Re: [tuning] Digest Number 723

Paul,

Thanks for the in depth discussion of the properties of noble numbers.
This is a big help to my futher studying the properties of the scales
generated by them. I missed the point of your previously reply because I
hadn't recognized 2^phi as the generator of the Fibonacci series scales.

>> 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?

I'm assuming propriety to be derived from relations in the scale
constructed by the listener, not relations between absolute pitches. If
the listener has 12 notes in his 8ve (which will presumably be the case for
Western music which is not completely diatonic), then a tritone is a
tritone, whether or not the scale is tuned as a Pythagorean or Just 12-tone
scale. Certainly a Pythagorean diatonic doesn't have the effect of an
improper scale, but in terms of absolute pitches, it is.

>> 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.

When I play a short repeating sequence of notes in the upper range of the
tin whistle, very clear difference tones result from the interaction of the
sounding note with the resonances of the preceeding note. But this is such
an unusual effect that you'd never design scales on the basis of it.
Obviously, with xylophone-type instruments, the same effect will not occur,
the decay time, range, and volume are all completely different. Rasch has
done psychoacoustical studies on difference tones and concluded that they
are so weak in ordinary musical circumstances that they couldn't possibly
have an influence on ordinary musical perception. The point I meant to
illustrate, which I didn't make clear, even the weakest aspect associated
with the vertical combination of tones can influence the perception of
melody in some circumstances, a con to the main point I was trying to make.

>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.

Absolutely. We're agreed on this point. My idea is that perhaps the
Kornerup golden fifth has certain special mathematical properties which are
musically relevant. The nearness of this interval to the harmonically
relevant interval of 3:2 may be a coincidence which confuses our sense of
why the fifth is an important interval in various respects.

>The question as to which generator is "best" in your sense is a
>difficult one, Jason, since all noble numbers share the property that
>the ratio of the size of the denominator of one convergent to the
>size
>of the denominator of the next convergent rapidly approaches phi as
>you
>go farther and farther out into the expansion.

>However, you may be more interested in the first few MOS scales you
>get
>than in the behavior of MOS scales out toward infinity. Observe that
>the last property I mentioned, dividing existing steps in the
>proportion 1:phi, starts happening at the point in the process of
>taking convergents where the rest of the continued fraction expansion
>contains all 1s. Since that happens right away with the "2^phi" or
>phi-
>of-an-octave generator, it would seem that this generator best
>captures
>the property you're after. I know that you are interested in
>propriety;
>I leave it as an exercise for you to catalogue the first few proper
>scales that each of these generators generates, and let me know how
>they compare. My bets are on the the phi-of-an-octave generator.
>
>I'm afraid this generator, and the scales it produces, do not agree
>with what we observe in scales around the world. In fact I'm unaware
>of
>any music that uses this generator and any of the Fibonacci series of
>MOSs that it produces.

I concur again. I'm way off the mark if my analysis points out the 2^phi
generated scales as special. And clearly, it's cardinality numbers are
lower with respect to their positions in the series than any other similar
series, since it begins with the lowest numbers. I've been looking at that
series and comparing it to Yasser's series and it turns out that Yasser's
scales are more efficient. Whether adding efficiency into the equation
brings out a third series as special, I haven't discovered yet. Efficiency
is a much more difficult property to generalize about. What I have
discovered is that MOS's in general are special with respect to efficiency,
because they always have two transpositions differing only in one note from
the original. All of the MOS scales are generated by this process are
equivalent with respect to stability, so another factor must be at work,
and most likely, that factor is the approximation of Kornerup's 5th to the
3/2, but efficiency is, at least, one more possible consideration.

jason

Jason wrote,

>I'm assuming propriety to be derived from relations in the scale
>constructed by the listener, not relations between absolute pitches.

Can you elaborate?

> If
>the listener has 12 notes in his 8ve (which will presumably be the case for
>Western music which is not completely diatonic), then a tritone is a
>tritone, whether or not the scale is tuned as a Pythagorean or Just 12-tone
>scale. Certainly a Pythagorean diatonic doesn't have the effect of an
>improper scale, but in terms of absolute pitches, it is.

Do we need to modify the definition of propriety? Would this modification
have any bearing on your definition of the "best" generator?

>>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.

>When I play a short repeating sequence of notes in the upper range of the
>tin whistle, very clear difference tones result from the interaction of the
>sounding note with the resonances of the preceeding note.

Due to room reverberations? If not, and there is no overlap in the decays,
then you must be hearing something other than difference tones.

>Absolutely. We're agreed on this point. My idea is that perhaps the
>Kornerup golden fifth has certain special mathematical properties

It certainly does.

>which are
>musically relevant.

That is what I dispute. Spud duBoise has some interesting web pages up (see
http://www.rev.net/people/aloe/music/golden.html, on Kornerup's golden
meantone tuning) and when I grilled him on his assumptions behind the page
http://www.rev.net/people/aloe/music/temperament.html, it turned out that he
felt much as you do (I'll ask him if I can forward his remarks). However he
did not discount the importance of approximating simple ratios -- if
anything, he over-zealously ascribes

>The nearness of this interval to the harmonically
>relevant interval of 3:2 may be a coincidence which confuses our sense of
>why the fifth is an important interval in various respects.

The Kornerup golden fifth is even nearer to the optimal meantone fifth
(i.e., the optimal fifth for triadic harmony). See the table about 3/5 of
the way down the page http://www.ixpres.com/interval/dict/meantone.htm --
observe how very close the optimal meantone fifths are to the Kornerup
Golden fifth at 696.21 cents. See also the table at the bottom of the page.
Anyhow, the Kornerup fifth (like the Harrison/LucyTuning fifth) is so near
as to be indistinguishable from fifths with musically relevant properties.
That's as close as I'm willing to come to saying it's musically relevant.

>I concur again. I'm way off the mark if my analysis points out the 2^phi
>generated scales as special. And clearly, it's cardinality numbers are
>lower with respect to their positions in the series than any other similar
>series, since it begins with the lowest numbers.

Well, as I showed in my post earlier today, that appears incorrect. Calling
the 2^phi case "Phibonacci":

Phibonacci 1 2 3 5 8 13 21 34 55 89 . . .
Kornerup 1 2 3 5 7 12 19 31 50 81 . . .
Lucas 1 2 3 4 7 11 18 29 47 76 . . .

So in fact, Phibonacci's points of propriety are higher numbers with respect
to their positions than those of Kornerup's golden fifth. However, the Lucas
generator of 868.33 cents has got both of them beat.

>I've been looking at that
>series and comparing it to Yasser's series and it turns out that Yasser's
>scales are more efficient. Whether adding efficiency into the equation
>brings out a third series as special, I haven't discovered yet. Efficiency
>is a much more difficult property to generalize about. What I have
>discovered is that MOS's in general are special with respect to efficiency,
>because they always have two transpositions differing only in one note from
>the original. All of the MOS scales are generated by this process are
>equivalent with respect to stability, so another factor must be at work,
>and most likely, that factor is the approximation of Kornerup's 5th to the
>3/2, but efficiency is, at least, one more possible consideration.

I'd be interested in seeing your efficiency analysis and also analyzing the
Lucas scales I posted about today on the basis of efficiency. Actually, I
was going to guess the opposite -- that the Phibonacci generator is best
with regard to efficiency.