My NExT experiment...

This too may have been done already, but I'm going to work out some new scales I'll call Natural Equal Temperaments, or NETs. These are equal-temperament scales, but not based on divisions of an octave (a 2:1 ratio), but on divisions of the constant e, used in natural logarithms:

e = 2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274274663919320030599218174136...

Thus there will be circles of octaves, fifths, thirds, natural sevenths, etc., depending on how many tones the scale is divided into. So far I have these equal temperaments, grouped according to which interval is best represented, using a Grade A logarithmic tolerance (±0.05) and going up to 101 (I'm planning to go to 1,001 and at least 13-limit):

Octaves (2/1): 13, 26, 36, 39, 49, 52, 62, 75, 88, 101
Fifths (3/2): 10, 20, 30, 41, 51, 61, 71, 81, 91, 101
Major thirds/ninths (5/4, 5/2) 5, 18, 23, 41, 46, 59, 64, 82, 87
Natural sevenths (7/4): 18, 19, 37, 55, 56, 74, 92, 93

The approximate value of e in relation to the common 12-tet scale is 1731.234049¢, making it a perfect eleventh one-third sharp. I'll try and get some MIDI files; they'll just be scales, chords and simple arrangements for now.

~DaW~

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

You might amuse yourself with the JI sequences
1,2,5/2,8/3,65/24,163/60... converging to e and
1/2,1/3,3/8,11/30,53/144,103/280... converging to 1/e, as well as
similar note sequences to 1/sqrt(e), etc. etc. deriving from the
power series for exp(x).

From: <genewardsmith@juno.com>

> --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
>
> You might amuse yourself with the JI sequences
> 1,2,5/2,8/3,65/24,163/60... converging to e and
> 1/2,1/3,3/8,11/30,53/144,103/280... converging to 1/e, as well as
> similar note sequences to 1/sqrt(e), etc. etc. deriving from the
> power series for exp(x).

You're using the limit definition of e, which is something like 1 + 1 + 1/2!
+ 1/3! + 1/4! + 1/5! ... right?

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> You're using the limit definition of e, which is something like 1 +
1 + 1/2!
> + 1/3! + 1/4! + 1/5! ... right?

This is a series which converges to e; the limit definition is
usually taken to be lim n--> infinity (1+1/n)^n, and gives you
another toy to play with: (3/2)^2, (4/3)^3, (5/4)^4 as musical
sequences.