Post from Brian McLaren

From: mclaren
Subject: a possible misunderstanding of the psychoacoustic data
---
In Tuning Digest 708, Marion McCoskey stated:
"[mclaren's] assertion that people don't prefer small whole number
ratios seems to me to rest on a somewhat shaky basis. Upon watching
archers shooting a targets, do we assume from the distribution of
their arrows that they don't prefer hitting the bull's eye?"

I'd like to point out why Marion's analogy is out of kilter.
First, Marion's analogy presumes that the psychoacoustic test
results for interval preference are distributed like arrows that
miss a bull's eye.
This is incorrect.
Arrows that miss a bull's eye have a symmetric radical
distribution centered around the bull's eye. The mean
value of the arrows' distribution is thus the center of
the bull's eye.
But all the psychoacoustic tests done over the last 130
years consistently show that the mean value for melodic
interval preference is skewed much higher than the target
small integer ratios--even though test subjects consistently
*hear the intervals as "pure" and "just."*
Vertical interval preference mean values are skewed slightly
less high, but still significantly higher than the values predicted
by the small integer ratio theory of consonance.
So let's change Marion McCoskey's analogy to reflect the actual
facts.
In that case, all the arrows would always be grouped toward the
top half of the bull's eye--but when the archers looked at them,
they would see *all* the arrows as being exactly dead center in the
bull's eye. The archers could walk up, examine the arrows closely,
and they'd still report all the arrows as being dead center of
the bull's eye.
This correct analogy is bizarre and counterintuitive enough that
it captures the striking nature of the psychoacoustic results.

But these are not the only problems with the theory of consonance
according to small integer ratios. J. Murray Barbour has pointed
out other evidence against the small integer theory in his 1938
paper "Just Intonation Confuted," in Music and Letters,
Vol. 19, 1938, pp. 48-60, when he adduces evidence that the
tonal fusion of which Helmholtz spoke does not occur in the
way or for the small integer ratios Helmholtz's theory predicts:
"This extraordinary conclusion received complete and necessary
confirmation through experiments conducted by Guthrie and Morrill
at the University of Washington. They used the impressive number
of 753 student observers and tested a far greater number
of intervals--forty-four within an interval slightly larger than
a fifth. The averages of such an array of judgments compel
respect. The best fused of all intervals tested was a major third,
approximately the third of equal temperament, 2/3 comma sharper
than the just third. The preferred values for fifth and minor third
were similarly sharp. This report states emphatically that no
preference whatever was shown for just or for equally temered
intervals, and that slight resemblance exists between curves of
consonance and Helmholtz's theoretical curves.
"These experiments prove conclusively that Helmholtz and his
followers are wrong, that singers have no predeliction for the so-
called natural or just intervals, not even the major third (5/4), the
interval which most surely distinguishes just intonation from equal
temperament or the Pythagorean tuning." [J. Murray Barbour, "Just
Intonation Confuted," Music and Letters, Vol. 19, 1938, pp. 58-63]
Further on, Barbour points out:
"Singers show no natural preference for these [just] intervals. They
are likely to overleap a desired pitch by a comma or more, after
having swooped up to it from a whole tone below. They are
very fond of portamento.
"Athough the great lack of accuracy revealed in artistic singing
is mentioned here to destroy the myth that signers do use or
can use just intonation, it by no means follows that the
popular antithesis 'singers and musicians' is a truism. It is pointed
out in [Carl Stumpf's] Iowa study that although 'singers in the
physical sense are never on pitch, in another sense, the
perceptual, they are heard as on pitch.'" [J. Murray Barbour, "Just
Intonation Confuted," Music and Letters, Vol. 19, 1938, pp. 58-63]
Other experiments from the 1990s show that "waves with the same
period can lead to average octave judgments that differ
consistently by more than half an octave, and that a substantial
component of many timbre differences is actually a tone-height
difference." [Patterson, R. D., Robert Milroy and Michael Allerhand,
"What is the octave of a harmonically rich note?" Contemporary
Music Review, 1993, Vol. 9, parts I & 2, pp. 69-81]
This flagrantly violates the small-integer doctrine.
You can try some simple experiments that destroy the credibility
of the theory of consonance according to small integer ratios
on your own if you have a precise tunable synthesizer or Csound:
[1] Tune up the dyad 1: (sqrt(3/2) i.e., the ratio
1: 1.2247449
Starting at 320 Hz this would give a dyad with note
frequencies of 320 Hz, 391.91836 Hz.
Now tune up the dyad 45/32 and compare the two.
(Starting at 450 Hz this would give a dyad with note
frequencies of 320 Hz, 450 Hz.)
Because the ratio 45/32 is a much smaller ratio than
the irrational number (anathema to the mystic
numerology of small whole numbers) square root of 3/2,
the small-number consonance myth predicts that
the just intonation 45/32 tritone will sound more consonant
than the geometric-mean neutral third.
However, the irrational-ratio neutral third clearly sounds
much more consonant.
[2] Tune up 11/9 and compare it to 9/8. Which sounds more
consonant?
The 9/8 uses smaller numbers and should please the ear more,
according to the smaller integer theory of consonance. But
the 11/9 is clearly a more consonant interval. (Various forum
subscribers will predictably claim that the ear hears the 11/9
as a 5/4 or a 6/5 in this context. That conflicts drastically
with Harry Partch's experience and prescriptions--namely, that
11-limit odentities and udentities are clearly and unmistakably
heard as unique intervals with distinctive sonic characteristics
entirely apart from the 5-limit odentities and udentities. Anyone
who has listened to Partch's music knows this to be true. If you
want to argue with Partch, you've got a real problem, because
you're now arguing against the acoustic reality of limits higher
than the 5-limit.)
[3] Tune up a second inversion 4:5:6 chord--that is, a just major
chord with a 4/3 in the bass, i.e., a 9:12:15 triad. Now compare it
with a 10:12:15 minor chord.
The 9:12:15 chord clearly wants to resolve to *something* because
the 4/3 in the bass sounds discordant, whereas the 10:12:15 triad
sits happily where it is, not requiring any resolution.
Harmonic theory for more than four hundred years has recognized
that a second inversion chord demands resolution--and yet
the progression of a 10:12:15 triad to a 9:12:15 triad clearly
sounds like a progression from consonance to dissonance, while
the progression of a 9:12:15 to a 10:12:15 triad clearly
sounds like a progression from dissonance to consonance.
Yet this violates the myth of consonance by small integer
ratios.
(If you want to *really* blow away the small whole number theory,
resolve a 9:12:15 second inversion major triad to an 18:22:27
root-position neutral triad. Again, a dissonance audibly
goes to a consonance--yet the numbers go from small to
pretty damn large.)
[4] Tune up the ratio 3:5:7 and then tune up the ratio 18:22:27.
Which one sounds more consonant?
Clearly the neutral triad 18:22:27 sounds quite consonant, while
the 3:5:7 just triad sounds quite dissonant. Again,
however, this violates the theory of consonance by small
integer ratios.
[5] Tune up the 7:5 and then tune up the ratio 11:9. Which
dyad sounds more consonant?
Clearly the 7:5 tritone sounds much less consonant than the
11:9 neutral third, yet this too violates the theory of
consonance by small integers.
Many forum subscribers will claim that the ear approximates large
ratios by smaller ones automatically--this does NOT solve the
problem, for it simply puts us on the slippery slope. What's
a "large" ratio? And why doesn't the ear reduce ad infinitum?
Where does the process of reductive approximation stop,
and why?
This also sidesteps the dilemma of the just perfect fourth.
In fact, the ratio 4/3 has bedeviled music theorists for thousands
of years. Lippius and Zarlino classed the just perfect fourth
as a vertical consonance, while Marchettus of Padua classed it
as a vertical dissonance. Franco of Cologne, along with Boethius,
classed the diatessaron (perfect fourth) as a consonance--yet
Rameau and Mersenne classed it as a dissonance.
Even today the disagreement continues, with the perfect fourth
sometimes classed as a consonance, sometimes as a dissonance.
My own music teacher told us "it depends on the circumstances,
and it's very complicated." Basically he threw his hands up
in the air.
Yet according to the consonance theory of small whole numbers,
the perfect just fourth should always be the third most
consonant interval (2:1 first, then 3/2, then 4/3).
Worse yet, Boomsliter and Creel reported in 1970 that
"Incidentally, all of them [i.e., the test subjects] find the misnamed
true [i.e., just intonation] scale to be dull and lifeless for all
melodies. They find it inferior to the tempered scale, while the
tuning chosen for each melody sounded better than the tempered
scale." [Boomsliter, Paul C. and Warren Creel, "Hearing With Ears
Instead of Instruments," Journal of the Audio Engineering Society,
August 9170, 18(4), pg. 407]
Yet Boomsliter and Creel's referential search organ produced
"preferred" pitches with far more complex ratios than the
simple small whole numbers of the natural (that is, JI) scale.
Again, this violates (for melodies, rather than for
vertical harmonies) the theory of consonance according
to small integer ratios.
So what have we got?
Over and over again, small integer ratios are heard as dissonant
both melodically and harmonically when the theory of
small whole numbers predicts that they should be heard
as consonant.
Too, as Norman Cazden points out:
"Traditional rules of harmony and counterpoint explicitly
proscribe parallel successions of perfect intervals, though
with the unaccountable excpetion of that same troublesome
fourth when it is not in the lowest placement. That this
notorious ruling relates to some more abstract quality than
is presented concretely in sonorous events is plain from its
inapplicability to unison or octave parallels used `merely'
for 'doublings.' It is difficult to reconcile the standing
proscription with the purportedly superior 'consonant'
status of favored combinations.
"The traditional rules for harmony and counterpoint alike
regularly proscribe that dissonances be resolved, and
further that they be resolved wherever possible to imperfect
consonances [i.e., triads], rather than to the supposedly 'better'
perfect combinations. Now the Law of Nature seems quite
unable to account either for the permissibility of dissonance
in music, or, if it be permitted, the need for such dissonances
to be resolved. That such resolution is then best satisfied by
second-rank or imperfect consonances therefore compounds
the difficulties extremely.
"Indeed, the recognized need for resolution appears thus to
arise independetly of that Natural Law. At the least, it
indicates that that Law cannot stand as the sole determinant
of auditory judgment. The resolution relation also flatly
contradicts the Law's evaluation of the relative status
of desirable intervals. Thus at least some of the fundamental
instructions for musical procedures, all of them
demonstrably derived from the practices of leading
composers over many centuries, contravene the
premise of an inherently superior status for those
special relationships expressible in the simplest number
ratios. " [Cazden, Norman, "The Definition of Consonance
and Dissonance," International Review of the Aesthetics
and Sociology of Music, 1980, Vol. 2, pp. 123-168]
Worse yet, the small whole number theory of consonance
has an shady and very unsavory past.
"In its most general axiomatic form, the Natural Law
theory states that consonance results from the ratios
of small whole numbers.
"Crystallization of that axiom was ascribed by his
disciples to Pythagoras, though undoubtedly its basis
has been widely known to acient civilizations. However,
inseparably from the strictly musical relevance of the
principles, the magic of 'numbers come to life as music'
has long been linked more to the marvels of mystic
cosmologies and numerological speculations
than to any indications bearing on the practice of music
as an art. It is accordingly noteworthy that for the purposes
of speculative ventures devoted to wonderment over the
'harmony of the spheres,' the musical relevance tends ever
to remain remote and elementary. Typically their relevations
have applied more to the raw ingredients that are later
transformed into music, to tuning formulas or to the derivation
and naming of scale degrees, than to musical relationships
proper. So tenuous and curiously pre-musical a handling is
retained down to the present day in the fanciful use of musical
metaphor that inspires the still extensive literature of
number mysticism, such as occasionally also entices
students with musical training and experience." [Cazden, Norman,
"The Definition of Consonance and Dissonance," International
Review of the Aesthetics and Sociology of Music, 1980, Vol. 2,
pp. 123-168]
As J. Murray Barbour put it, the theory of small whole number
ratios "has always been a beautiful theory. Its devotees have been
drawn chiefly from the ranks of mystics and philosophers--
mathematicians who knew no music and musicians who knew
no mathematics." [Barbour, J. M., "Just Intonation Confuted,"
Music and Letters, Vol. 19, 1938, pg. 60]
--mclaren




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