Strict JI considered undesirable (was Re: brass quintets ...)

In TD450.25 "Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote:
>Daniel Wolf wrote,
>
>>Contrary to your viewpoint,
>>it is when instruments with harmonic spectra use an exact harmonic (as
>>opposed to subharmonic) JI that the identities of the individual
>instruments
>>are masked or blended into the composite sound.
>
>Very good point! Johnny Reinhard has made the same point. That is one reason
>why when using "adaptive JI", I think one should stop a couple of cents
>short of getting the simulaneities _exactly_ in JI.

I totally agree. This is of course unlikely to be a problem except with electronic instruments, and even then... But I thought:

If you are going to introduce deliberate mistunings into adaptive JI they might as well be useful ones, such as distributing the septimal kleisma (224:225) to eliminate at least one small source of drift. But now, when distributing this kleisma, instead of wanting to minimise the maximum absolute error as is usual, we want to MAXimise the MINimum absolute error.

I found the following simultaneous distribution of the 224:225 and 384:385 by trial and error. It is at least at a _local_ maximum in the minimum absolute error function over the (11-limit) intervals listed. Its minimum error is alas only 0.24 cents (in the 7:8, 4:7, 3:5 and 3:10). It does not appear to correspond to any rational fractions of the septimal kleisma.

Intvl Error
9:10 1.14
8:9 -3.25
7:8 0.24
6:7 1.63
5:6 0.72
9:11 -0.28
4:5 -2.11
7:9 -3.01
3:4 1.87
8:11 -3.54
5:7 2.35
7:10 -1.87
2:3 -1.39
7:11 -3.30
5:8 2.59
3:5 -0.24
4:7 0.24
5:9 -0.66
6:11 -1.67
1:2 0.48
5:11 -0.95
4:9 -2.77
3:7 2.11
2:5 -1.63
3:8 2.35
4:11 -3.06
1:3 -0.90
3:10 0.24
2:7 0.72
3:11 -1.19
1:4 0.96

Note that no 7-(odd)limit interval has an error greater than 2.59 c (in the 5:8 minor sixth).

It is not very far from the minimum in the max abs error function. Here it is for comparison. Maximum error 2.71 c (in the 8:9, 4:5, 8:11, 5:8, 4:9, 2:5 and 4:11).

Intvl Error
9:10 0.00
8:9 -2.71
7:8 0.43
6:7 0.93
5:6 1.36
9:11 0.00
4:5 -2.71
7:9 -2.28
3:4 1.36
8:11 -2.71
5:7 2.28
7:10 -2.28
2:3 -1.36
7:11 -2.28
5:8 2.71
3:5 -1.36
4:7 -0.43
5:9 0.00
6:11 -1.36
1:2 0.00
5:11 0.00
4:9 -2.71
3:7 0.93
2:5 -2.71
3:8 1.36
4:11 -2.71
1:3 -1.36
3:10 -1.36
2:7 -0.43
3:11 -1.36
1:4 0.00

Regards,

-- Dave Keenan
http://dkeenan.com