RE: [tuning] RE: Re: Holy smokes. . .

Carl wrote,

>>>I don't see how the minimum-entropy scales are better than, say,
>>>harmonics 5-10 under these circumstances.

I wrote,

>>You may not have noticed the line
>>
>>0 268 498 703 885 40.473
>>
>>in my results, but that corresponds to the scale you're talking about.
Would
>>you like me to show, interval by interval, how this number is arrived at,
as
>>opposed to how the 39.506 of
>>
>>0 195 390 699 891 39.506
>>
>>is arrived at; or should we step outside this framework and talk about its
>>validity?

Carl wrote,

>Show first, please.

OK. Each scale has 10 intervals:

0 268 498 703 885 0 195 390 699 891

Interval Harmonic Entropy Interval Harmonic Entropy
498¢/702¢ 3.6803 501¢/699¢ 3.6917
497¢/703¢ 3.6811 501¢/699¢ 3.6917
315¢/885¢ 4.0065 504¢/696¢ 3.7230
387¢/813¢ 4.0885 504¢/696¢ 3.7230
230¢/970¢ 4.1326 309¢/891¢ 4.0246
583¢/617¢ 4.1510 309¢/891¢ 4.0246
182¢/1018¢ 4.1685 390¢/810¢ 4.0900
268¢/932¢ 4.1811 192¢/1008¢ 4.1767
205¢/995¢ 4.1857 195¢/1005¢ 4.1805
435¢/765¢ 4.1994 195¢/1005¢ 4.1805

total 40.4747 total 39.5062

I'd like to develop a true odd-limit harmonic entropy measure, but with that
the general pattern would be the same. Both lists start with two ratios of
three. But then, my scale has one ratio of 3 for every ratio of 5 in your
scale and one ratio of 5 for every ratio of 7 in your scale. Finally, both
scales have 3 intervals that are dissonant in the 7-limit. So my scale
clearly wins, and the small about of tempering required doesn't change that
result.