From: mclaren
Subject: non-octave scales and octave
equivalence
--
With typical insight, Paul Erlich made a particularly
interesting comment about non-octave scales in topic
6 of digest 797. He wrote: "The point is that if a note
comes close enough to an octave or a multiple octave,
it will sound equivalent, especially in the case of
harmonic partials. For example, an interval of 33 Pierce
steps exhibits equivalence, even though it is a very
harmonic partials. For example, an interval of 33 Pierce
steps exhibits equivalence, even though it is a very
different pitch class in the tritave scheme. Even when
the even partials are removed, I believe the virtual
pitch sensation is not very octave-specific."
My ears agree with Paul Erlich's here. Removing
odd or even partials doesn't seem to affect my
perception of the Bohlen-Pierce scale.
However, Paul E. did not mention whether he was
talking about "an interval of 33 Pierce steps"
*melodically* or *harmonically.* That is,
sounding the interval as a vertical dyad or
as two sequential notes one after the other.
Now, my experience is that this makes a *huge*
difference in the perception of octave
equivalence in non-octave scales. My ears
hear sequential (melodic) intervals as
being octave equivalent even if they are
significantly off from the octave--upwards
being octave equivalent even if they are
significantly off from the octave--upwards
of 30 or 40 cents in many cases, especially
if the interval is a multiple of an octave--say,
2 octaves, 3 octaves, etc.
However, the range of detuning within which my
ears will accept an interval as octave equivalent
is much smaller when the interval is a vertical
dyad (harmonic): somewhere in the range of 0-18
cents.
As a concrete example, take the Bohlen-Pierce
scale. Play melodically the interval of 8 scale
steps; if you play the melodic interval reasonably
quickly, you'll find that your ear accepts it as
a melodic octave. But if you sound that 8-step
interval as a vertical dyad, it will not sound
like an octave at all since the interval is
1170.4338 cents, outside the acceptable vertical
range for octave equivalence (except at very
1170.4338 cents, outside the acceptable vertical
range for octave equivalence (except at very
low fundamental frequencies).
This brings up an interesting point with regard
to non-octave scales: as most of you know, Enrique
Moreno has a very different conception of non-octave
scales than Gary Morrison or Your Humble E-Mail
Correspondent. Enrique believes that it is pointless
and meaningless to try to assign to the intervals of
non-octave scales familiar categories such as
"third" or "fifth" or "octave." Instead, Enrique
suggests that we accept the intervals of non-octave
scales on their own merits, rather than misguidedly
trying to jam them into familiar but conceptually
and musically limiting categories.
This view has merit. It recognizes the fact that
non-octave scales sound different in a basic way
from octave = 2.0 scales; Gary Morrison has
described non-octave scales as sounding like
from octave = 2.0 scales; Gary Morrison has
described non-octave scales as sounding like
"the musical equivalent of thick rich chocolate
milk shakes" and this is true--there's something
unutterably exotic and gorgeously alien about
most non-octave scales. They all share a very
sultry foreign "sound" which renders, say,
the 12th root of 3, the 15th root of 3 and the
13thr oot of 3 and the 25th root of 5 and the
37th root of 31 much more akin to one another
in "sound" or what Ivor Darreg called "mood"
than any trivial considerations of audible
octave equivalence.
On the other hand, there are problems with
Enrique's view of non-octave scale. For one
thing, there exist infinitely many non-octave
scales which are audibly identical to familiar
octave = 2.0 divisions of the octave. For
example: I defy anyone to tell the difference
octave = 2.0 divisions of the octave. For
example: I defy anyone to tell the difference
audibly between 12-TET and the 51st root
of 19, or the 105th root of 431, or the 114th
root of 727, the 122nd root of 1153, or the 126th
root of 1453. THe difference between a 2/1
and the equivalent interval in each of these
"non-octave" scales is less than 1/3 cent--
you *cannot* hear the difference between these
intonations and 12.
There exist infinitely many non-octave scales
audibly identical (not close, *identical* to
the ears, with a 2/1 less than 0.1 cents off
from 1200 cents) to 13-TET, 14-TET, 15-TET,
and so on.
This being the case, we are forced to recognize
that for a significant sub-class of non-octave
Nth root of K scales, there is *no audible
difference whatsoever* between these and
Nth root of K scales, there is *no audible
difference whatsoever* between these and
some N-TET octave = 2.0 scale. This being the
case, it would obviously be perverse in a
tuning audibly identical to 12-TET to try to
describe the intervals in exotic Nth root of
K terms rather than in terms of the familiar
fifth, major and minor third, fourth, major
and minor second, and so on.
Thus the situation for non-octave scales is
more complicated than anyone has mentioned
to date.
On the one hand, listeners will tend to hear
intervals in these scales *very* differently
melodically than harmonically if the interval
is slightly off from a familiar interval.
On the other hand, there exist a large class
of non-octave scales which sound audibly
*identical* to familiar dvisions of the octave.
of non-octave scales which sound audibly
*identical* to familiar dvisions of the octave.
Lastly, there's the question: In a given Nth
root of K non-octave scale, what is the most
consonant interval? That is, what is the
interval which takes the musical and acoustic
place of the 2:1 octave in ordinary divisions
of the octave with harmonic series timbres?
--
There is no simple answer to this question.
A superficial answer is: obviously, if we're
talking about the Nth root of K, then K is the
most consonant interval in all cases.
This is sometimes true, and sometimes clearly
false.
In the 13th root of 3, the 3:1 ratio is clearly the
primary consonant interval. It functions musically
in the same way that a 2:1 does. If you "double"
pitches at an interval of 13 scale steps in the
in the same way that a 2:1 does. If you "double"
pitches at an interval of 13 scale steps in the
13th root of 3, you'll get much the same result
as when you double pitches at an interval of 12
scale steps in 12/oct.
In the 21st root of 17, however, the interval
of 21 scale steps is not nearly as great a
point of acoustic rest as the interval of 3 scale
steps.
Moreover, all Ks are not created equal.
Intervals which are low members of the harmonic
series multplied by small integers tend to sound
more consonant than Ks which are high members
of the harmonic series. Thus , an interval of 17:1
sounds less consonant than 6:1 since 6:1 is 3:1 times
2, while 17 is relatively far up the harmonic series.
Even this statement must be qualified, for the
harmonic series exhibits the property that
consonance decreases as one climbs the harmonic
harmonic series exhibits the property that
consonance decreases as one climbs the harmonic
series, then suddenly it begins to increase as one
climbs further, then consonance decreases again,
then it suddenly increases, and so on.
For example: 2, 3, 4, 5, 6 are highly consonant.
7 is less so, intermediate in fact between consonance
and dissonance; 8, 9, 10 are highly consonant, 11
is much less consonant; 12 is highly consonant;
13 is relatively dissonant; 14, as a multple of 7,
is intermediate in consonance; 15, 16 are highly
consonant; 17 is relatively dissonant; 18 is highly
consonant; 19 is quite consonant, ditto
20 and 21; but 22 and 23 are relatively dissonant;
24, 25 are highly consonant; 26 is dissonant,
27 is extremely consonant; 28 intermediate;
29 is dissonant...and so on.
Thus the particular K is question must be considered,
in addition to the issue of whether the Nth root
Thus the particular K is question must be considered,
in addition to the issue of whether the Nth root
of K scale contains an interval interval within
the N:1 span that sounds more consonant than
N:1.
One last point is that the absolute size of the
musical interval in question is very important.
Paul Erlich mentioned that an interval of 33
scale-steps of the Bohlen-Pierce scale sounds
like an interval of 4 octaves. However, this
interval comes out to 4828.0396 cents, 28 cents
away from 4 octaves. The ear doesn't tend to
notice this discrepancy for very large intervals
because the two notes are so greatly separated
from one another than there is little opportunity
for the harmonics of the lower and the upper
note to beat with one another. Most acoustic timbres
exhibit very little energy above the 16th harmonic,
and the 16th harmonic is the fundamental of a
exhibit very little energy above the 16th harmonic,
and the 16th harmonic is the fundamental of a
pitch 4 octaves above the base note of a dyad.
Thus, while an interval of 33 scale steps in
the Bohlen-Pierce scale is about as far away
from the octave as an interval of 8 scale
steps (28.039 cents for the former as opposed to
29.567 cents for the latter), 8 scale-steps in
the 13th root of 3 sounds very far from octave
equivalence while 33 scale-steps sounds
reasonably close to octave equivalence because
many harmonics of both notes fall within the
critical band in the case of the 8-step interval
while almost no harmonics of both notes fall
within the same critical band in the case of the
33-step interval.
In short, octave equivalence and the question of
which intervals will most tend to function and
sound as points of acoustic and musical rest in
which intervals will most tend to function and
sound as points of acoustic and musical rest in
intervals formed from the notes of non-octave
scales are issues more complex than anyone on
this forum appears to have suggested.
--
Paul Erlich goes on to write that "In the case
of inharmonic partials, octave equivalence may
play less of a role, but still exists, and is less
demanding as to intonation." Both my experiments
with additive synthesis inharmonic timbres in
Csound and William Sethares' experiments with
resynthesized Fourier-analyzed timbres with
stretched partials strongly contradict this
statement. In particular William Sethares has
a set of instrument timbres resynthesized with
all harmonics stretched so that the octave is a
ratio of 2.1 instead of 2.0, etc.
Playing a vertical octave dyad with such timbres
ratio of 2.1 instead of 2.0, etc.
Playing a vertical octave dyad with such timbres
produces unbearable dissonance; but playing a
vertical octave whose ratio is 2.1 rather than 2.0
produces the familiar sensation of octave
equivalence. So the evidence *strongly* indicates
that 2:1 octave equivalence goes away when the
timbre becomes inharmonic, and this is confirmed
by William Sethares' mathematical procedure for
finding scale pitches from an inharmonic timbre.
--mclaren
