RE: [tuning] Re: phibonacci

>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.

Are we assuming the concert involves a single voice or monophonic instrument
only?

>>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. . . 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.

I don't think that criterion could help you decide whether n for a
Pythagorean diatonic scale should be 12 or 17 (or 29, . . .)

>>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.

hm -- see below

>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.

OK, why is that property important?

>Then we consider sequences
>where the latter number must be the number only two places previous in the
>series.

Have you been brainwashed by Yasser? :) :) :)

>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.

And the Yasser/Kornerup series would go 2, 5, 7, 12, 19 . . .

> 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 . . .

The fact that it satisfies the weaker property is intimately intwined with
the continued fractions I used to derive the sequence, in response to
Pierre's challenge. Think about it! It should work for _any_ generator.

Also -- I thought you wanted the scales to be proper, but the 7-, 17-, etc.
-tone scales of the pure 5th generator are not proper. The fact that we had
to use incremental values between the continued fraction convergents to get
these scales is closely related to their impropriety.

>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.

Though, once again, I wonder why you're attracted to the strong criteria,
I'm happy to note that all sequences that satisfy them will turn out to be
the denominantors of the continued fraction convergents of noble numbers,
and conversely.