Yahoo Tuning Groups Ultimate Backup tuning-math powers of 3/2 mod 1

Hi, I read an interesting thing in a footnote to Stephen Wolfram's book _A New Kind of Science_ - it is that while the powers of 3/2 mod 1 are suspected to be uniformly distributed in the range 0 to 1, you can find values "u" such that the set of values { u((3/2)^n) mod 1 } is always higher than a certain quantity. For example, if you choose u =0.38906669065... then u(3/2)^n mod 1 is always greater than roughly 0.3. This footnote can be found online at http://www.wolframscience.com/nksonline/page-903c-text

This sounded to me like it was a statement about chains of fifths and the harmonic series, so I tried to translate it. I believe the first statement says that the difference between the pitch p1, n perfect fifths higher than pitch p0, and the closest harmonic of pitch p0 that is lower than p1, is uniformly distributed between 0 and 1. Ok; but how is this difference expressed? I get:

(3/2)^1 mod 1 = 1/2
(3/2)^2 mod 1 = 1/4
(3/2)^3 mod 1 = 3/8
(3/2)^4 mod 1 = 1/16
(3/2)^5 mod 1 = 19/32
(3/2)^6 mod 1 = 25/64
(3/2)^7 mod 1 = 11/128

and so on. The numbers make some immediate, but approximate, sense, as follows:

G is 1/2 of the way between the 1st and 2nd harmonic of C
D is 1/4 of the way between the 2nd and 3rd harmonic of C
A is 3/8 of the way between the 3rd and 4th harmonic of C
E is 1/16 of the way between the 5th and 6th harmonic of C
B is 19/32 of the way between the 7th and 8th harmonic of C
F# is 25/64 of the way between the 11th and 12th harmonic of C
C# is 11/128 of the way between the 17th and 18th harmonic of C

and so on (pitches in the left-hand column are Pythagorean).

Is there an exact sense in which these numbers correspond to familiar ways of measuring musical proportions? For example, G can be said to be exactly 1/2 of the way between C and c if we understand "halfway" to refer to the harmonic mean. Is there a corresponding sense in which Pythagorean A can be said to be exactly 3/8 of the way between the 3rd and 4th harmonic of C?

For the second statement, it would seem to say that we can choose an interval u such that if we look at the set of pitches that are all interval u above a Pythagorean pitch, the resulting pitches are never close upper approximations to harmonics of the starting pitch. Alternatively, we could say that the powers of 3/2 from a starting point p0 contain no close upper approximations to the harmonic series of p1, which is related to p0 by the interval u. (Close _lower_ approximations are not ruled out.) Does that sound right? It seems pretty surprising to me!

Using Wolfram's example where u = 0.389... (an interval of roughly an octave and 434 cents below), we would get the result that the Pythagorean set of powers of 3/2 above p0 is always at least 0.3 of the way between members of the harmonic series of p1, ~1634 cents above p0.

Anyway, I have no greater point to make, but I thought it was surprising, and it does concern something quite important to tuning: approximating harmonics with stacks of fifths. Has anyone come across related phenomena?

Best --Jon Wild

--- In tuning-math@yahoogroups.com, Jon Wild <wild@m...> wrote:

> Is there an exact sense in which these numbers correspond to
familiar ways
> of measuring musical proportions? For example, G can be said to be
exactly
> 1/2 of the way between C and c if we understand "halfway" to refer
to the
> harmonic mean. Is there a corresponding sense in which Pythagorean
A can
> be said to be exactly 3/8 of the way between the 3rd and 4th
harmonic of
> C?

Yes, it's the same sense. Just look at the frequencies or ratios to
the fundamental frequency. The 3rd harmonic, 3/1, can be written as
24/8; the 4th harmonic as 32/8. 27 is clearly 3/8 of the way from 24
to 32; so 27/8 is also 3/8 of the way from 24/8 to 32/8.

I'll have to think about the other part of your post later, but
hopefully Gene will beat me to it.

powers of 3/2 mod 1

🔗Jon Wild <wild@music.mcgill.ca>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Jon Wild <wild@music.mcgill.ca>