wilson mos/ rothenberg

Carl,

to explain this point:

>>In fact, I think this is true of any series-of-fifths-gerenated scale tuned
>>in the next highest ET in the series 3, 5, 7, 12, 19, 31.
>
>Eh? The diatonic scale doesn't contain all of the intervals of 19-tET.
>

I could be completely wrong, but I'm thinking of Yasser's series of
5th-generated scales, where the pentatonic is tuned in 7-tET, the diatonic
scale is tuned to 12-tET, the chromatic scale of 12 notes is tuned in
19-tET, and so on. These scales are always strictly proper when tuned to
an ET two places higher in the series 5, 7, 12 . . . (3 isn't on the list,
actually), such as the pentatonic 22323, but always proper but ambiguous in
one degree (and it's inversion) when tuned in an ET one place higher: as in
11212. Also, in later case, I observed that the scale always exhausts the
interval set of the tuning system. I'm not positive, however, whether it's
always the proper subset of lowest cardnality in the system to do so, but
that's the claim I was making.

>>Wilson's diagrams of scales generated by series of equal intervals
>>elegantly show, I think, that the series of fifths (approximate 3/2's) are
>>special in generating the most stable (in Rothenberg's sense) scales.
>
>Can you demonstrate this? I would be surprised if there are any special
>generators here -- all generators should turn out their fair share of both
>proper and improper scales, if the chaining is carried out far enough.

Again, I could be wrong, but I have the rudiments of a proof in this case.
The property I'm looking for is the greatest number of high-stability
scales in the sequence of scales generated by the interval. It's
important, then, that the sequence produces stable scales very quickly,
that is, with relatively few iterations (discarding the 1 and 2 note
cases). Restricting consideration to generators between the unison and the
tritone (1/2 8ve in log freq), since their inversions generate equivalent
scales in Rothenberg's sense, we first notice that small intervals fail the
criterion because it takes too many repetitions to cycle through the 8ve.
This has two consequences: there's an imbalance in the scale until just
before the interval cycles through the 8ve, so proper scales come only at
such a point, and if this occurs fewer times, then there are fewer
potential proper scales. Second, after (or just before) the generator
cycles through the 8ve, the newest tone will create an interval with the
original tone smaller than any which existed in the scale before. The
first such interval divides the generator itself (which had previously been
the smallest interval). The generator which creates the greatest number of
proper scales must have the largest number of succesively smaller minimum
intervals. But if the generator, which itself is the first minimum
interval, is already small, then the next smallest interval must be smaller
still, and the total number of intervals on the list of succesive minimum
intervals will be fewer than in the case of a larger generator, if we stop
the list at some arbitrary interval, such as 1 step of a given ET.
However, this doesn't mean that the very largest interval is the best. The
interval nearest to the tritone in an odd-numbered ET will always generate
the interval of 1 step after its second iteration. So the best interval
will always be a step below this interval in lower numbered ET's > 9, and
as we get into higher ET's it will fall a greater number of steps below it
(but since the steps are growing increasingly smaller, this translates into
a generator of relatively stable absolute frequency).
Paul also contends this point. I see two possible reasons for disagreeing
with it. Any generator, as you say, excluding intervals that divide the
8ve by an interger in log freq, will produce an infinite number of proper
scales. But this disregards the number of tones in these scales. I'm
interested in the generator which produces proper scales the most
frequently, that is, produces x number of proper scales with the fewest
number of iterations. You also might contend that the argument above
depends upon the use of equal tunings. If we allow that an interval in
some equal tuning serves as an adequate approximation of an interval
otherwise defined (say, by a rational freq ratio) then the reasoning above
finds the best generators (most frequent proper scales) given some
restricted range in the cardinality of the ET's. In actuality, the
argument doesn't necessarily depend on ET's, but depends on the abelian
groups represented by ET's. The prelude to Tristan und Isolde is in a
twelve note cyclic system whether or not the orchestra performing it
intones it to an accurate 12tET within some margin of error (unless that
margin is > 50 cents). Are cyclic groups perceptually real? I can't say,
but from experience I feel that every piece of music sets up an expectation
of a minimum step, which doesn't mean minimum with reference to some
absolute standard in freq or log freq ratio (we might say, Western music
sets up a 100 cent minimum, but that would be wrong, because a raised
leading tone doesn't upset the expectation), but a minimum with reference
to the number of distinct tones per 8ve, tuned however. If this is
correct, then my line of reasoning sets a limit on a perceptually real
factor: the minimum interval, or saturation of the 8ve.
You also responded to my point that this analysis might explain the use of
5ths in the generation of scales in many musical cultures, saying that this
fact is better explained by the concordance of the 3/2. The stumbling
block of this latter reasoning (this is not a point of my own invention) is
that such scales exist in cultures where harmony doesn't exist. You might
counter this by saying that the concordance of the 3/2, because it's so
powerful, can be percieved in melodic situations. I would agree: I have
found that difference tones (a relatively weak phenomenon of harmony) are
clearly perceptable in certain melodic situations, rapid sequences of high
pitched tones. I don't know if this is also true ourdoors.
Psychoacoustical evidence, however, shows that while the mistuning of
certain rational intervals is perceptable on the order of a cent when heard
vertically, such intervals are not tunable when heard melodically and
separated by a short pause. Notes separated by fifths and fourths tend not
to occur in direct succession in melodies (there may be some exceptions:
Venda music?). In any case, a stronger point is that many musical systems
using fifth-generated scales employ non-harmonic and rapid decay toned
idiophones such as xylophone-type instruments, where the consonance of the
3/2 is irrelevant.

jason

--- In tuning@egroups.com, Jason_Yust <jason_yust@b...> wrote:

> The property I'm looking for is the greatest number of
high-stability
> scales in the sequence of scales generated by the interval. It's
> important, then, that the sequence produces stable scales very
quickly,
> that is, with relatively few iterations (discarding the 1 and 2 note
> cases). Restricting consideration to generators between the unison
and the
> tritone (1/2 8ve in log freq), since their inversions generate
equivalent
> scales in Rothenberg's sense, we first notice that small intervals
fail the
> criterion because it takes too many repetitions to cycle through
the 8ve.
> This has two consequences: there's an imbalance in the scale until
just
> before the interval cycles through the 8ve, so proper scales come
only at
> such a point, and if this occurs fewer times, then there are fewer
> potential proper scales. Second, after (or just before) the
generator
> cycles through the 8ve, the newest tone will create an interval
with the
> original tone smaller than any which existed in the scale before.
The
> first such interval divides the generator itself (which had
previously been
> the smallest interval). The generator which creates the greatest
number of
> proper scales must have the largest number of succesively smaller
minimum
> intervals. But if the generator, which itself is the first minimum
> interval, is already small, then the next smallest interval must be
smaller
> still, and the total number of intervals on the list of succesive
minimum
> intervals will be fewer than in the case of a larger generator, if
we stop
> the list at some arbitrary interval, such as 1 step of a given ET.
> However, this doesn't mean that the very largest interval is the
best. The
> interval nearest to the tritone in an odd-numbered ET will always
generate
> the interval of 1 step after its second iteration. So the best
interval
> will always be a step below this interval in lower numbered ET's >
9, and
> as we get into higher ET's it will fall a greater number of steps
below it
> (but since the steps are growing increasingly smaller, this
translates into
> a generator of relatively stable absolute frequency).
> Paul also contends this point.

I don't know if you understood my replies, Jason. I suggested you try
a
generator of 2^(phi), as well as the other noble generators shown as
convergence point of the zigzags in Wilson's scale tree (http://
www.anaphoria.com/scaletree.html). The 2^(phi) generator, whose
location on the horizontal axis in the scale tree is given by the
number phi = 1.6180339887, produces proper scales with 1, 2, 3, 5, 8,
13, and, as you can see in http://www.anaphoria.com/ST22.html, 21,
34,
55, 89, 144, etc. notes (i.e., all the Fibonacci numbers).
Mathematically, the noble generators have the property that when
their
logarithmic size, as a proportion of the octave, is expressed as a
continued fraction, the expansion ends in a infinite succession of
1s.
Your observation about Yasser's series is not irrelevant, since the
same series of proper scales is produced by chaining Kornerup's
golden
fifth, which I believe is also a noble proportion of the octave. It
does not appear in the scale tree because the first two steps, 2/5
and
5/7, Yasser's "sub-infra diatonic" and "infra-diatonic" scales, are
not
adjacent entries in the tree. However, the corresponding series 2, 5,
7, 12, 19, 31, 50, 81, 131, etc. does not compare unfavorably with
the
Fibonacci series generated by the 2^(phi) generator, except that the
"degree of propriety" at each stage is greater with the latter.

> I see two possible reasons for disagreeing
> with it. Any generator, as you say, excluding intervals that
divide the
> 8ve by an interger in log freq, will produce an infinite number of
proper
> scales. But this disregards the number of tones in these scales.
I'm
> interested in the generator which produces proper scales the most
> frequently, that is, produces x number of proper scales with the
fewest
> number of iterations. You also might contend that the argument
above
> depends upon the use of equal tunings. If we allow that an
interval in
> some equal tuning serves as an adequate approximation of an interval
> otherwise defined (say, by a rational freq ratio) then the
reasoning above
> finds the best generators (most frequent proper scales) given some
> restricted range in the cardinality of the ET's. In actuality, the
> argument doesn't necessarily depend on ET's, but depends on the
abelian
> groups represented by ET's. The prelude to Tristan und Isolde is
in a
> twelve note cyclic system whether or not the orchestra performing it
> intones it to an accurate 12tET within some margin of error (unless
that
> margin is > 50 cents). Are cyclic groups perceptually real? I
can't say,
> but from experience I feel that every piece of music sets up an
expectation
> of a minimum step, which doesn't mean minimum with reference to some
> absolute standard in freq or log freq ratio (we might say, Western
music
> sets up a 100 cent minimum, but that would be wrong, because a
raised
> leading tone doesn't upset the expectation), but a minimum with
reference
> to the number of distinct tones per 8ve, tuned however.

With no restrictions on propriety?

If this is
> correct, then my line of reasoning sets a limit on a perceptually
real
> factor: the minimum interval, or saturation of the 8ve.
> You also responded to my point that this analysis might explain
the use of
> 5ths in the generation of scales in many musical cultures, saying
that this
> fact is better explained by the concordance of the 3/2. The
stumbling
> block of this latter reasoning (this is not a point of my own
invention) is
> that such scales exist in cultures where harmony doesn't exist.
You might
> counter this by saying that the concordance of the 3/2, because
it's so
> powerful, can be percieved in melodic situations. I would agree: I
have
> found that difference tones (a relatively weak phenomenon of
harmony) are
> clearly perceptable in certain melodic situations, rapid sequences
of high
> pitched tones. I don't know if this is also true ourdoors.
> Psychoacoustical evidence, however, shows that while the mistuning
of
> certain rational intervals is perceptable on the order of a cent
when heard
> vertically, such intervals are not tunable when heard melodically
and
> separated by a short pause. Notes separated by fifths and fourths
tend not
> to occur in direct succession in melodies (there may be some
exceptions:
> Venda music?).

I don't know what Venda music is, but most melodies do use fourths or
fifths, if not in succession, then at least as framing intervals, and
in the memory of the listener these intervals have a strong
relationship, especially when sung or produced by another instrument
with harmonic partials.

> In any case, a stronger point is that many musical systems
> using fifth-generated scales employ non-harmonic and rapid decay
toned
> idiophones such as xylophone-type instruments, where the consonance
of the
> 3/2 is irrelevant.

What happened to the difference tones you mentioned above? They occur
even between sine waves when played harmonically, so certainly they
occur with xylophon-like timbres, but I'm suspicious about your claim
that they occur melodically -- you must have overlapping decays or
something. In any case, you will not find that the 3:2 is at all a
special generator when considered on the basis of how often it
produces
proper scales. And in fact, in cultures where these xylophone-like
timbres are used, the generator often deviates quite far from 3:2 (or
7/12 octave): in Thailand it is 4/7 octave, and in Indonesia it is
either approximately 3/5 octave, or approximately 5/9 octave, for
slendro and pelog, respectively. The fact that all these generators
are
within 35 cents of a 3:2 ratio is, I believe, rooted in the vocal
basis
of these musical cultures. If the octave were truly the only
acoustically relevant interval, and propriety at various stages of
the
construction of a scale from a single generator was the only
criterion,
there would be no reason to choose a 3:2 ratio or anything near it.

>>>In fact, I think this is true of any series-of-fifths-gerenated scale
>>>tuned in the next highest ET in the series 3, 5, 7, 12, 19, 31.
>>
>>Eh? The diatonic scale doesn't contain all of the intervals of 19-tET.
>
>I could be completely wrong, but I'm thinking of Yasser's series of
>5th-generated scales, where the pentatonic is tuned in 7-tET, the diatonic
>scale is tuned to 12-tET, the chromatic scale of 12 notes is tuned in
>19-tET, and so on.

Ah- I thought you meant only the diatonic scale. You meant the 'Yasser
generalized diatonic scale' of the ET in question.

>but always proper but ambiguous in one degree (and it's inversion) when
>tuned in an ET one place higher: as in 11212. Also, in later case, I
>observed that the scale always exhausts the interval set of the tuning
>system. I'm not positive, however, whether it's always the proper subset
>of lowest cardnality in the system to do so, but that's the claim I was
>making.

I don't think so -- in 19-tET, a 10-tone MOS of generator 17...

0 1 3 5 7 9 11 13 15 17 19

...is proper and exhausts the intervals of the temperament. Also the
11-tone MOS of generator 12...

0 1 3 5 6 8 10 12 13 15 17 19

...Both are smaller than the 12-note Yasser chain-of-fifths scale in
19-tET.

>Any generator, as you say, excluding intervals that divide the
>8ve by an interger in log freq, will produce an infinite number of proper
>scales. But this disregards the number of tones in these scales. I'm
>interested in the generator which produces proper scales the most
>frequently, that is, produces x number of proper scales with the fewest
>number of iterations.

Perhaps the range of a "fifth" is better than most here, but many other generators do produce proper scales with reasonable number of tones. And
the fact that the fifth has been a historically popular generator isn't
really explained its utility here, since a given culture usually uses only
a single scale. Western music is overwhelmingly diatonic, for example.
Why not use one of the proper MOSs of the minor third? And, as you pointed
out, Rothenberg's model does not imply that improper scales are "bad".

I think we must attribute the popularity of the fifth to its strong position
in the harmonic series, one way or another. Paul Erlich's idea of
tetrachordality as a sort of 2nd-order octave equivalence is one good option.

>You also responded to my point that this analysis might explain the use of
>5ths in the generation of scales in many musical cultures, saying that this
>fact is better explained by the concordance of the 3/2. The stumbling
>block of this latter reasoning (this is not a point of my own invention) is
>that such scales exist in cultures where harmony doesn't exist.

Octaves exist in such cultures as well.

>I would agree: I have found that difference tones (a relatively weak
>phenomenon of harmony) are clearly perceptable in certain melodic
>situations, rapid sequences of high pitched tones.

Yes, but I don't think we need difference tones to explain the 3/2 in
melody.

>In any case, a stronger point is that many musical systems using fifth-
>generated scales employ non-harmonic and rapid decay toned idiophones
>such as xylophone-type instruments, where the consonance of the 3/2 is
>irrelevant.

I'm not so sure. And in fact, my position would be supported if one
could find evidence that cultures using such instruments subject their
fifths to greater mis-tuning than cultures using instruments with more
harmonic timbres.

-Carl