Just [pure]
intonation.
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.
2. Instruments.
BIBLIOGRAPHY
MARK LINDLEY
Just intonation
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).
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 (fig.1a). 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 (fig.1b),
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 (fig.2a). Zarlino eventually became aware that this would
entail a sour 5th in any diatonic scale consisting of seven rigidly fixed pitch
classes (see fig.2b, where D–A is labelled
‘dissonant’); 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 1650s 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 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.
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 the scheme shown in fig.2b,
it does rather overlook the fact that the tuning of the ‘fundamental sounds’
was normally tempered on keyboard instruments and lutes.
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
195 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 (see fig.3) or the amount by which 21 octaves theoretically exceed
the difference between 37 5ths and two major 3rds.
Just intonation
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 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).
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 cistern (fig.4a),
according to which, if the open-string intervals were tuned as in fig.4b, then 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 (see fig.5).
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 (see fig.6a).
(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 as
indicated in fig.6b, 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 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.
Just intonation
BIBLIOGRAPHY
to 1800
MersenneHU
B. Ramis de
Pareia:
Musica practica (Bologna, 1482, 2/1482/R; Eng. trans., 1993)
G. Spataro: Errori de
Franchino Gafurio da Lodi (Bologna, 1521)
G. Zarlino: Le
istitutioni harmoniche (Venice, 1558/R, 3/1573/R; Eng. trans. of pt iii, 1968/R, as
The Art of Counterpoint; Eng.
trans. of pt iv, 1983, as On the Modes)
G. Zarlino: Dimostrationi
harmoniche (Venice, 1571/R)
F. de Salinas: De musica
libri septem (Salamanca, 1577, 2/1592; Sp. trans., 1983)
V. Galilei: Dialogo
della musica antica et della moderna (Florence, 1581/R; partial Eng. trans. in StrunkSR1)
G.B. Benedetti: Diversarum
speculationum mathematicarum & physicarum liber
(Turin, 1585), 282
G. Zarlino: Sopplimenti
musicali (Venice, 1588/R)
V. Galilei: Discorso
intorno all'opere di Messer Gioseffo Zarlino da Chioggia (Florence, 1589/R)
S. Stevin: Vande
spiegheling der singconst, c1600, ed. D. Bierens
de Haan (Amsterdam, 1884); ed. A.
Fokker in The Principal Works of Simon Stevin, v (Amsterdam, 1966), 413–64 [incl. Eng. trans.]
G.M. Artusi: ‘Considerationi musicali’, appx to Seconda parte dell'Artusi (Venice, 1603/R)
G.B. Doni: Compendio
del trattato de' generi e de' modi della musica (Rome, 1635)
M. Mersenne: Harmonicorum
libri, in quibus agitur de sonorum natura (Paris, 1635–6)
G.B. Doni: Annotazioni
sopra il Compendio de' generi e de' modi della musica
(Rome, 1640)
D.R. van Nierop: Wis-konstige
musyka (Amsterdam, 1659)
I. Newton: unpubd MSS
on music, 1665 (GB-Cu Add.4000)
C. Huygens: writings on
music, ed. in Oeuvres complètes,
xx (The Hague, 1940)
W.C. Printz: Exercitationes
musicae theoretico-practicae curiosae, iii–v (Dresden, 1687–8)
T. Salmon: A Proposal
to Perform Musick in Perfect and Mathematical Proportions (London, 1688)
T. Salmon: ‘The Theory of Musick Reduced to Arithmetical and Geometrical
Proportion’, Philosophical
Transactions of the Royal Society, xxiv (1705), 2072–77
J. Mattheson: Grosse
General-Bass-Schule, oder, Der exemplarischen Organisten-Probe zweite,
verbesserte und vermehrte Auflage (Hamburg, 1731/R)
P. Prelleur: The Modern
Musick-Master, or The Universal Musician, v: The Art of Playing on the Violin (London, 1731/R)
L. Euler: Tentamen
novae theoriae musicae (St Petersburg, 1739) repr. in Opera
omnia, III/i (Leipzig, 1926), 197–427; Eng. trans. in
C.S. Smith: Leonhard Euler's ‘Tentamen novae theoriae
musicae’ (diss. Indiana U., 1960)
F. Geminiani: The Art of
Playing the Violin (London, 1751/R1952 with introduction by D.D. Boyden
G. Tartini: Trattato di
musica secondo la vera scienza dell'armonia (Padua, 1754/R; Eng. trans. in F.B. Johnson: Tartini's ‘Trattato di musica secondo la vera scienza dell'armonia’ (diss., Indiana U., 1985)
G.B. Doni: Lyra
Barberina amphicordos, ed A.F. Gori and G.B. Passeri
(Florence, 1763/R)
L. Euler: ‘Conjecture sur la raison de quelques dissonances généralement reçues
dans la musique’; ‘Du véritable
caractère de la musique moderne’, Mémoires de l'Academie royale des sciences et des belles lettres de
Berlin, xx (1764), 165, 174–99; repr. in Opera omnia, iii/1, 508–15, 516–39
L. Euler: Lettres à
une princesse d'Allemagne (Berne, 1775, many later edns; Eng. trans., 1975)
M. Stratico: MS
treatises on music (I–Vsm it. iv 341–3)
since 1800
H. Liston: An Essay on
Perfect Intonation (Edinburgh, 1812)
J. Farey: ‘On Different Modes of Expressing the Magnitudes and Relations of
Musical Intervals’, American
Journal of Science, ii (1820), 74
T.P. Thompson: Instructions
to my Daughter for Playing on the Enharmonic Guitar
(London, 1829)
H.W. Poole: ‘An Essay on Perfect Intonation in the Organ’, American Journal of Science and the Arts, 2nd
ser., ix (1850), 68–83, 199–216
M. Hauptmann: Die Natur
der Harmonik und der Metrik (Leipzig, 1853, 2/1873; Eng trans., 1888/R)
C.E.L. Delezenne: Table de
logarithmes acoustiques (Lille, 1857)
T.P. Thompson: On the
Principles and Practice of Just Intonation with a View to the Abolition of
Temperament (London, 1860)
H. von
Helmholtz: Die Lehre von den Tonempfindungnen als
physiologische Grundlage für die Theorie der Musik
(Brunswick, 1863, 6/1913/R;
Eng. trans., 1875, as On the Sensations of Tone, 2/1885/R)
G. Engel: Das
mathematische Harmonium (Berlin, 1881)
S. Tanaka: ‘Studier im Gebiete der reinen Stimmung’, VMw, vi (1890), 1–90
C.A. Eitz: Das
mathematisch-reine Tonsystem (Leipzig, 1891)
M. Planck: ‘Die naturliche Stimmung in der modernen Vokalmusik’, VMw, ix (1893) 418–40
E.P.L. Atkins: ‘Ear-Training and the Standardization of Equal Temperament’, PMA, xli (1914–15), 91–111
E.H. Pierce: ‘A Colossal Experiment in Just Intonation’, MQ, x (1924), 326–32
N.L. Norton: ‘A New Theory of Untempered Music’, MQ, xxii (1936), 217–33
J.M. Barbour: ‘Just Intonation Confuted’, ML, xix (1938), 48–60
E. Groven: Temperering
og renstemning [Temperament and tuning] (Oslo, 1948; Eng. trans., 1970)
J.M. Barbour: Tuning and
Temperament: a Historical Survey (East Lansing, MI, 1951/R, 2/1953)
D. Boyden: ‘Prelleur, Geminiani, and Just Intonation’, JAMS, iv (1951), 202–19
E. Groven: ‘My Untempered Organ’, Organ
Institute Quarterly, v/3 (1955), 34
W. Kok: Harmonische
orgels (The Hague, 1955) [with Eng. summary]
H. Stephani: Zur
Psychologie des musikalischen Hörens: Hören wir naturrein, quintengestimmt,
temperiert? (Regensburg, 1956)
C.V. Palisca, ed.: Girolamo Mei
(1519–1594): Letters on Ancient and Modern Music to Vincenzo Galilei and
Giovanni Bardi, MSD, iii (1960, 2/1977)
C.V. Palisca: ‘Scientific Empiricism in Musical Thought’, Seventeenth Century Science and the Arts, ed.
H. Rhys (Princeton, 1961), 91–137;
repr. in C. Palisca: Studies in the History of Italian
Music and Music Theory (New York, 1994), 200–35
Ll.S. Lloyd: Intervals,
Scales and Temperaments, ed. H. Boyle (London, 1963, 2/1978)
C. Dahlhaus: ‘Die “Reine Stimmung” als musikalisches Problem’,
Festskrift til Olav Gurvin, ed.
F. Benestad and P. Krømer (Drammen and Oslo, 1968), 49–55
R. Dammann: ‘Die Musica mathematica von Bartolus’, AMw, xxvi (1969), 140–62
E. Groven: Equal
Temperament and Pure Tuning (Oslo, 1969)
D.P. Walker: Studies in
Musical Science in the Late Renaissance (London, 1978)
M. Lindley: ‘Pythgorean Intonation and the Rise of the 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, xxxiii (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.