Re: phibonacci

Paul,

Thanks for the message you forwarded from Spud Dubois. His ideas about
the relationship between scales and vowel sets are very interesting and I
think point to a information theory based description of musical scales and
intonational systems. This seems like a fruitful avenue for further
research.
From our discussion it tends to seem like we're retreading beaten paths
but I'm not so sure. Certainly many people have noticed the mathematical
properties of 5 7 12 and its extension to 19 31, but I think looking at
these scales in new ways may lead to new insights about the nature of
scales and the music based on them. Using Rothenberg's calculations is one
way of looking at these scales in a new way, but in our discussion its
becoming increasingly evident to me that I'm beginning to diverge from
Rothenberg in my use of his ideas. Hopefully this will all become more
clear further on . . .

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

This is simply to say that the human ear and mind do not necessarily act
like a frequency analyser. There's only a certain degree to which actual
frequencies are relevant to the percieved relationships of pitches. In
fact, "actual" frequency is an ambiguous term--we have to decide what
determines what we call the actual freq: maybe a reading on a guitar tuner
sort of device. In that case I would say taking an electric tuner to a
concert and trying to experience the music by watching the needle go up and
down (presumably to make sure your hearing correctly) would be like trying
to do a simple algebra problem with Russel's predicate calculus.

>> 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?

I think you may be right. I'm taking the basic concepts which make the
calculation of propriety valuable to be those ideas of Helmholtz's which
Rothenberg discusses at the beginning of his Math. Sys. Theory article. A
musical scale acts in some sense like a machinist's scale: it allows us to
measure the distance between two tones on that scale. If the greatest
precision I could want in measuring intervals is such that there are twelve
distinct intervals in an 8ve, then I can think about actual tones in terms
of twelve categories. But if, at a particular moment, the music I hear is
completely diatonic, then, because the diatonic scale is proper in the
system of twelve tones, I can simplify my representation of tonal space
into a scale of seven tones where the unit of measurement is the diatonic
step. As we extend the scale to include more tones, we get a series of
embedded scales. 19 as a subset of 31 is 2122122121221221221 which
contains a 12-note subset, 323233232323 expressed in the scale of 31 or
212122121212 in the scale of 19, which contains a 7-note subset 5535553 or
3323332 or 2212221 or 1111111. The tonal interest of the music comes from
the existence of such embedded scales. The method of calculating propriety
doesn't need to change, I don't think, but if faced with a set of real
pitches converted by means of an electronic freq analyzer, the first
question to ask would not be "is this scale proper" but "is this scale
proper in a system of n pitches per 8ve." The same goes for a set of
pitches described in a different mathematical system, such as the
pythagorean diatonic. This scale would be proper in a 12 tone system but
improper in a 17 note system. I'm taking for granted a method of
translation which I need to define more precisely. The criteria for a good
translation from, say, a set of heard pitches to a scale of n tones would
be that intervals between pitches which are heard as the same get the same
measurement on the scale. If we choose any arbitrary number for n we're
likely to get many bad fits. A bad fit could correspond to an impression
of out-of-tuneness: if I insist on representing a set of heard pitches in a
twelve-note system into which they refuse to fit (and the contradictory
intervals are made obvious to me by the music) then I will characterize the
music as out-of-tune. That's not to say that "out-of-tune" doesn't
sometimes refer to a deviation from just intervals: we may use the same
term to describe two different phenomena.

>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.

So what you're saying here is, the Kornerup fifth is relevant only to such
a degree that it's replaceable by the pure fifth or the common meantone
fifths. If it were especially relevant musically than only an extension to
50 or 81 pitches would distinguish it's properties from those of the
meantone fifths. And on the implicit premise that systems of 50 or 81
pitches don't seem like a payoff in terms of the musical richness of the
systems, I would agree with you. I like the Yasser sequence up to 31 but
if we extend it to 50 or 81 pitches I start to question the usefulness of
this. I'm not settled on this point though. Systems of 50 and 81 tones
are purely theoretical to me, so my intuitions about them are as irrelevant
as they are to any music I know.

>>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 would propose a different way of looking at this problem. I must say
that continued fractions don't strike me as the most useful way to
represent it. They're useful for finding a noble number which the
successive generators approach, but for looking at proper scales which are
members of the sequence, I find the following representation simpler and
more descriptive. We're considering sequences of numbers which follow the
rule: the last member of the series at any point is the sum of the previous
number and some earlier number in the series. Then we consider sequences
where the latter number must be the number only two places previous in the
series. In this case the first two numbers of the series define the entire
series, Phibonacci's minimizes the magnitude of the numbers with respect to
their place in the series (we have to add certain constraints which exclude
irrelevant sequences: the second number must be greater than the first, the
first number is never 1, in which case the Phibonacci sequence is 2, 3, 5 .
. . but still has the property in question), and the Lucas sequence fails
the stronger criteria but can be replaced by 3, 4, 7, 11 . . . which
satisfies them. The sequence with the least numbers per place satifying
the weaker criteria is the natural numbers. An interesting sequence
failing the stronger but satisfying the weaker criteria is the sequence of
MOS's of the pure 5th generator which Pierre mentioned, 2, 3, 5, 7, 12, 17,
29, 41 . . .

>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.

It's still in the works, but I'll post my results when I come up with
anything. There may also be other properties to look at, because with the
method of beginning with two arbitrary numbers and making a sequence which
satisfies the strong criteria, some predictable patterns develop in the
scales.

jason