Grove Music Online
Just intonation [pure]
Mark Lindley
https://doi.org/10.1093/gmo/9781561592630.article.14564
Published in print: 20 January 2001
Published online: 2001
When pitch can be intoned with a modicum of flexibility, the term
‘just intonation’ refers to the consistent use of harmonic intervals
tuned so pure that they do not beat, and of melodic intervals derived
from such an arrangement, including more than one size of whole
tone. On normal keyboard instruments, however, the term refers to a
system of tuning in which some 5ths (often including D–A or else G–
D) are left distastefully smaller than pure in order that the other
5ths and most of the 3rds will not beat (it being impossible for all the
concords on a normal keyboard instrument to be tuned pure; see
Temperaments, §1). The defect of such an arrangement can be
mitigated by the use of an elaborate keyboard.
1. General theory.
In theory, each justly intoned interval is represented by a numerical
ratio. The larger number in the ratio represents the greater string
length on the traditional Monochord and hence the lower pitch; in
terms of wave frequencies it represents the higher pitch. The ratio
for the octave is 2:1; for the 5th 3:2; for the 4th 4:3. Pythagorean
intonation shares these pure intervals with just intonation, but
excludes from its ratios any multiples of 5 or any higher prime
number, whereas just-intonation theory admits multiples of 5 in
order to provide for pure 3rds and 6ths.
To find the ratio for the sum of two intervals their ratios are
multiplied; the ratio for the difference between two intervals is
found by dividing their ratios. In Pythagorean intonation the whole
tone normally has the ratio 9:8 (obtained by dividing the ratio of the
5th by that of the 4th), and so the major 3rd has the ratio 81:64
(obtained by squaring 9:8). But a pure major 3rd has the ratio 5:4,
which is the same as 80:64 and thus smaller than 81:64. (The
discrepancy between the two (81:80) is called the syntonic comma
and amounts to about one ninth of a whole tone.) Since 5:4 divided
by 9:8 equals 40:36, or rather 10:9 (a comma less than 9:8), just
intonation has two different sizes of whole tone – a feature that
tends to go against the grain of musical common sense and gives
rise to various practical as well as theoretical complications. Some
18th-century advocates of just intonation and others since have
admitted ratios with multiples of 7 (such as 7:5 for the diminished
5th in a dominant 7th chord; see Septimal system).
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Two medieval British theorists, Theinred of Dover and Walter
Odington, suggested that the proper ratio for a major 3rd might be
5:4 rather than 81:64, and some 15th-century manuscript treatises
on clavichord making include quintal and, in one instance, septimal
ratios (see Lindley, 1980). Quintal ratios were introduced into the
mainstream of Renaissance musical thought by Ramis de Pareia,
whose famous theoretical monochord (1482) provided just intonation
for the notes of traditional plainchant, but with G–D, B♭–G and D–B
implicitly left a comma impure. Thence Ramis derived the 12-note
scale by adding two 5ths on the flat side (A♭ and E♭) and two on the
sharp (F♯ and C♯); in this scheme, C♯–A♭ would make a good 5th,
hardly 2 cents smaller than pure. Ramis did not intend or expect this
tuning to be used in any musical performances, however, for in his
last chapter (giving advice to ‘cantors’ and describing what he called
‘instrumenta perfecta’) he said that G–D was a good 5th but C♯–A♭
must be avoided (see Temperaments, §2).
Gioseffo Zarlino (1558) argued that although voices accompanied by
artificial instruments would match their tempered intonation, good
singers when unaccompanied would adhere to the pure intervals of
the ‘diatonic syntonic’ tetrachord which he had selected (following
the example of Ramis's disciple, Giovanni Spataro) from Ptolemy's
various models of the tetrachord. Zarlino eventually became aware
that this would entail a sour 5th in any diatonic scale consisting of
seven rigidly fixed pitch classes; but he held that the singers'
capacity to intone in a flexible manner would enable them to avoid
such problems without recourse to a tempered scale – and that they
must do so because otherwise the ‘natural’ intervals (those with
simple ratios) ‘would never be put into action’, and ‘sonorous
number … would be altogether vain and superfluous in Nature’. This
metaphysically inspired nonsense was to prove a stimulating irritant
in the early development of experimental physics, and during the
next three centuries a number of distinguished scientists paid a
remarkable amount of attention to the conundrum of just intonation
(as well as to various attempts to explain the nature of consonance
by something more real than sonorous numbers).
In the 1560s Giovanni Battista Benedetti, a mathematician and
physicist, pointed out in two letters to the distinguished composer
Cipriano de Rore (who had been Zarlino's predecessor as maestro di
cappella at S Marco, Venice) that if progressions such as that shown
in ex.1 were sung repeatedly in just intonation, the pitch level would
change quite appreciably, going up or down a comma each time. In
1581 Vincenzo Galilei, a former pupil of Zarlino, denied that just
intonation was used in vocal music, and asserted that the singers'
major 3rd ‘is contained in an irrational proportion rather close to
5:4’ and that their whole tones made ‘two equal parts of the said
3rd’. In the ensuing quarrels, Vincenzo Galilei's search for evidence
against Zarlino's mystical doctrine of the ‘senario’ (the doctrine that
the numbers 1–6 are the essence of music) led him to discover by
experiment that for any interval the ratio of thicknesses between two
strings of equal length is the square root of the ratio of lengths
between two strings of equal thickness. This undermined the
theoretical status of the traditional ratios of just intonation as far as
the eminent Dutch scientist Simon Stevin was concerned; it might
have had further consequences had not Galilei retracted in 1589 his
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1581 account of vocal intonation, and had not his son Galileo's
generation devised the ‘pulse’ theory of consonance, according to
which the eardrum is struck simultaneously by the wave pulses of
the notes in any consonant interval or chord (thus mistakenly
assuming that the waves are always in phase with one another).
Such a theory tended rather to undermine the concept of tempered
consonances, where the wave frequencies are theoretically
incommensurate.
Ex.1 One of Benedetti's demonstrations that just intonation, if used
consistently, will disturb the pitch
Descartes found Stevin's dismissal of simple ratios ‘so absurd that I
hardly know any more how to reply’, but Marin Mersenne advanced
the real argument that the superiority of justly intoned intervals is
shown by the fact that they do not beat (1636–7). (He probably
gained this argument from Isaac Beeckman, who seems to have
invented the ‘pulse’ theory of consonance.) 50 years later, however,
Wolfgang Caspar Printz wrote that a 5th tempered by 1/4-comma
remains concordant because ‘Nature … transforms the confusion
into a pleasant beating [which] should be taken not as a defect but
rather as a perfection and gracing of the 5th’. Andreas Werckmeister
agreed (Musicalische Temperatur, 2/1691/R).
About this time Christiaan Huygens developed Benedetti's point
(although he did not associate it with Benedetti) in his assertion that
if one sings the notes shown in ex.2 slowly, the pitch will fall (just as
in ex.1); ‘but if one sings quickly, I find that the memory of the first C
keeps the voice on pitch, and thus makes it state the consonant
intervals a little falsely’. Rameau stated (Génération harmonique,
1737) that an accompanied singer is guided by the ‘temperament of
the instruments’ only for the ‘fundamental sounds’ (the roots of the
triads), and automatically modifies, in the course of singing the less
fundamental notes, ‘everything contrary to the just rapport of the
fundamental sounds’. While this represents a musicianly departure
from the common error that there is something natural about
Zarlino’s model of the octave, it does rather overlook the fact that
the tuning of the ‘fundamental sounds’ was normally tempered on
keyboard instruments and lutes.
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Ex.2. Huygens's example of a succession of notes for which the use of
just intonation would make the pitch fall
The most eminent scientist among 18th-century music theorists,
Leonhard Euler, developed an elaborate and remarkably broad
mathematical theory of tonal structure (scales, modulations, chord
progressions and gradations of consonance and dissonance) based
exclusively upon just-intonation ratios. He failed to observe that a
5th tuned a comma smaller than pure sounds sour, and so allowed
himself to be misled by an inept passage in Johann Mattheson's
Grosse General-Bass-Schule (1731) into supposing that keyboard
instruments of his day were actually tuned in just intonation. Euler
at first rejected septimal intervals, saying in 1739 that ‘they sound
too harsh and disturb the harmony’, but declared in 1760 that if they
were introduced, ‘music would be carried to a higher degree’ (an
idea previously voiced by Mersenne and Christiaan Huygens). He
published two articles in 1764 to demonstrate that ‘music has now
learnt to count to seven’ (Leibnitz had said that music could only
‘count to five’).
Another extreme of theoretical elaboration was reached in the early
19th century by John Farey, a geologist, who reckoned intervals by a
combination of three mutually incommensurate units of
measurement derived from just-intonation ratios. Farey's largest unit
was the ‘schisma’, which was the difference between the syntonic
and Pythagorean commas. (The Pythagorean comma is the amount
by which six Pythagorean whole tones exceed an octave; the schisma
is some 1.95 cents and has the ratio 32805:32768.) His smallest unit
was the amount by which the syntonic comma theoretically exceeds
11 schismas (or by which 11 octaves theoretically exceed the sum of
42 Pythagorean whole tones and 12 pure major 3rds; this is some
1/65-cent, and its ratio would require 49 digits to write out). His
intermediate unit (some 0·3 cent) was the amount by which each of
the three most common types of just-intonation semitone (16:15,
25:24 and 135:128) theoretically exceeds some combination of the
other two units or the amount by which 21 octaves theoretically
exceed the difference between 37 5ths and two major 3rds.
2. Instruments.
Rameau reported (1737) that some masters of the violin and basse
de viol tempered their open-string intervals – an idea also found in
the writings of Werckmeister (1691) and Quantz (Versuch einer
Anweisung die Flöte traversiere zu spielen, 1752). But Boyden has
shown (1951) that evidence from the writings of 18th-century
violinists, particularly Geminiani and Tartini, points to a kind of just
intonation flexibly applied to successive intervals with adjustments
when necessary both melodically and harmonically on each of the
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four strings, tuned in pure 5ths, as points of reference. In the 1760s
Michele Stratico, a former pupil of Tartini, worked out a fairly
efficient system of notation for this kind of just intonation, including
septimal intervals (ex.3).
Ex.3 Stratico's pragmatic notation for just intonation: (a) the basic signs;
(b) the harmonic series and its inversion: (c) some chord progressions
To model a fretted instrument upon just intonation entails the use of
zig-zag frets. Dirck Rembrandtsoon van Nierop, a mathematician
who favoured just intonation for all sorts of instruments as well as
voices, worked out (1659) an exact fretting scheme for a cittern,
according to which each position on the highest course could be
supplied with one or more justly intoned chords as shown in ex.4.
Some other devotees of just intonation who designed fretted
instruments were Giovanni Battista Doni, Thomas Salmon and
Thomas Perronet Thompson (fig.1).
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Ex.4 Pure chords available on Nierop's justly intoned cittern (the
fourth fret, ‘e’ in the tablature notation, is a whole tone above the third
fret, ‘d’; the others are a semitone apart)
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5. Guitar with frets placed for just intonation, from Thomas Perronet
Thompson's ‘Instructions to my Daughter for Playing on the Enharmonic
Guitar’ (1829)
The simplest way to provide all possible pure concords among the
naturals of a keyboard instrument with fixed intonation is to have
two Ds, one pure with F and A and the other, a comma higher, pure
with G and B. (The concept of a diatonic scale in just intonation with
two Ds a comma apart goes back to Lodovico Fogliano's Musica
theorica, 1529.) If this group of eight notes is then provided with a
complement of ten chromatic notes each natural will have available
all six of its possible triadic concords. This scheme was described by
Mersenne and employed by Joan Albert Ban for a harpsichord built
in Haarlem in 1639 (for illustration see Ban, Joan Albert). Mersenne
stated that on a keyboard instrument of this type the ‘perfection of
the harmony’ would abundantly repay the difficulty of playing,
‘which organists will be able to surmount in the space of one week’.
The ‘justly intoned harmonium’ of Helmholtz (in mathematical terms
not exactly embodying just intonation, but deviating from it
insignificantly from a practical and acoustical point of view)
combined two normal keyboards for the scheme shown in fig.7. The
12 pitch classes shown to the left are on the upper manual, the 12 to
the right on the lower manual. No justly intoned triadic note is
present beyond the lines along the top and bottom of the diagram,
but the three notes at the right end (A, C♯ or D♭, and E) make justly
intoned triads with the three at the left (E, G♯ or A♭, and C). Thus the
major and minor triads on F, A, and D♭ or C♯ require the use of both
manuals at once. The 12 pitch classes shown in the upper half of the
diagram are each a comma lower in intonation than their equivalents
in the lower half of the diagram. Every 5th except C♯–G♯ or D♭–A♭ is
available at two different pitch levels a comma apart, and the same
is true of six triads: the major ones on E, B and F♯, and the minor
ones on G♯, D♯ or E♭, and B♭. In the case of triads on C, D, F, G and
A, however, the major triad is always intoned a comma higher than
its parallel minor triad.
Various other elaborate keyboard instruments capable of playing in
just or virtually just intonation have been built by Galeazzo
Sabbatini, Doni, H.W. Poole, H. Liston, R.H.M. Bosanquet, S. Tanaka,
Eitz, Partch, the Motorola Scalatron Corporation and others (see
Microtonal instruments). Playing such an instrument involves
choosing which form of each note to use at which moment. If the
proper choice is consistently made, impure vertical intervals will be
avoided and the occurrence of impure melodic ones minimized. The
criteria for choosing, which differ in detail with each kind of
elaborate keyboard pattern, are intricate but capable of being
incorporated in a pattern of electric circuits amounting to a simple
computer programme. In 1936 Eivind Groven, a Norwegian
composer and musicologist, built a harmonium with 36 pitches per
octave tuned to form an extension of Helmholtz's quasi-just-
intonation scheme, but with a normal keyboard, the choice of pitch
inflections being made automatically while the performer plays as on
a conventional instrument. He later (1954) devised a single-stop pipe
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organ of the same type, now at the Fagerborg Kirke in Oslo, a
complete electronic organ with 43 pitches per octave (1965), now at
the Valerencen Kirke in Oslo, and a complete pipe organ
incorporating his invention (c1970, built by Walcker & Cie.).
Groven's work has made just intonation practicable on keyboard
instruments that are no more difficult to play than ordinary ones.
While the distinctive quality of justly intoned intervals is
unmistakable, their aesthetic value is bound to depend upon the
stylistic context. In 1955 Kok reported, on the basis of experiments
with an electronic organ capable of performing in various tuning
systems, that musicians, unlike other listeners, heard the difference
between equal and mean-tone temperaments, giving preference to
the latter, ‘and a fortiori the just intonation, but only in broad
terminating chords and for choral-like music. However, they … do
not like the pitch fluctuations caused by instantaneously corrected
thirds’. According to McClure (‘Studies in Keyboard Temperaments’,
GSJ, i, 1948, pp.28–40), George Bernard Shaw recalled that in the
1870s the progressions of pure concords on Bosanquet's harmonium
(with 53 pitches in each octave) had sounded to him ‘unpleasantly
slimy’. E.H. Pierce (1924), describing the 1906 model of the
Telharmonium, which was capable of being played in just intonation
with 36 pitches in each octave, reported:
The younger players whom I taught … at first followed out my
instructions, but as time went on they began to realize (as in fact I
did myself) that there is a spirit in modern music which not only
does not demand just intonation, but actually would suffer from its
use, consequently they relapsed more and more into the modern
tempered scale.
The composer and theorist J.D. Heinichen remarked (Der General-
Bass in der Composition, 1728, p.85) that because keys with two or
three sharps or flats in their signature were so beautiful and
expressive in well-tempered tunings, especially in the theatrical
style, he would not favour the invention of the ‘long-sought pure-
diatonic’ keyboard even if it were to become practicable. These
remarks suggest that the recently achieved technological feasibility
of just intonation on keyboard instruments is but a step towards its
musical emancipation and that further steps are likely to depend on
the resourcefulness of composers who may be inclined in the future
to discover and exploit its virtues.
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Triad’, RMARC, no.16 (1980) 4–61
M. Lindley: ‘Der Tartini-Schuler Michele Stratico’, GfMKB:
Bayreuth 1981, 366–70
M. Lindley: ‘Leonhard Euler als Musiktheoretiker’, GfMKB
: Bayreuth 1981, 547–53
R. Rasch: ‘Ban's Intonation’, TVNM, 33 (1983), 75–99
M. Lindley: Lutes, Viols and Temperaments (Cambridge,
1984)
M. Lindley: ‘Stimmung und Temperatur’, Hören, Messen
und Rechnen in der frühen Neuzeit, ed. F. Zaminer
(Darmstadt, 1987), 109–331
For further bibliography see Temperaments.
Page 12 of 12
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