Some interesting expansions and scales can be derived from figurate

numbers.

One simple example would be square numbers using the 4:5:6 where both

the syntonic comma and the diesis are tempered out. These would be

symmetric scale expansions of the Tcherepnin scale.

Can anyone think of any others?

take care,

--Dan Stearns

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Some interesting expansions and scales can be derived from figurate

> numbers.

Numbers of the form n/(n-1) where n is figurate show up a lot; you could look at my discussion of "jacks", for instance. The fact that

triangle and square demomenators lead to other triangle and square denomenators allows us to create series of scales.

The most interesting of these I found were generalized pentagonal

numbers in the form of n(3n-1)/2 where n = 0, ï¿½ 1, ï¿½ 2, ...,.

The interpretation I particularly like is one where positive integers

correspond to harmonic series that convert JI superparticulars to

uniquely articulated ET stepsizes by way of a 2n-1, ..., n sequence,

and negative integers correspond to subharmonic series that convert JI

superparticulars to uniquely articulated ET stepsizes by way of n+1,

..., 2n.

So if n = 4, then n(3n-1)/2 = 22, and if 2n-1, ..., n = 7, 6, 5, 4,

then in 22-tet, a 4 - 8 harmonic series' 5/4, 6/5, 7/6, 8/7 are

uniquely articulated. If n = -4, then n(3n-1)/2 = 26, and if n+1, ...,

2n = 5, 6, 7, 8, then a 4 - 8 subharmonic series' 8/7, 7/6, 6/5, 5/4

is uniquely articulated in 26-tet.

This all seemed too elegant not to be important to me!

take care,

--Dan Stearns

----- Original Message -----

From: "genewardsmith" <genewardsmith@juno.com>

To: <tuning-math@yahoogroups.com>

Sent: Saturday, June 15, 2002 12:32 PM

Subject: [tuning-math] Re: figurate number expansions as scales

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

>

> > Some interesting expansions and scales can be derived from

figurate

> > numbers.

>

> Numbers of the form n/(n-1) where n is figurate show up a lot; you

could look at my discussion of "jacks", for instance. The fact that

> triangle and square demomenators lead to other triangle and square

denomenators allows us to create series of scales.

>

>

>

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