious or the harsh, which [in turn] makes us distinguish one key from another by ear.
Yet Ozanam’s distinction between ‘natural’ and ‘transposed’ modes confirms that the inequalities in question were distributed not at random but in a pattern consistent enough to allow the
keys to be identified by ear according to their intonational inflections. Rousseau wrote, in 1768,
that the keys could thus be identified by ear, in the course of discussing a circulating form of the
‘ordinary’ temperament as described by Rameau in 1726. Rameau’s instructions were accompanied with the following remarks:
The excess of the last two 5ths and the last four or five major 3rds is tolerable, not only because it is
almost insensible, but also because it occurs in modulations little used – except for when one might
choose them on purpose to render the expression more keen etc. For it is good to note that we receive
different impressions from intervals in keeping with their different [degrees of] alteration. For example
the major 3rd, which [in its] natural [state] excites us to joy, as we know from experience, impresses upon
us ideas even of fury when it is too large; and the minor 3rd, which [in its] natural [state] transports us to
sweetness and tenderness, saddens us when it is too small. Knowledgeable musicians know how to exploit these different effects of the intervals, and give value, by the expression they draw therefrom, to the
alteration which one might [otherwise] condemn.
In 1737 Rameau changed his views (see §8 below); but in that respect d’Alembert (1752) took
a neutral position in what had evidently become a controversial matter. Remarkably lucid comments on the relation between irregular temperaments and modulation (in virtually the modern
classroom sense of the term) appeared in the Histoire de l’Académie royale des sciences at Paris
in 1742. The preference of some composers for the ‘transposed modes’ had been discussed in
an equally urbane article in the Mémoires de Trevoux (1718) which suggested that while it was
arbitrary to give specific affective labels to particular keys, the extreme keys were, because of
their tuning, comparable with a ragout with more vinegar and spice than in an ordinary one and
more than is conducive to good health.
Mattheson in 1720 published a book of reflections on this article and mentioned favorably the
work of Neidhardt, who soon became the most elaborate technician of subtle shades of ‘good’
temperament (1724 and 1732). Later French and English advocates of a fairly subtle irregular
tuning included Mercadier de Belesta (1776), Suremain-Missery (1793), Thomas Young (1800)
and Jean Jousse (1832). Sorge (1744 and 1746) and J.H. Lambert (1774) also described such
tunings, but were themselves content with equal temperament. These finer tunings are usually
the most appropriate for late 18th-century music. F.A. Vallotti, maestro at S Antonio, Padua,
from 1730 to 1779 and a highly regarded church composer and organist, designed a moderately
subtle tuning (with the six 5ths among the diatonic notes each tempered by 1/6-comma) which is
one of the simplest of its type (Lindley, 1981), although Werckmeister’s equally simple scheme
for the organ (in which the 5ths G-D-A-E and B-F are each tempered by 1/4-comma) sounds
more fitting in some late 17th- and early 18th-century German organ music.
There is evidence, however, that a less subtle degree of inequality in some Italian and English
as well as some French tuning, persisted in the 18th century. Giordano Riccati, a physicist who
corresponded extensively with Vallotti, published in 1762 a scheme which he suggested was
in keeping with current practice, and in which A –E was larger than pure by some 2/5 of the
Pythagorean comma. (The six diatonic 5ths were each tempered 3/17 of the syntonic comma,
while E -B -F and B-F -C -G were as in Equal Temperament.) A contemporary set of manuscript instructions preserved at the Conservatory of Padua refers to the sequence of 4ths B -E A as tempered smaller than pure, and that of E-B-F -C as pure. Metastasio referred in a letter
of 1770 to the use of one 5th, ‘which they call allegra’, tuned larger than pure by ear. In England, irregular schemes of similar crudity were prescribed by Arthur Wood (ca.1730), J.C. Petit
(ca.1740) and William Hawkes (1805).
Among the late 18th- and early 19th-century Italian and English accounts of irregular temperaments may be mentioned those of Gervasoni (1800 and 1812), Barca (1809), Serassi (1816),
Asioli (1816), William Jones (i) (1784), Cavallo (1788 and 1803), Robison (1801), Stanhope
(1806), Kollmann (1806 and 1823) and Callcott (1807). The American Journal of Science began
publication in 1818 with an account of the attempt by Alexander Fisher, a Yale mathematician,
to derive an ideal model of temperament from a statistical reckoning of the use of different key
signatures in 1600 compositions, and of the use of each major and minor triad in 200 compositions ‘taken promiscuously from all the varieties of music for the organ’. Fisher acknowledged,
but made too little allowance for, the crucial fact that ‘making the aggregate of dissonance … the
least possible … [would] render the harmony of the chords very unequal … a disadvantage’.
More important than the work of such theorists is the evidence, which can be outlined here only
briefly, that none of the masterworks of late Baroque and Classical German keyboard music was
in fact created in a pervading ambience of equal-tempered intonational sameness among the various triads and keys, although some forward-looking composers did tend, like C.P.E. Bach (see
§8 below), to think in such terms. In 1697 Werckmeister had written:
Now if all semitones, tones, 3rds, 5ths, etc. had the same size and [equivalent] beating, people would
take little pleasure in transpositions: for example, if the Dorian is transposed a second into either C or E:
such transpositions produce notable alterations and excitement. This is brought about not so much by
the change in pitch level as by the reordering of the tones and semitones, and also the [varied] beating
of the concords.
Heinichen in 1728, while ridiculing Mattheson’s propensity to ‘assign specifically to this or that
key the affect of love, sadness, joy, etc.’, acknowledged that
in general it may well be said that one key is more fitting than another for expressing affects [at large];
and in today’s good temperaments (I am not referring to old organs) the keys with two or three flats or
sharps in their signatures emerge, especially in the theatrical style, as the most beautiful and expressive.
For this reason I would not even support the invention of the long-sought clavier in just intonation were
it to become practicable.
Kirnberger held that a good temperament ‘must not injure the variegation of the keys’ (1776–9),
and his disciple (in this regard) Tempelhof in 1775, while acknowledging that any key could express any affect, held nonetheless that in a good temperament each key would do so in its own
particular way (‘auf eine ihnen angemessene Art’) and that without such expressive resources
music would be ‘nothing more than a harmonious noise that tickles the ear but leaves the heart
slumbering away in a disgusting indifference’. In 1780 the polymath J.J. Engel, in a book dedicated to Reichardt, placed the choice of key before melody and harmony as a resource of musical portrayal, and said that among the major keys, C and A differed most since the steps of their
scales differed most.
In 1784 Cramer’s Magazin der Musik reported that Clementi used a tuning in which C-E was
tempered ‘ein klein wenig hoch schwebend’ (‘beating, slightly high’), E-G ‘sehr hoch’ (‘very
high’) and A -C ‘noch höher’ (‘even higher’). In 1785 Mozart’s pupil Thomas Attwood recorded
in his notebook that G was a note which ‘the Harpsichord has not, but all other instruments have’
(Chesnut 1977). According to A.F. Schindler (1860), Beethoven in his last years maintained a
keen interest in the expressive characteristics of different keys and suggested that they were most
apparent in piano music. It is unclear to what extent Beethoven may have attributed the differences to acoustical factors, but his piano music does in fact benefit from an 18th-century unequal
temperament (Lindley and others 2007, Chapter 5).
In 1826 the leading champion of Viennese Classical music in Italy, Peter Lichtenthal, wrote that
equal temperament ‘cannot subsist’ or else the keys would lose their character and ‘one could
equally [well] compose a nocturne in A minor or a military blare in A ’ – an opinion that was
excised, however, in Dominique Mondo’s French translation 13 years later. Yet the late Baroque
associations of different qualities with different keys cast their shadows far into the 19th century.
This legacy warrants investigation. A good point of departure is that much (though not all) of
Schubert’s piano music benefits from an unequal well-tempered tuning if the nuances are subtle
enough that C-E is tempered more than half as much as in equal temperament (rendering E suitable to such melodious uses as shown in Ex.15) and D -F is no nearer in size to a Pythagorean
3rd than to an equal-tempered one (since Schubert used very freely the key of D major). Equally
telling is the curious fact that D minor, the most eminent of keys in Baroque keyboard music and
one that to a large extent retained its old meantone-like sound in the 18th-century irregular temperaments, was the key least favored by Chopin, except in the last and most magnificent of his
24 preludes.
Ex.15. Schubert, Piano sonata in C (D840), beginning
8. Equal temperament from 1735. In his Génération harmonique (1737) Rameau endorsed
equal temperament and, by way of retracting his own views of 11 years before, introduced a new
argument in its favor:
He who believes that the different impressions which he receives from the differences caused in each
transposed mode by the temperament [now] in use heighten its character and draw greater variety from it,
will permit me to tell him that he is mistaken. The sense of variety arises from the intertwining of the keys
[l’entrelacement des Modes] and not at all from the alteration of the intervals, which can only displease
the ear and consequently distract it from its functions.
Distracting the musical ear from its proper functions is an unpardonable fault in a tuning. Rameau’s argument might well have applied more palpably in France than in Germany, if French
unequal tunings were, as they generally appear to have been, less subtle than their German counterparts. Rameau’s authority as a musician was such that the 1749 register of the Paris Académie
Royale des Sciences could state, ‘M. Rameau assures us that experience is not opposed to the
temperament that he proposes; and in this regard he has earned the right to be taken at his word’.
Equal temperament continued to be identified with his name throughout the 18th century in
France and occasionally in Italy as well.
In Germany J.N. Ritter, perhaps the most important organ builder in Franconia at the middle of
the century, is said to have used equal Temperament. Among theorists advocating it Barthold
Fritz is especially important, despite the crudeness of his tuning instructions, because in the preface to the second edition (1757) of his Anweisung (1757: the title may be translated as ‘Method
for tuning claviers, harpsichords and organs, in a mechanical way, equally pure in all keys’), he
reported that C.P.E. Bach had found ‘in my few pages everything … that was necessary and possible’ for a good tuning. C.P.E. Bach’s own advice (1762) for improvising a fantasia mentioned
temperaments: on the organ, he said, ‘one must restrain oneself in chromatic passages; at least
they should not be advanced sequentially, because organs are seldom well tempered. The clavichord and the piano are the most fitting instruments for our fantasia. Both of them can, and must,
be tuned pure’. For Marpurg and others ‘rein’ (‘pure’), which C.P.E. Bach himself here distinguished from ‘gut’ (‘good’ or ‘well’), became a catchword in arguments favoring equal temperament (or at least not a purposefully unequal one). Fritz’s title shows how it came to serve in that
capacity; once equal-tempered 3rds were considered acceptable, then a tuning with certain 3rds
tempered more heavily could be described as relatively impure. Since C.P.E. Bach was sufficiently concerned to give not only a warning about the limitations of meantone but also emphatic
advice about the tuning of the clavichord and piano, the fact that he did not recommend exploiting the inflections of a circulating unequal temperament in a genre which, by his own definition,
‘modulates into more keys than is customary in other pieces’ suggests an indifference to those
inflections. When C.P.E. Bach spoke of ‘remote’ keys, he meant keys remote from the tonic key,
not keys remote from C major or D minor. His compositions, for instance the great rondos from
the collections für Kenner und Liebhaber, reflect this neglect of the concept expressed in Fig.2
above; and his favorite instrument, the clavichord, was the least likely of all normal keyboard
instruments to display to advantage the niceties of an irregular temperament. If the music of any
leading 18th-century German composer ought to be performed in equal Temperament, C.P.E.
Bach is the best candidate.
No simple conclusion can be established as to the attitude of his father, J.S. Bach, towards the
relative merits of equal temperament and a mildly unequal one. On the basis of evidence such as
applied above to C.P.E. Bach, Barbour showed (1932) that J.S. Bach would probably not have
held a dogmatic opinion (a view rejected in Rasch 1981). Barbour’s later statement (1951, p.196)
that ‘much of Bach’s organ music would have been dreadfully dissonant in any sort of tuning
except equal temperament’ is a silly exaggeration, due perhaps to the fact, which he mentioned
in a letter of 1948 to A.R. McClure, that he (Barbour) had never heard any keyboard temperament other than equal temperament. John Barnes (1979) investigated the ‘48’ in a fairly subtle
type of irregular temperament (see WELL-TEMPERED CLAVIER) and found that the peculiarities of
the various keys in that tuning are nicely suited to or accommodated by the music. According to
Marpurg (1776), Kirnberger scrupulously reported that Bach, his teacher, had instructed him to
tune all major 3rds larger than pure – thus ruling out any unsubtle irregular temperament (such as
used by Kirnberger himself). One could readily believe that Bach sometimes exploited the qualities of a particular key as inflected in a typical irregular temperament, sometimes merely accommodated what he knew was likely to be the kind of tuning his published music would be played
on, and sometimes – for instance, in the concluding ricercar of the Musical Offering – ignored
completely the possibility of intonational shadings.
The most vigorous and articulate late 18th-century champion of equal temperament seems to
have been F.W. Marpurg, whose Versuch über die musikalische Temperatur was published in
1776 but who had already advocated equal temperament in his Principes du clavecin (1756).
Although capable of meretricious reasoning, he presented, in greater detail than Rameau, numerous forceful arguments, some of which were rendered so valid by historical circumstance that
during the 19th century equal temperament became the standard keyboard tuning and, in the
West, a widely followed norm of intonation in general. Marpurg (1776) knew that a composer
might select a key for reasons ‘that have nothing to do with temperament’; and he saw (as
did Tiberius Cavallo in 1788) the advantages of equal temperament in ensemble music, where
so long as not all the instruments playing together, and the vocal parts as well, are intoned in the most
perfect agreement in one kind of [irregular] temperament, the composer must obtain the character of his
piece, the building up of an emotion, and the strength of expression, from sources quite other than the
creative powers of the tuning hammer or cone.
His encounters with the tuning schemes of Kirnberger and C.L.G. von Wiese (see Eitner), as
well as his knowledge of the writings of Werckmeister, Neidhardt, Lambert and others, showed
him that:
There is only one kind of equal temperament but countless possible types of unequal temperament. Thus
the latter opens up to speculative musicians an unstinting source of modifications, and since every musician will readily invent one, the result will be that from time to time we shall be presented with a new
type of unequal temperament, and everyone will declare his own the best.
In short, he gave equal Temperament decisive preference on both of the counts envisaged by
Fontenelle, who had written in 1711: ‘After these motley combats, one system will become victorious. If fortune favors the best system, music will gain thereby a real advantage; and in any
case it will at least profit from the convenience of having the same ideas and the same language
accepted everywhere.’
D.G. Türk (1802) extended Rameau’s argument of 1737 cited above by suggesting that a sameness of quality among the various keys would contribute to unity of character in a composition.
Influential musicians supported equal temperament in Italy (Asioli, 1816) and England (Crotch,
1812 and, invoking J.S. Bach, 1833). Hummel’s Anweisung zum Piano-Forte Spiel (1828) concluded with a discussion of tuning that justified ignoring the old, unequal temperaments on the
grounds that they presented, particularly for the many novice tuners brought into the trade by
the popularity of the piano, greater difficulties than equal temperament and that these difficulties
were aggravated critically by the burden of tuning, on modern pianos, three heavy strings for
each note instead of two thin ones as on older instruments. Jousse, in a book on piano tuning
(1832) dedicated to W.F. Collard, expressed preference for a subtly unequal temperament, but
Claude Montal (1834) gave instructions solely for equal temperament on the piano. In the 1840s
Alfred Hipkins persuaded the Broadwood firm to tune their pianos in equal temperament, which
he must have used when tuning for Chopin in London in 1848. Cavaillé-Coll (in his maturity)
and contemporary German organ builders used it; almost all the English organ builders resisted
until after the Great Exhibition of 1851, but their notebooks show that from the mid-1850s until
the 1870s, rebuilt or reconditioned church organs were usually raised to the current concert pitch
and converted to equal temperament (see Mackenzie 1980).
A relatively late development was the widespread use of more refined tuning procedures than
those of Rameau or Fritz, in order to guarantee uniformity among the 3rds and 6ths. (In reality
this can seldom be achieved by tempering all the 5ths and 4ths alike or by matching unisons
with a monochord. See EQUAL TEMPERAMENT and TUNING.) Sorge’s method (1749, republished
by Marpurg, 1756, Roesner, ca.1765, and Bossler, 1782) was to temper a chain of major 3rds (CE-G /A -C) before tempering any of the 5ths, and after setting the 5ths in three chains of four
each (rather than one chain of 12), to check whether all 12 major triads, ascending chromatically,
‘are equally sharp to one another’. A sufficiently exact yardstick for ‘equally sharp’ gained currency during the 19th century – that each major 3rd or 6th should beat no slower than the one
below and no faster than the one above.
Equal temperament in this more exact sense is virtually considered an inherent characteristic
of the modern concert piano. Indeed the ideals of sonority in the acoustic design of the modern
piano and in all but the more radical forms of modern pianism are as intimately bound to the
acoustic qualities of equal temperament as any previous keyboard style ever was to its contemporary style of intonation. The enharmonic facility of Brahms or Fauré, the hovering sonorities
of Debussy, the timbral poise of Webern, the slickness of the most urbane jazz chord progressions, all rely implicitly on the hue of equal temperament as much as on the other normal characteristics of the instrument’s tone. An 18th-century tuning usually sounds as inappropriate for this
music as the piano would seem visually if its glossy black finish were replaced by an 18th-entury
décor.
9. Fretted instruments. Insofar as the relation between performing practice and compositional
style is concerned, the history of temperaments on fretted instruments in Western art music
since the mid-16th century has been simpler than on keyboard instruments, for two reasons:
the placing of the frets and tuning of the open strings does not impose an exact intonation of the
scale on the player as definitively as the harpsichord or organ tuner’s handiwork does; and the
use of distinctly unequal semitones in the fretting scheme is likely to be problematical as each
fret runs under all the strings at once.
Lute and viol strings were traditionally made of gut (although from the late 17th century the
lowest strings were likely to be overspun). Inconsistencies in this material are greater than in
carefully drawn wire for harpsichord strings. Even Hubert Le Blanc acknowledged, in his enthusiastic Défense de la basse de viole (1740), that ‘the rules for gut strings are variable. Two
strings of the same thickness [grosseur], as clear as rock crystal, make the 5th at a considerably
different degree forward and back’. Practical considerations of this kind tend to overshadow the
embodiment of any precise model of a tempered scale to a greater extent than on the harpsichord
(where imperfections in the strings are less telling) or the organ. Moreover, all the mathematical
schemes for determining the position of the frets were based on measuring off certain portions
of the neck, as if an alteration in string length were the only effect produced when the player
presses the string against the fret: but the concomitant increase in the string’s tension is significant enough to reduce even the most precise geometrical division to a mere preliminary. Several Renaissance and Baroque musician-theorists who discussed fret placing remarked that one
must make, or that players did make, further adjustments by ear.
Thus a mathematical procedure for determining the position of frets on a lute or viol had a rather
different kind of significance from that of a monochord prescription. Monochord schemes often
embodied a precise model of intonation which the performer or instrument maker or tuner was
supposed to transfer into practice by whatever techniques were appropriate. The fact that Renaissance and Baroque theorists developed the use of irrational numbers in their monochord schemes
is, in large part, evidence of this precise intention. Formulae for marking where to tie a fret on
the neck of a lute or viol often involved exclusively rational numbers. The reason is not only that
an accomplished player, a person often of quite different educational and social background from
many music theorists, would be less likely to know or care about irrational numbers, but also that
for the player the numerical measuring was but the first stage in determining the intonation of the
instrument. Its function was merely to help place the frets close enough to their final position so
that the player could go on to the next stage, that of tuning the open strings by ear and concurrently making slight adjustments in the frets. The intonation of the notes might then be further
shaded ad hoc during performance.
In this light one may distinguish four kinds of late Renaissance fretting prescriptions: those
with exclusively Pythagorean ratios (e.g. Finé, 1530); those that embody a precise mathematical model of equal temperament but are too elaborate to be of much practical use (e.g. Zarlino
1588); those that seem to betray by their complexity an erudite intention, but fail to embody precisely any feasible model of intonation (notably Dowland 1610; see Poulton 1972, Appendix 1);
and those simplified for practicability. Among the last type, equal temperament is best represented by Vincenzo Galilei’s rule – that the ratio 18:17 should be used for placing each successive fret down the neck of the instrument – a rule that Mersenne said was used by many instrument makers. A less exact approximation, but perhaps adequate for its purpose, could be gained
by dividing the 9:8 whole tones of a Pythagorean diatonic scheme into 18:17 and 17:16 semitones, a rule given by theorists as late as Nassarre (1723-4). Ganassi’s equally simple viol prescription (1542-43) would have produced, from the nut to the eighth fret, the ratios 24:22 :21 :
20:19:18:17:16:15, had he not specified that certain frets be shifted up or down by their own
width or by half that amount. His accompanying remarks not only emphasize the need for refinements by ear but also refer to the tempering of 5ths for the sake of distinguishing major and
minor semitones and for a better sound in chords and in ensemble music. There are corroborating
indications that before the second half of the 16th century, approximations of meantone temperament may have competed with equal temperament on fretted instruments, and a distinguished
modern lutenist (see Dombois 1974) has found them musically advantageous in some instances.
The historical evidence is inconsistent but suggestive.
In meantone temperament as in Pythagorean intonation there is a distinction in size between
diatonic and chromatic semitones (A-B , for example, being larger than B -B♮ in meantone temperament, but smaller in Pythagorean intonation). The inconvenience of maintaining this distinction on fretted instruments is illustrated by the fact that most 16th-century lute music requires the
first fret on the third lowest string to provide the note a chromatic semitone above the open string
(G on the G string on a lute with the highest string tuned to a′) but calls on the same fret to provide a diatonic semitone above the fourth open string (C on the B string).
It would have been difficult for early Renaissance theorists to recognize or approve of the use
of equal semitones because until the publication in Latin of Euclid’s Elements in 1482, the myth
that a whole tone could not be divided into two equal parts was virtually unchallenged among
scholars. To demonstrate this premise, theorists would sometimes explain that if a whole tone
of ratio 9:8 were divided into two semitones of ratio 18:17 and 17:16, then obviously one of the
semitones (18:17) would be smaller than the other. The fact that an 18:17 semitone amounts to
99 cents, virtually the same as the semitone of equal temperament, might give particular significance, however, to certain early 16th-century assertions such as that of Spataro (1521) that
‘the lute has minor semitones [for] all its frets’ (‘el leuto ha tuti li soi tasti semitonii minori’; see
Lowinsky 1956) or of Martin Agricola (1545) that ‘almost the majority of lute and viol players
make all the frets equal to one another … a fret produces the minor semitone’ (‘fast das gröste
part / Der Lautnisten und Geiger art / Alle bünd machen gleich von ein / … ein bund / Der semiton minus/thut kund’). Yet Agricola’s statement implies that in his day some players of fretted
instruments did not use equal semitones. Arnolt Schlick (Tablaturen etlicher Lobesang und Lidlein, 1512) and, for the most part, Luys Milán (El maestro, 1536) seem to have avoided those
combinations of fingerings that would render inconvenient the use of unequal semitones. The
only one of Schlick’s 15 extant lute pieces that uses the first fret on the third string (Nach lust
hab ich) does so on no other, while in other respects his music accommodates a fretting with
the following succession of semitones as one ascends from the open strings: diatonic, chromatic;
diatonic, chromatic; diatonic; diatonic, chromatic; diatonic. (Above each string the intervals provided are: minor 2nd, whole tone, minor 3rd, major 3rd, perfect 4th, diminished 5th, perfect 5th,
minor 6th; Fig.3.) To judge by his organ tuning instructions, Schlick’s ear was probably extraordinarily perceptive of such intricacies; he might well have sought to accommodate the use
of a meantone temperament in his lute compositions.
Figure 3. Partly schematic representation of a typical 16th-century lute fretting
Luys Milán did not altogether avoid using the first fret to provide G on the G string, but he did
exercise circumspection: whereas he would use the second fret for the A in Ex.16a or the fourth
fret for the G in Ex.16b, the first fret would not be required to provide G in any context more
conspicuous vertically than that shown in Ex.16c. It is true that occasionally (see Ex.16d) he required the fourth fret to provide not only the B shown in Fig.3 but also an E (instead of the D
in Fig.3), and similarly with regard to C and F at the sixth fret. Yet a rejection of equal temperament, in which major 3rds are the same size as diminished 4ths, is indicated by the prefatory
instructions to one of his fantasias (1536, p.30) to ‘raise the fourth fret a little so that the note of
that fret will be strong and not feeble’. It is difficult to be certain just what ‘raise’ and ‘strong’
meant in this instance; but a clearer inference can be drawn from the instruction to adjust the
same fret ‘up towards the pegs’ in his setting of Con pavor recordo el moro, in which the fret
is used exclusively for notes a major 3rd above the open string. This inference is confirmed
obliquely by an instruction of Enríquez de Valderrábano (1547, f.74v) to move the fourth fret
‘slightly down towards the rose’, indicating unequivocally that the pitch of the notes affected,
which in this case are all a diminished 4th above the open string, was to be higher than if they
had been a major 3rd above the open string: the major 3rd, one may thus infer, was a smaller
interval than in equal temperament. Evidently, however, Valderrábano did not consistently
reject the use of equal semitones, as some of his duets require the two vihuelas to be tuned a
minor 3rd apart from each other, an arrangement for which the use of unequal semitones in the
fretting would be bound to entail some sour unisons.
Ex.16. Milán (El maestro, 1536)
Bermudo’s comments (1555, ff.103–9) on fret placement and intonation reflect a confusion –
which is evident elsewhere in his book and which he shared with Gaffurius, Aaron, Lanfranco
and a number of other early 16th-century theorists – between meantone temperaments in practice
and Pythagorean intonation in theory. Bermudo indicated unmistakably, however, that rather
than using equal semitones many vihuelists would set some of the frets at a slant or else ‘not
press down the finger equally’ when playing. The latter technique had been referred to ten years
earlier by Aaron (1545, f.35v): ‘the [intonation of a] lute can be aided with the finger of the player, by the intensione and remissione of a minute space for the reintegration of its consonance’.
Well before the end of the 16th century it became a commonplace of Italian musical writings
that the use of equal temperament was normal for fretted instruments while keyboard instruments
were tuned with unequal semitones (i.e. in some form of regular meantone temperament); and
this distinction was carried on through most of the 17th century. Mersenne wrote, when discussing the characteristics of equal temperament, that musicians called the lute the charlatan of
instruments ‘because it passes off as good [il fait passer pour bon] that which, on good instruments, is bad’. The legerdemain in question, however, may often have been of the kind referred
to by Bermudo and Aaron: a matter of left-hand artfulness to draw chords more resonant than
those of equal temperament from instruments fretted in equal semitones. Every modern viol consort of any degree of finesse does this, and Marin Marais (1689) implied the use of such a technique on the solo viol when he said that his compositions in F minor would, to their detriment,
sound less ‘piercing’ (‘perçant’) if the viol were tuned up a semitone (taking its F from the
harpsichord’s G) for the benefit of an accompanist unfamiliar with that key. In all probability
equal temperament was, then as now, a theoretical norm from which players departed flexibly
for acoustical reasons – much as Renaissance keyboard musicians, confronted with instruments
of rigid intonation, departed from the theoretical norm of just intonation for reasons of practicability.
Two potential sources of systematic data on the spacing of frets in old instruments are the surviving, unaltered specimens of the cittern and the like, in which the frets were usually inlaid as
they are on a modern guitar; and depictions in works of art where the artist can be shown to have
treated such minutiae with sufficient care to give the rendition documentary value.
10. Difficulties in interpreting theoretical writings. A casual reading of Renaissance and
Baroque treatises will yield misleading clues as to contemporary practices. The prestige of the
traditional, neo-Pythagorean concept of music as ‘sounding number’ led quite a few writers
who were not musicians – for instance, mathematicians like Henricus Grammateus (1518),
Stevin (ca.1600), Caramuel (1667–70), Wallis (1698) and Euler (1739) – to describe theoretical
schemes that one should beware of taking as a mirror of contemporary musical norms. Other,
more musicianly writers, who may have been alert to very fine nuances of current tuning, were
ill-equipped mathematically to describe them in a quantitatively coherent and accurate way.
Some 15th- and 16th-century music theorists regarded ‘the comma’ as an indivisible quantity,
and Gaffurius in 1518 attributed to the syntonic comma a ‘remarkable power to wander about’,
adding itself to and subtracting itself from the various intervals of the scale. (This is feasible
if a computer is interposed between a keyboard and some tone-generators, but one can hardly
credit Gaffurius with predicting that possibility.) Several 16th-century theorists, including
Aaron (1545) and Bermudo (1555) as well as Gaffurius, confused the non-identity of enharmonic ‘twins’ (such as D and E ) in Pythagorean intonation, where the sharp is theoretically
higher than the flat, with their non-identity in a meantone temperament, where the flat is higher
than the sharp.
Because of the logarithmic relation between musical intervals and the corresponding stringlength distances on a monochord, theorists who wrote about tempered tuning before the development of logarithms in the 1620s may be regarded, with hindsight, as having been mathematically handicapped. When Artusi in 1603 criticized Monteverdi’s theory of tempered intonation
in madrigals, he said that in any case the science of music could not embrace irrational ratios.
(But in fact such ratios are proper to monochord schemes for tempered scales.) Some later theorists who were just as naive as Artusi in this regard were Thomas Salmon (1705), who would
divide the 5:4 major 3rd into 20:19:18:17:16 semitones, and Kirnberger (1776-79), who designed
his musically very crude temperaments exclusively in terms of integer ratios. Bédos de Celles
was more charmingly naive when he said (1770, p.430) that the octave is divided into 58 commas of four different sizes, which, however, he would not specify, because ‘that is useless to our
purpose’.
Even the 18th-century theorists who dealt in detail with subtle nuances in a ‘good’ style of unequal temperament often failed in their calculations to distinguish between the syntonic and
Pythagorean commas. Vallotti in his theoretical scheme (see WELL-TEMPERED CLAVIER) pro-
posed to temper the six 5ths among the diatonic notes, F-C-G-D-A-E-B, smaller than pure by
1/6 comma and make the other six pure. This obviously refers to the Pythagorean comma (the
wrinkle in the circle of 5ths; see Table 1 above); but also said that the diatonic major 3rds would
thereby be made larger than pure by 1/3 comma, which would be true only if his 1/6 and 1/3
were taken as fractions of the syntonic comma (the amount by which two pure 5ths minus two
pure 4ths exceeds a pure major 3rd). Werckmeister, Neidhardt and Sorge (in his earlier writings)
also used this approximative kind of reckoning.
Modern scholars have often sown worse confusions. Because Bermudo (1555, ff.103-09) discussed not only Pythagorean intonation but also an approximation to equal temperament, it has
been inferred (in Ward 1953, p.33) that in equal temperament itself, ‘the mi frets were a comma
higher than the fa frets’. Several early theorists have been misinterpreted as having advocated
equal temperament, among them Werckmeister (mainly because he designed temperaments with
no wolf 5th), Salmon (because of his 20:19:18:17:16 semitones), Lanfranco (because he said to
temper the major 3rds larger than pure), Grammateus (because he would divide each of the five
9:8 whole tones of the traditional diatonic monochord into two equal parts, thereby leaving sour
5ths at B –F and B–F and Ramis de Pareia (because his ideas were new, his name occurs in the
title of Barbour’s doctoral dissertation, and Barbour favored equal temperament). According to
one scholar (Fose 1992), Ramis’s placing of the wolf 5th was radically mistaken and all his other
advice as to which intervals must be avoided in practice was based merely on ‘notational spellings’ unrelated to their sound. In reality his choice of notational spellings was due to his having
observed that certain intervals which looked promising on the keyboard sounded sour. Since
there was, for example, a euphonious major 3rd above A but a sour one below F, he called A-C
good and C -F bad. Another scholar has argued that non-keyboard musicians are ‘beyond temperament’ and therefore 1/6-comma meantone temperament was the norm in 18th-century nonkeyboard music (Haynes 1991). Also notable is Kellner’s alleged discovery, which he patented
in 1975) of Bach’s ‘secret’ tuning from clues that Kellner thought were to be found in the design
of Bach’s seal and in some elaborate anagrams. Kellner argued that, since Bach was a musician,
the fact that the seal has seven points and five dashes conveys a music-treatise type message, and
the message is not that the keyboard has seven diatonic and five chromatic keys in each octave,
but that there should be seven untempered and five tempered 5ths (even though Marpurg said in
1756 (p.5) that ‘the best of the unequal temperaments in use’ had seven tempered and five untempered 5ths). In order to make fun of Kellner’s claim, Sparschuh published in 1999 a one-page
paper saying that a tuning of the scale can be derived from decorative loops on the title page of
Part I of Das Wohltemperierte Clavier, but avoided claiming that this could have anything to
do with how Bach tuned; Lehman then concluded (2005) that, although Kellner’s claim about
the seal is indeed spurious, the loops are tantamount to a theoretical writing by Bach and show
that a certain tuning which he, Lehman, had already devised and adopted matches Bach’s secret
numerical formula for tempered tuning. The journal Early Music declared that this ‘brilliant’
decoding ‘clearly establishes the temperament Bach had in mind’ (Knighton 2005), and Duffin
(2007) agrees; but various other schemes derived from the same loops have been declared by
other writers to be what Bach must have intended, and meanwhile the notion that the loops have
an occult meaning has been discredited (Lindley and Ortgies 2006).
If a treatise includes a description explicitly about tuning but entirely non-mathematical, one
should take care not to infer arbitrarily a rigid scheme. Schlick’s account of organ tuning has
been subjected to an exact reduction (Lindley 1974), and the same thing has been done to
Rameau and other 18th-century theorists (Asselin 1984). An over-exact interpretation should
not be imposed upon a rough rule of thumb. Sorge said (1748, pp.32f) that Gottfried Silbermann
tempered the 5ths twice as much as in equal temperament and Praetorius three times as much.
But since Praetorius had clearly meant 1/4-comma meantone temperament (in which the 5ths
are tempered 2¾ as much as in equal temperament), it is imprudent to assign, on the basis of
Sorge’s remark, an exact fraction of this or that comma to ‘Silbermann’s temperament’. Sorge
was only saying that it was a regular meantone temperament intermediate in quality between
1/4-comma meantone and equal temperament.
One has always to distinguish between speculative calculations, innovative proposals and descriptive references to current practice. It is instructive to compare Werckmeister’s novel monochord schemes with his non-mathematical account (1698) of good keyboard tuning; when he
said that an organ could be reset to his temperament by adjusting only some of the pipes, he was
clearly referring to his best-known scheme (see Fig.1c), which Huygens, Sorge and Marpurg also
tell us was the one he advocated, and allotting to a lesser practical status his own mathematical
schemes involving 5ths tempered by the unhealthy amount of 1/3 comma or based on mystical
virtue in the number 7. Neidhardt’s method of presentation was to describe a long series of
speculative possibilities and then specify three or four as being of practical value (musicians
close to J.S. Bach praised Neidhardt as a theorist of tempered tuning without ever referring to
any one of his schemes in particular). Marpurg, championing equal temperament and irritated by
the inept schemes of Kirnberger (‘which some [people] praised so much but nobody used’) and
von Wiese, printed in 1790 a number of new schemes of unequal temperament, none of which
favored the diatonic 3rds and none of which he intended for use.
These difficulties are aggravated by the fact that, perhaps because fine shades of tuning are
subtle and yet sometimes aesthetically telling, writers on the subject have often been involved
in quarrels emitting more heat than light. Some well-known examples are associated, one way
or another, with Ramis, Vicentino, Doni, Rameau and Kirnberger – who went so far as to say
(1781, p.3) that in a ‘good’ temperament, not a single piece by J.S. Bach, C.P.E. Bach, Handel
‘and other composers’ could be transposed to any other key without making it ‘unpracticable’.
A sensible but very time-consuming way to overcome the most significant difficulties is to combine a careful study of the treatises with musical probing using historically suitable repertory.
The revised neo-Baroque temperaments of the 1990s (see WELL-TEMPERED CLAVIER) were
arrived at in this way.
11. Temperament and harmony. From time to time there have been disagreements as to where
the limit between ‘heavily tempered’ and ‘sour’ may lie for this or that interval. Such discussions
show that a tempered interval is not the same as an out-of-tune one. Even stronger proof is the
fact that tempering the chromatic scale somehow or other has for more than 500 years been a
fundamental, albeit distinctive (until the 20th century) premise of Western harmony. Using terms
like ‘5th’ and ‘3rd’ for relations between pitch-classes rather than between specific pitches (thus
taking ‘5th’, for instance, to mean the relation of any G to any C, of any D to any G etc.), one
can say that a basic purpose of tempering is to obtain, physically and therefore conceptually, one
or both of the following two kinds of equation, which would otherwise, in the raw world of untempered keyboard scales before the advent of the computer (see JUST INTONATION), be unobtainable: (1) a chain of 5ths acceptably in tune = a consonant 3rd acceptably in tune; (2) a chain
of consonant major 3rds or minor 3rds acceptably in tune = a pure unison.
The meantone temperaments satisfy the first but not the second type of equation. A chain of
three or four 5ths (such as C-G-D-A or C-G-D-A-E) makes a good consonant 3rd, but then three
major 3rds (such as A -C-E-G ) do not make a unison, nor do four minor 3rds (although they
would if three major ones did, since the minor 3rd in a triad is the difference between a 5th and
a major 3rd). One consequence of this is that meantone temperament is inimical to double en-
tendre between a sharp and a flat. There is no sense of identity between, for instance, the E and
the D in Ex.17; instead, they are distantly related pitch classes which, although approaching each
other in pitch more closely than by a semitone, cannot together take a pivotal role in intertwining
of keys – as they do in Ex.18, which is by a lute composer and was conceived in terms of equal
temperament.
Ex.17. Schütz, Die so ihr den Herren fürchtet (swv364), beginning of the last section
Ex.18. S.L. Weiss, isolated enharmonic modulation, captioned ‘He who is knowledgeable will
understand’ (Chi sá / L’intenderá; from the Stammbuch of C.A. Schmid, 15 August 1742; originally lute tablature)
As equal temperament gradually came into widespread use on keyboard instruments during the
late 18th century and the 19th, enharmonic modulations (such as in Ex.18) gradually became less
‘experimental’ and then finally, in the music of composers like Wagner and Franck, perfectly
routine. Nuanced well-tempered tuning was not inimical to such modulations and they can be
found occasionally in the keyboard music of composers like Haydn (for instance in the Fantasia,
HXVII:4, bar 303) and even Rameau (in his L’Enharmonique, composed a few years before he
endorsed equal temperament), yet the nuances did to some extent direct the composer’s attention
away from the possibilities of enharmonic modulation. This was one cause of the profound difference between Schubert’s uncanny sensitivity to all sorts of non-enharmonic relations between
keys and Liszt’s apparently daring but, to a modern ear, relatively insipid exploitation, in equal
temperament, of ambiguous diminished 7th chords (e.g. in the opening section of his Prometheus).
Schubert was, on the other hand, more at home in formerly ‘remote’ keys like D major than had
been composers like Bach whose musical formation went back to the days of meantone temperament when, because there was no circle of 5ths, notes such as D , G , A and E were regarded
as inherently remote. Their mitigated remoteness in the ‘good’ unequal temperaments was due to
the fact that not all the imperfect consonances were equally euphonious in those tunings. A composer sensitive to the nuances would therefore tend to use the various leading-notes and triads in
various more or less subtly different ways. A proper understanding of lateBaroque and Classical
tonal structures must take this into account.
Without such nuances, meantone temperament had provided the basis for the high-Renaissance
and Baroque transformation of the medieval ‘church modes’, which had originally been monophonic, into vehicles of triadic harmony (see §5 above). Some curious byways in an early 15thcentury phase in the development of triadic harmony are due, however, to the fact that in a quasiPythagorean temperament, a chain of three or four acceptably tuned 5ths does not yield a 3rd
tuned acceptably for a stable triad such as the D- and A-major chords in Exx.1 and 2 above.
Some apparent musical consequences of this can be observed in one of Du Fay’s earliest songs,
Belle, plaissant et graciese, where the first three phrases, comprising the first of the two main
sections of the piece, cadence on open 5th sonorities based on F, C and A respectively, and in
the second section the main voice-part, the tenor, returns from A to F through a series of rising
4ths and falling 5ths (A-D-G-C-F) and then emphasizes again A as well as C (Ex.19), and yet
no F-major triad ever occurs as a vertical sonority, nor a C triad as a stable sonority (presumably
because on a keyboard instrument familiar to Du Fay, the 5ths F-C-G-D-A-E were untempered
and so the 3rds F-A-C-E-G were impure by an entire comma). Du Fay’s somewhat later music,
however, routinely includes stable triads on F and on C (as well as on G, B etc.); he evidently
participated in a 15th-century probing (see Ex.1 above) and subsequent abandonment of quasiPythagorean temperament. At stake was not how keys might be intertwined as in 18th- and 19th-
century modulations, but how triads might be used within one key.
Ex.19. Du Fay, Belle, plaissant et graciese (GB-Ob can.misc.213, fasc.2, p.22), tenor part, the
end of the first section and the entire second section
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