optimal meantone revisited 1

Dave Keenan wrote:

>I propose the following weighting formula, given a (non octave equivalent)
>ratio a:b
>
>Max( 6/Max(a,b), (a*b)/30) )

If we take 1/(odd limit) to represent this for 5-limit intervals considered in octave-invariant terms, the
meantone tunings ordered by mean dissonance are, from best to worst,

Meantone Size of Fifth Average error/limit
1/4-comma 696.58 0.956
Golden 696.21 1.021
31-equal 696.77 1.025
7/26-comma 696.16 1.029
50-equal 696.00 1.059
2/7-comma 695.81 1.092
LucyTuning 695.49 1.149
2/9-comma 697.18 1.168
3/14-comma 697.35 1.229
1/3-comma 694.79 1.274
19-equal 694.74 1.303
1/5-comma 697.65 1.338
43-equal 697.67 1.346
55-equal 698.18 1.526
1/6-comma 698.37 1.593
12-equal 700.00 2.172
26-equal 692.31 2.706
Pythagorean 701.96 2.868

Paul Erlich wrote:

>Dave Keenan wrote:
>
>>I propose the following weighting formula, given a (non octave equivalent)
>>ratio a:b
>>
>>Max( 6/Max(a,b), (a*b)/30) )
>
>If we take 1/(odd limit) to represent this for 5-limit intervals
considered in octave-invariant terms, the
>meantone tunings ordered by mean dissonance are, from best to worst,
>
>Meantone Size of Fifth Average error/limit
>1/4-comma 696.58 0.956
>Golden 696.21 1.021
>31-equal 696.77 1.025
>7/26-comma 696.16 1.029
>50-equal 696.00 1.059
>2/7-comma 695.81 1.092
>LucyTuning 695.49 1.149
>2/9-comma 697.18 1.168
>3/14-comma 697.35 1.229
>1/3-comma 694.79 1.274
>19-equal 694.74 1.303
>1/5-comma 697.65 1.338
>43-equal 697.67 1.346
>55-equal 698.18 1.526
>1/6-comma 698.37 1.593
>12-equal 700.00 2.172
>26-equal 692.31 2.706
>Pythagorean 701.96 2.868

This looks ok to me.

Choosing to use 1/odd-limit is equivalent to using my proposed formula but
deciding that 3:5 is more important than 5:6. This is the difference
between weighting major and minor thirds equally (as above) and weighting
them in the proportion 6 is to 5. Not a big difference.

I disagree with calling Pythagorean a kind of meantone tuning. Which
prompted my next post.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

>>>I propose the following weighting formula, given a (non octave
equivalent)
>>>ratio a:b

>>>Max( 6/Max(a,b), (a*b)/30) )

I wrote,

>>If we take 1/(odd limit) to represent this for 5-limit intervals
>>considered in octave-invariant terms, the
>>meantone tunings ordered by mean dissonance are, from best to worst,

Dave Keenan wrote,

>Choosing to use 1/odd-limit is equivalent to using my proposed formula but
>deciding that 3:5 is more important than 5:6. This is the difference
>between weighting major and minor thirds equally (as above) and weighting
>them in the proportion 6 is to 5. Not a big difference.

Since according to your formula, 3:5 is more sensitive to mistuning than
5:6, it makes sense to put the major sixth in and automatically get good
minor thirds.

>I disagree with calling Pythagorean a kind of meantone tuning.

Yeah, that's questionable, since it's a schismic tuning. It was really just
there for comparison.

Moving on to your next post . . .

>Can't we agree that
>Pythagorean and meantone are disjoint or at least share only 12-tET as the
>borderline case of both. We can go further.

>If I were asked to determine the other limit of the Pythagorean region I'd
>put it at 17-tET, 705.8c (also 34-tET).

You should use the term schismic or schismatic instead of Pythagorean.
Pythagorean refers only to the case with just 3:2s.