Yahoo Tuning Groups Ultimate Backup tuning 17\29

I have no doubt that this has been discussed before -- and in fact, I just
found the post by Margo Schulter about "Gentle Temperaments" -- but the
12-note MOS scale that's generated by using 17\29 as a generator is
outstanding for a lot of 13-limit ratios. Here's the scale:

! C:\Program Files (x86)\Scala22\canton-esque.scl
!
17\29 (3/2) generator, accurate 2.3.7/5.11/5.13/5 canton-like scale
12
!
124.13796
206.89656
289.65516
413.79312
496.55172
620.68968
703.44828
786.20688
910.34484
993.10344
1117.24140
2/1

29 EDO is lousy for primes 5, 7, 11, and 13, but in approximately the same
amounts (11 and 5, for instance, have errors of -13.4 cents and -13.9
cents, respectively), so ratios of these primes are often really good. (I
think that means it's good for the 2.3.7/5.11/5.13/5 subgroup, but I'm not
quite bold enough to say that definitively.) The just approximations (all
with less than 5 cent error) for the above scale are:

15/14 or 14/13 (approximately the same error for each)
9/8
13/11
14/11
4/3
10/7
3/2
11/7
22/13
40/21
2/1

If you look at the modes, other intervals include close approximations for
21/20, 11/10, 26/21, 15/11, 7/5, 21/13, 20/11, and 13/7.

Only three intervals didn't have a close just (and not-too-complex)
approximation that I could quickly see: 331, 662, and 868 cents.

The scale reminds me of Canton, which Gene Smith identified a while back,
but it has only one wolf fifth.

Regards,
Jake

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@> wrote:
> >
> > I have no doubt that this has been discussed before -- and in fact, I just
> > found the post by Margo Schulter about "Gentle Temperaments" -- but the
> > 12-note MOS scale that's generated by using 17\29 as a generator is
> > outstanding for a lot of 13-limit ratios. Here's the scale:
>
> Is this jake temperament?
>
> Subgroup: 2.3.7/5.11/5.13/5
> Commas: 196/195, 352/351, 364/363
>
> Gencom: [2 3/2; 196/195 352/351 364/363]
> Gencom map: [<1 1 -5 -1 2 4|, <0 1 29/4 5/4 -11/4 -23/4|]
>
> Map: [<1 0 10 17 22|, <0 1 -6 -10 -13|]
> EDOs: 12, 17, 29, 70e, 99e, 128e, where "e" is the 13/5

This is the same as "pepperoni" with a single dimension added to the subgroup.

Keenan

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

> This is the same as "pepperoni" with a single dimension added to the
subgroup.

Keenan's right. Pepperoni is defined as being 2.3.11/7.13/7. Pepperoni[12]
in 271 EDO is almost the same scale as the 17\29 example I've given, with
the intervals varying by up to about 5 cents. I told you I was sure it had
been discussed before. :)

Without 5 in the subgroup, I'm not sure what Margo would say the 576-ish
cent interval would be; It's very close to a 7/5 (583 cents), and I don't
see anything else simple in that area. It almost feels like Pepperoni has
an accidental 5/7 in it, even though it's not documented.

That's the only interval in the scale you need 5 for, though. After looking
again, 1117 cents could just as easily be 21/11 as 40/21. (21/11 / 40/21 =
441/440.) Now every interval is in the 2.3.11/7.13/7 subgroup except the
tritone.

---

Gene, I'm not sure how you figure out which commas determine the
temperament.

196/195 seems right because it means 15/14=14/13. For a similar reason,
would you want to include 441/440, as noted above?

364/363 is good because it ensures that 13/11 * 14/11 = 3/2, which is an
important feature.

352/351 doesn't make sense to me yet: it means 11/9 * 16/13 = 3/2, but
there are no good approximations to 11/9 in the scale, so why would this
comma be important?

My spreadsheet tells me that a bunch of additional commas are tempered out,
and I'm not sure which really matter. I presume that, based on the
subgroup, we'd only want commas where the exponents for 5, 7, 11, and 13
add up to zero, which would leave out, e.g., "Beta 2", but I'll leave them
here for completeness.

| 3 -2 0 -1 3 -2 >
0.16
10648/10647
harmonisma

| -15 8 1 >
1.95
32805/32768
schisma

| 3 0 2 0 1 -3 >
2.36
2200/2197
Parizek comma, petrma

| 2 -3 -2 0 0 2 >
2.56
676/675

| 5 -1 3 0 -3 >
3.03
4000/3993
undecimal schisma, Wizardharry

| 25 -14 0 -1 >
3.80
33554432/33480783
Beta 2, septimal schisma

| -3 2 -1 2 -1 >
3.93
441/440
Werckmeister's undecimal septenarian schisma

| 2 -1 0 1 -2 1 >
4.76
364/363

| 10 -6 1 -1 >
5.76
5120/5103
Beta 5, Garibaldi comma

| -5 2 2 -1 >
7.71
225/224
septimal kleisma

| 2 -1 -1 2 0 -1 >
8.86
196/195

| 7 -4 0 1 -1 >
9.69
896/891
Pentacircle, undecimal semicomma

| 5 -4 3 -2 >
13.47
4000/3969
septimal semicomma

| -3 -1 -1 0 2 >
14.37
121/120
undecimal seconds comma

| -4 1 -5 5 >
14.52
50421/50000

| -3 1 1 1 0 -1 >
16.57
105/104
small tridecimal comma, animist comma

| 2 -2 2 0 -1 >
17.40
100/99
Ptolemy's comma

| -1 -2 -1 1 0 1 >
19.13
91/90

| 0 -2 5 -3 >
21.18
3125/3087
major BP diesis

| -1 0 1 2 -2 >
21.33
245/242

| -9 3 -3 4 >
22.23
64827/64000

| -6 0 1 0 0 1 >
26.84
65/64
13th-partial chroma

| 1 -3 -2 3 >
27.99
686/675
senga

| -4 -1 0 2 >
35.70
49/48
slendro diesis, septimal 1/6-tone

| -9 1 2 1 >
43.41
525/512
Avicenna enharmonic diesis

| 1 -5 3 >
49.17
250/243
maximal diesis

| -14 3 4 >
51.12
16875/16384
double augmentation diesis

Regards,
Jake

🔗Jake Freivald <jdfreivald@...>

352/351 makes sense to me now. Three fifths plus a minor third = two
octaves; equivalently, since a major third plus a minor third = a
fifth (because 364/363 is tempered out), four fifths = a major third
(octave equivalent). I was thrown by the fact that the major third has
a 7 in it, but 352/351 doesn't.

Speaking of which, if I actually used 896/891 instead of 352/351, I
would directly get four fifths = a major third. Clearly both commas
are tempered out, but I don't know why one would be in the definition
and the other would not. I'm guessing there are equivalencies or
factors I'm not seeing yet. I'll play with that and see.

I don't know how you guys keep this stuff straight. :)

Regards,
Jake

On 6/20/12, Jake Freivald <jdfreivald@...> wrote:
>> This is the same as "pepperoni" with a single dimension added to the
> subgroup.
>
> Keenan's right. Pepperoni is defined as being 2.3.11/7.13/7. Pepperoni[12]
> in 271 EDO is almost the same scale as the 17\29 example I've given, with
> the intervals varying by up to about 5 cents. I told you I was sure it had
> been discussed before. :)
>
> Without 5 in the subgroup, I'm not sure what Margo would say the 576-ish
> cent interval would be; It's very close to a 7/5 (583 cents), and I don't
> see anything else simple in that area. It almost feels like Pepperoni has
> an accidental 5/7 in it, even though it's not documented.
>
> That's the only interval in the scale you need 5 for, though. After looking
> again, 1117 cents could just as easily be 21/11 as 40/21. (21/11 / 40/21 =
> 441/440.) Now every interval is in the 2.3.11/7.13/7 subgroup except the
> tritone.
>
> ---
>
> Gene, I'm not sure how you figure out which commas determine the
> temperament.
>
> 196/195 seems right because it means 15/14=14/13. For a similar reason,
> would you want to include 441/440, as noted above?
>
> 364/363 is good because it ensures that 13/11 * 14/11 = 3/2, which is an
> important feature.
>
> 352/351 doesn't make sense to me yet: it means 11/9 * 16/13 = 3/2, but
> there are no good approximations to 11/9 in the scale, so why would this
> comma be important?
>
> My spreadsheet tells me that a bunch of additional commas are tempered out,
> and I'm not sure which really matter. I presume that, based on the
> subgroup, we'd only want commas where the exponents for 5, 7, 11, and 13
> add up to zero, which would leave out, e.g., "Beta 2", but I'll leave them
> here for completeness.
>
> | 3 -2 0 -1 3 -2 >
> 0.16
> 10648/10647
> harmonisma
>
> | -15 8 1 >
> 1.95
> 32805/32768
> schisma
>
> | 3 0 2 0 1 -3 >
> 2.36
> 2200/2197
> Parizek comma, petrma
>
> | 2 -3 -2 0 0 2 >
> 2.56
> 676/675
>
> | 5 -1 3 0 -3 >
> 3.03
> 4000/3993
> undecimal schisma, Wizardharry
>
> | 25 -14 0 -1 >
> 3.80
> 33554432/33480783
> Beta 2, septimal schisma
>
> | -3 2 -1 2 -1 >
> 3.93
> 441/440
> Werckmeister's undecimal septenarian schisma
>
> | 2 -1 0 1 -2 1 >
> 4.76
> 364/363
>
> | 10 -6 1 -1 >
> 5.76
> 5120/5103
> Beta 5, Garibaldi comma
>
> | -5 2 2 -1 >
> 7.71
> 225/224
> septimal kleisma
>
> | 2 -1 -1 2 0 -1 >
> 8.86
> 196/195
>
> | 7 -4 0 1 -1 >
> 9.69
> 896/891
> Pentacircle, undecimal semicomma
>
> | 5 -4 3 -2 >
> 13.47
> 4000/3969
> septimal semicomma
>
> | -3 -1 -1 0 2 >
> 14.37
> 121/120
> undecimal seconds comma
>
> | -4 1 -5 5 >
> 14.52
> 50421/50000
>
> | -3 1 1 1 0 -1 >
> 16.57
> 105/104
> small tridecimal comma, animist comma
>
> | 2 -2 2 0 -1 >
> 17.40
> 100/99
> Ptolemy's comma
>
> | -1 -2 -1 1 0 1 >
> 19.13
> 91/90
>
> | 0 -2 5 -3 >
> 21.18
> 3125/3087
> major BP diesis
>
> | -1 0 1 2 -2 >
> 21.33
> 245/242
>
>
> | -9 3 -3 4 >
> 22.23
> 64827/64000
>
>
> | -6 0 1 0 0 1 >
> 26.84
> 65/64
> 13th-partial chroma
>
> | 1 -3 -2 3 >
> 27.99
> 686/675
> senga
>
> | -4 -1 0 2 >
> 35.70
> 49/48
> slendro diesis, septimal 1/6-tone
>
> | -9 1 2 1 >
> 43.41
> 525/512
> Avicenna enharmonic diesis
>
> | 1 -5 3 >
> 49.17
> 250/243
> maximal diesis
>
> | -14 3 4 >
> 51.12
> 16875/16384
> double augmentation diesis
>
> Regards,
> Jake
>