non-octave

Alpha [78.0 cents/step]

15tET (x1) = 80.0
31tET (x2) = 77.1
46tET (x3) = 78.3
77tET (x5) = 77.9

Beta [63.8 cents/step]

19tET (x1) = 63.2
56tET (x3) = 64.3
75tET (x4) = 64.0

Gamma [35.1 cents/step]

34tET (x1) = 35.3
103tET (x3) = 35.0

88CET [88.0 cents/step]

14tET (x1) = 85.7
27tET (x2) = 88.9
41tET (x3) = 87.8

Equalized Bohlen-Pierce [146.3 cents/step]

8tET (x1) = 150.0
25tET (x3) = 144.0
33tET (x4) = 145.6

Carl

Carl Lumma wrote:

> Alpha [78.0 cents/step]
> 31tET (x2) = 77.1
> 88CET [88.0 cents/step]
> 27tET (x2) = 88.9

Ah, interesting. Those are two I hadn't noticed before.

>>Alpha [78.0 cents/step]
>>31tET (x2) = 77.1
>>
>>88CET [88.0 cents/step]
>>27tET (x2) = 88.9
>
>Ah, interesting. Those are two I hadn't noticed before.

Just as the 7th root of 3/2 is related to 12tET, so are Alpha and Beta
related to 19 and 31tET. Gamma can be considered two interlaced 10th root
of 3/2 scales, much as 34tET can be considered two interlaced 17tET scales.

The idea is that when tempering an MOS, the generator usually takes the
punishment. But why not give the interval of equivalence a taste of the
medicine? The difference between the versions can be measured, for a given
MOS, by D/(G+IE) where D is the size difference between the two chains and
G and IE are the number of members in the chain of generators and intervals
of equivalence, respectively.

Of course any tuning can be explained as a retempering of any other tuning,
and so much the better if it helps us think about them.

88CET is then the "other" version of the 11-tone 7/4 vs. 2/1 MOS. 27 and
41 are both higher MOS's of this interval pair. I don't know how to
classify BP, which I find to be a scale of limited (I should say specific)
usefulness.

For sufficiently low values of the above formula I think various temperings
of an MOS ought to be quite similar conceptually, differing mainly in what
Ivor called "mood". My experience with 12tET vs. the 7th root of 3/2 backs
this up.

Carl

> 88CET is then the "other" version of the 11-tone 7/4 vs. 2/1 MOS. 27 and
> 41 are both higher MOS's of this interval pair.

I generally work with 88CET as an MOS (2 1 2 2 1 2 1) subset of 11 steps
per 7:4. 2 2 1 2 1 within a 3:2 is probably useful as well, although I
haven't used it much myself.