figurate number expansions as scales

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, � 1, � 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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