Secor29htt in tolerant temperament

What the heck is "tolerant temperament", you ask? I propose it for a name for the 13-limit rank 3 temperament tempering out 325/324, 352/351 and 364/363. There's not much tuning difference between that and tempering out 625/624 instead of 325/324, but there is a touch more complexity. You can do them both and get metakleismic temperament, but analyzing George's scale in terms of metakelismic doesn't seem to make much sense. Below I give a tolerant temperament version of secor29htt, which seems to me to work pretty well.

! secor29tolerant.scl
!
Version of George Secor's secor29htt in tolerant temperament, POTE tuning
29
!
57.71785
98.86459
137.91534
179.06208
207.92100
265.63886
294.49778
345.83634
386.98308
415.84201
473.55986
496.03950
553.75735
594.90409
633.95484
681.48086
703.96050
761.67835
802.82509
841.87584
883.02258
911.88151
969.59936
992.07900
1049.79685
1090.94359
1119.80251
1177.52036
1200.00000
!
!! secor29trans.scl
!!
!5-limit transversal for secor29tolerant
! 29
!!
! 6561/6400
! 135/128
! 27/25
! 10/9
! 9/8
! 59049/51200
! 4782969/4096000
! 243/200
! 5/4
! 81/64
! 531441/409600
! 4/3
! 2187/1600
! 45/32
! 36/25
! 4782969/3276800
! 3/2
! 19683/12800
! 405/256
! 81/50
! 5/3
! 27/16
! 177147/102400
! 16/9
! 729/400
! 15/8
! 243/128
! 1594323/819200
! 2/1

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> What the heck is "tolerant temperament", you ask? I propose it for a name for the 13-limit rank 3 temperament tempering out 325/324, 352/351 and 364/363.

Hi Gene,

I've been severely strapped for time lately, so I have not been able read the entire discussion between you, Margo, and Mike. (Also, I had a very busy weekend, during which time I have not been on the internet, so I'll need to play catch-up over the next couple of days.) Anyway, I saw the above on Friday and spent some time checking out the details.

I agree with your proposal for the name "tolerant temperament", as defined above. Thank you for suggesting that 325/324 be tempered out; since I found that the difference between its optimal tuning and my rank 4 high-tolerance temperament is, for all practical purposes, inaudible. I played around with the numbers you gave for the POTE tuning of this temperament and found by trial and error that the tuning can be significantly improved. Details below.

> There's not much tuning difference between that and tempering out 625/624 instead of 325/324, but there is a touch more complexity. You can do them both and get metakleismic temperament, but analyzing George's scale in terms of metakelismic doesn't seem to make much sense. Below I give a tolerant temperament version of secor29htt, which seems to me to work pretty well.
>
> ! secor29tolerant.scl
> !
> Version of George Secor's secor29htt in tolerant temperament, POTE tuning
> 29
> !
> 57.71785
> 98.86459
> 137.91534
> 179.06208
> 207.92100
> 265.63886
> 294.49778
> 345.83634
> 386.98308
> 415.84201
> 473.55986
> 496.03950
> 553.75735
> 594.90409
> 633.95484
> 681.48086
> 703.96050
> 761.67835
> 802.82509
> 841.87584
> 883.02258
> 911.88151
> 969.59936
> 992.07900
> 1049.79685
> 1090.94359
> 1119.80251
> 1177.52036
> 1200.00000
> !
> !! secor29trans.scl
> !!
> !5-limit transversal for secor29tolerant
> ! 29
> !!
> ! 6561/6400
> ! 135/128
> ! 27/25
> ! 10/9
> ! 9/8
> ! 59049/51200
> ! 4782969/4096000
> ! 243/200
> ! 5/4
> ! 81/64
> ! 531441/409600
> ! 4/3
> ! 2187/1600
> ! 45/32
> ! 36/25
> ! 4782969/3276800
> ! 3/2
> ! 19683/12800
> ! 405/256
> ! 81/50
> ! 5/3
> ! 27/16
> ! 177147/102400
> ! 16/9
> ! 729/400
> ! 15/8
> ! 243/128
> ! 1594323/819200
> ! 2/1

Errors (in cents) for POTE tuning, my improvement on it, and my original (rank 4) high-tolerance-temperament of 1975 are:

Odd POTE Improved Rank 4 HTT
3 2.00550 1.81348 1.6237
5 0.66937 0.00000 0.0000
7 0.77343 0.00000 0.0000
9 4.01100 3.62697 3.2474
11 2.43940 2.43402 3.1932
13 1.34818 1.91885 3.2474
15 2.67487 1.81348 1.6237

Generators for POTE are 2/1, 703.96050 c, and 386.98308 c. Generators for my improved tuning are 2/1, 703.76849 c, and 5/4. Generators for the original (rank 4) HTT of 1975 are 2/1, 703.5785 c, 5/4, and 7/4. (Note that tolerant temperament is supported by 208-EDO, which, BTW, also has less overall error than the POTE tuning. Perhaps you should check the POTE algorithm to determine how it could be improved.)

Following are listings for the improved tuning of tolerant temperament subsets mapped to a 17-, 29-, and 41- tone octave, respectively:

! Tolerant-Secor-17.scl
!
Tolerant temperament (rank 3: 324:325, 351:352, 363:364), Secor 17 "triple delight" mapping
17
!
28.76023
138.67803
207.53697
265.05742
346.21500
5/4
496.23152
553.75197
611.27242
703.76849
732.52871
842.44651
882.54523
7/4
1049.98348
1090.0822
2/1

! Tolerant-Secor-29.scl
!
Tolerant temperament (rank 3: 324:325, 351:352, 363:364), Secor 29 mapping
29
!
57.52045
97.61917
138.67803
178.77674
207.53697
265.05742
295.87007
346.21500
5/4
415.07394
472.59439
496.23152
553.75197
593.85068
634.90954
680.13136
703.76849
761.28894
801.38765
842.44651
882.54523
911.30546
7/4
992.46303
1049.98348
1090.08220
1118.84243
1176.36288
2/1

! Tolerant-Secor-41.scl
!
Tolerant temperament (rank 3: 324:325, 351:352, 363:364), Secor 41 mapping
41
!Secor
28.76023
57.52045
97.61917
115.0409
138.67803
178.77674
207.53697
236.29720
265.05742
295.87007
322.57787
346.21500
5/4
415.07394
443.83417
472.59439
496.23152
524.99174
553.75197
593.85068
611.27242
634.90954
680.13136
703.76849
732.52871
761.28894
801.38765
818.80939
842.44651
882.54523
911.30546
940.06568
7/4
992.46303
1026.34636
1049.98348
1090.08220
1118.84243
1138.31659
1176.36288
2/1

If you want to try these out in Scala, I suggest that you enter "set notation sahtt". I think that, when the error gets this low, it's very difficult to tell the difference between this and JI.

--George

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

> Generators for POTE are 2/1, 703.96050 c, and 386.98308 c. Generators for my improved tuning are 2/1, 703.76849 c, and 5/4. Generators for the original (rank 4) HTT of 1975 are 2/1, 703.5785 c, 5/4, and 7/4.

Apparently my reply to this is never going to appear. I pointed out that this tuning is the 7, 9, 11, 13 and 15 limit minimax tuning, which is pretty impressive, but also that not everyone agrees minimax uniquely defines the meaning of optimal tuning. It depends on what you optimize.

>(Note that tolerant temperament is supported by 208-EDO, which, BTW, also has less overall error than the POTE tuning. Perhaps you should check the POTE algorithm to determine how it could be improved.)

208 also gives what I call the "optimal patent val" for tolerant, which however is defined in terms of closeness to POTE.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > Generators for POTE are 2/1, 703.96050 c, and 386.98308 c. Generators for my improved tuning are 2/1, 703.76849 c, and 5/4. Generators for the original (rank 4) HTT of 1975 are 2/1, 703.5785 c, 5/4, and 7/4.
>
> Apparently my reply to this is never going to appear. I pointed out that this tuning is the 7, 9, 11, 13 and 15 limit minimax tuning,

Okay, that's good to know. I simply fiddled around with the ~3/2 and ~5/4 generators on a spreadsheet until I couldn't improve my table of errors any further. (I was rather surprised that I could make both 5 and 7 exact without severely damaging everything else.)

> which is pretty impressive, but also that not everyone agrees minimax uniquely defines the meaning of optimal tuning. It depends on what you optimize.

Yes, of course, but this brings up a question. Here's my table of errors comparing POTE with my improved (15-limit minimax) tolerant tuning (use fixed-width font option to view properly):

Odd POTE Improved
3 2.00550 1.81348
5 0.66937 0.00000
7 0.77343 0.00000
9 4.01100 3.62697
11 2.43940 2.43402
13 1.34818 1.91885
15 2.67487 1.81348

My improved (minimax) tolerant tuning has less error than the POTE tuning for everything but 13 -- clearly no contest, particularly if you want to give added weight to the lower primes. What is it that POTE is supposed to be optimizing?

> >(Note that tolerant temperament is supported by 208-EDO, which, BTW, also has less overall error than the POTE tuning. Perhaps you should check the POTE algorithm to determine how it could be improved.)
>
> 208 also gives what I call the "optimal patent val" for tolerant, which however is defined in terms of closeness to POTE.

Okay, I found it listed here:
http://xenharmonic.wikispaces.com/Optimal+patent+val#x13-limit rank three

I found the POTE algorithm here:
http://xenharmonic.wikispaces.com/POTE+tuning
Is there something that's not being taken into account when "optimizing" higher-rank temperaments? Or is POTE only supposed to give you ballpark values for the generators (sufficient for finding the optimal patent val)?

--George

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

> Yes, of course, but this brings up a question. Here's my table of errors comparing POTE with my improved (15-limit minimax) tolerant tuning (use fixed-width font option to view properly):

You can't really do it like that. Comparing the 24 15-limit intervals
1 < x < sqrt(2), 13 were better with minimax and 11 with POTE: 4/3, 5/4, 7/5, 8/7, 9/8, 11/8, 11/9, 13/11, 13/12, 15/13, 15/14, 16/15, 18/13 versus 6/5, 7/6, 9/7, 10/9, 11/10, 12/11, 13/10, 14/11, 14/13, 15/11, 16/13.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > Yes, of course, but this brings up a question. Here's my table of errors comparing POTE with my improved (15-limit minimax) tolerant tuning (use fixed-width font option to view properly):
>
> You can't really do it like that. Comparing the 24 15-limit intervals
> 1 < x < sqrt(2), 13 were better with minimax and 11 with POTE: 4/3, 5/4, 7/5, 8/7, 9/8, 11/8, 11/9, 13/11, 13/12, 15/13, 15/14, 16/15, 18/13 versus 6/5, 7/6, 9/7, 10/9, 11/10, 12/11, 13/10, 14/11, 14/13, 15/11, 16/13.
>

The 15-limit least squares tuning also splits 13 vs 11 with minimax, with 4/3, 5/4, 7/5, 8/7, 9/8, 11/8, 11/9, 12/11, 13/11, 13/12, 15/13, 16/15, 18/13 vs 6/5, 7/6, 9/7, 10/9, 11/10, 13/10, 14/11, 14/13, 15/11, 15/14, 16/13. As you can surmise from that, it's closer to POTE.

Least squares map = <1200.0, 1903.9289098702692411, 2786.7949793448626946, 3369.6280498832362624, 4153.9124104021592981, 4442.1256807913515750|