Linear temperaments with small generators

These seem to turn up when people consider possible nonoctave tunings,
so I thought a look at temperaments with generators under an octave
and octave period was warrented.

Valentine 31&46
Poptimal generators
7-limit: 9/129 77.6978 cents
9-limit: 17/262 77.8626 cents
11-limit: 8/123 78.0488 cents. 13/200 of 78 cents is also 11-limit
poptimal.
16 note maximal MOS with no octave reductions
78 cent nonoctave scale

Octacot 41&68
Poptimal generators
7-limit: 13/177 88.1356 cents
9-limit: 8/109 88.0734 cents. 11/150 of 88 cents is also 9-limit poptimal.
11-limit: 8/109 88.0734 cents. 11/150 of 88 cents is also 11-limit
poptimal.
14 note maximal MOS with no octave reductions
88 cent nonoctave scale

31&94 temperament
Poptimal generators
7-limit: 5/156 38.4615 cents
9-limit: 4/125 38.4 cents
11-limit: 3/94 38.2979 cents
32 note maximal MOS with no octave reductions
38.4 cent nonoctave scale

What's a good name for the 31&94 temperament?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> These seem to turn up when people consider possible nonoctave tunings,
> so I thought a look at temperaments with generators under an octave
> and octave period was warrented.

Generators under 100 cents!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> These seem to turn up when people consider possible nonoctave
tunings,
> so I thought a look at temperaments with generators under an octave
> and octave period was warrented.
>
> Valentine 31&46
> Poptimal generators
> 7-limit: 9/129 77.6978 cents
> 9-limit: 17/262 77.8626 cents
> 11-limit: 8/123 78.0488 cents. 13/200 of 78 cents is also 11-limit
> poptimal.
> 16 note maximal MOS with no octave reductions
> 78 cent nonoctave scale

Ozan and you may be interested in this post by Robert Valentine,
where he mentions the 9-fold division of the perfect fifth, noticed
that it's also an equal division of other consonant intervals:

/tuning/topicId_25746.html#25746

and some subsequent discussion between Robert and me on this topic
can be accessed by following the links below that message.

> Octacot 41&68
> Poptimal generators
> 7-limit: 13/177 88.1356 cents
> 9-limit: 8/109 88.0734 cents. 11/150 of 88 cents is also 9-limit
poptimal.
> 11-limit: 8/109 88.0734 cents. 11/150 of 88 cents is also 11-limit
> poptimal.
> 14 note maximal MOS with no octave reductions
> 88 cent nonoctave scale

As you probably know, this is one of the most famous nonoctave scales
around here, being associated with the name Gary Morrison, often
considered the inventor of 88-cent (not 88-tone) equal temperament.

> 31&94 temperament
> Poptimal generators
> 7-limit: 5/156 38.4615 cents
> 9-limit: 4/125 38.4 cents
> 11-limit: 3/94 38.2979 cents
> 32 note maximal MOS with no octave reductions
> 38.4 cent nonoctave scale

This reminds me of, but is clearly different from, the Wendy
Carlos 'Gamma' scale, 35.1 cents/step or 34.188 steps/octave.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> This reminds me of, but is clearly different from, the Wendy
> Carlos 'Gamma' scale, 35.1 cents/step or 34.188 steps/octave.

The Gamma scale appears to make use of the 5-limit comma |-29 -11 20>,
which I don't recall seeing before. Could this be dubbed "wendy"? The
35.1 generator is at most 0.001 cents above the poptimal range, BTW.