back to list

Kees integral tuning vs poptimal

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

3/25/2007 8:02:37 PM

Looking at 5-limit examples, it seems that for higher limit
temperaments the Kees integral tuning is either in, or close to (even
in a relative sense) the poptimal range.but low limit temperaments are
another story, and I don't quite see what tendency it is exhibiting.

For 5-limit father (16/15) temperament, poptimal is a "fifth" of 757.82
cents. Kees integral is 743.935 cents. You might surmise it is favoring
3 over 5. For dicot (25/24) temperament, poptimal is exactly half of a
pure fifth for the generator. Kees integral is 348.508, which means the
fifth is a little flat. Why a little flat? I don't know.

🔗Graham Breed <gbreed@gmail.com>

3/26/2007 7:36:01 AM

Gene Ward Smith wrote:
> Looking at 5-limit examples, it seems that for higher limit > temperaments the Kees integral tuning is either in, or close to (even > in a relative sense) the poptimal range.but low limit temperaments are > another story, and I don't quite see what tendency it is exhibiting.

What's poptimal?

> For 5-limit father (16/15) temperament, poptimal is a "fifth" of 757.82 > cents. Kees integral is 743.935 cents. You might surmise it is favoring > 3 over 5. For dicot (25/24) temperament, poptimal is exactly half of a > pure fifth for the generator. Kees integral is 348.508, which means the > fifth is a little flat. Why a little flat? I don't know.

Your poptimal is exactly (to the precision you give) the unweighted 7-odd limit RMS optimum for father. The Kees integral is the Tenney-weighted prime STD optimum.

For Dicot, it looks like your poptimal is the Kees-max. (I don't have anything wired up to calculate this.) The Kees integral is my STD optimum again. It roughly makes the weighted error of 3:1 double that of 5:1, but not quite.

I can't follow the explanation of the Kees integral. So at this point I can't say if it's identical to my STD or not.

Graham

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

3/26/2007 11:33:47 AM

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> What's poptimal?

http://tonalsoft.com/enc/p/poptimal.aspx

> Your poptimal is exactly (to the precision you give) the unweighted
> 7-odd limit RMS optimum for father. The Kees integral is the
> Tenney-weighted prime STD optimum.

Interesting. I would have thought the Tenney integral, if anything,
would give that.

> For Dicot, it looks like your poptimal is the Kees-max. (I don't
have
> anything wired up to calculate this.) The Kees integral is my STD
> optimum again. It roughly makes the weighted error of 3:1 double
that
> of 5:1, but not quite.

Begans to be a pattern.

> I can't follow the explanation of the Kees integral. So at this
point I
> can't say if it's identical to my STD or not.

Hmmm. You could give a table of tunings for various 5-limit commas,
and I also.

🔗Graham Breed <gbreed@gmail.com>

3/26/2007 4:50:59 PM

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:> > >>What's poptimal?> > > http://tonalsoft.com/enc/p/poptimal.aspx

I'll have a look.

>>Your poptimal is exactly (to the precision you give) the unweighted >>7-odd limit RMS optimum for father. The Kees integral is the >>Tenney-weighted prime STD optimum.
> > Interesting. I would have thought the Tenney integral, if anything, > would give that.

I don't see why a Tenney metric in should give a standard deviation out. Hopefully it will give the TOP-RMS, and be easier to prove.

>>For Dicot, it looks like your poptimal is the Kees-max. (I don't > have >>anything wired up to calculate this.) The Kees integral is my STD >>optimum again. It roughly makes the weighted error of 3:1 double > that >>of 5:1, but not quite.
> > Begans to be a pattern. > > >>I can't follow the explanation of the Kees integral. So at this > point I >>can't say if it's identical to my STD or not.
> > > Hmmm. You could give a table of tunings for various 5-limit commas, > and I also.

Here are the tuning maps for the temperaments in Paul's paper, and some more I added. I can only do tuning maps because this library doesn't use the canonical generators.

5-limit
Temperament STD Error Optimal Octave-Equivalent Tuning Map
Father 13.4022255182 [743.93459218111104, 456.06540781888947]
Bug 11.5723187974 [678.699849851723, 418.04977477758473]
Dicot 7.0565596926 [697.01697353731618, 348.50848676865809]
Meantone 1.5803785988 [696.23543086425241, 384.94172345700923]
Augmented 2.4063634571 [706.62617271307329, 399.99999999999966]
Mavila 6.0227607261 [679.81650695671669, 360.55047912984935]
Porcupine 2.6793093003 [708.12266369080533, 380.20443948467585]
Blackwood 4.6479585762 [719.99999999999989, 399.53143682162704]
Dimipent 3.1072167492 [699.46611416044391, 399.46611416044391]
Srutal 0.8358526179 [704.89828972433304, 390.20342055133381]
Magic 1.1087491460 [700.28912013329887, 380.05782402665994]
Ripple 2.8179057247 [695.77996059472866, 393.24793695156541]
Hanson 0.2734646213 [702.03912658922798, 385.03260549102333]
Negripent 1.6870270975 [696.97883868910765, 377.26587098316935]
Tetracot 1.5803785988 [696.23543086425241, 384.94172345700923]
Superpyth 2.1161729331 [710.07566351907269, 390.68097167165234]
Helmholtz 0.0570287861 [701.73589554305966, 386.11283565552162]
Sensipent 0.3562210560 [701.40221428445079, 387.51713265143655]
Passion 1.5694412716 [706.64844439219667, 394.68124448624258]
Wuerschmidt 0.2619736591 [702.39424350297281, 387.79928043787191]
Compton 0.5035612898 [699.99999999999966, 384.88170161444748]
Amity 0.1404090667 [702.40590781468882, 386.25536031819081]
Orson 0.2152652177 [701.39078167820901, 385.11823642362464]
Vishnu 0.0470748042 [701.99757630576642, 386.5703898453292]
Luna 0.0154758267 [701.98886879844133, 386.40148416020816]

7-limit
Temperament STD Error Optimal Octave-Equivalent Tuning Map
Blacksmith 5.4052144688 [720.00000000000011, 392.69226360296375, 960.00000000000034]
Dimisept 4.9335219930 [699.38555147782949, 399.38555147782927, 999.38555147782927]
Dominant 4.7332149744 [701.56364373989618, 406.2545749595854, 996.87271252020651]
August 4.7434758480 [695.9553117708283, 399.99999999999966, 991.91062354165638]
Pajara 2.5772581650 [707.05134493759601, 385.8973101248086, 985.89731012480809]
Semaphore 2.7461724997 [694.72111269146058, 378.88445076584213, 947.36055634573017]
Meantone 1.3802759322 [696.49360839947747, 385.97443359790981, 964.93608399477398]
Injera 3.1343086130 [694.35834905840329, 377.43339623361186, 977.43339623361192]
Negrisept 2.5677503080 [697.56242389555939, 376.82818207833077, 948.78121194777987]
Augene 2.2345195024 [709.25722072672966, 400.00000000000017, 981.48555854654069]
Keemun 2.5769495555 [698.80647396744962, 382.33872830620771, 949.40323698372447]
Catler 2.6936797409 [700.00000000000023, 399.99999999999966, 973.18750140596558]
Hedgehog 2.7819631251 [706.90522878650518, 378.175381310842, 978.175381310842]
Superpyth 1.9214282682 [710.29007769488271, 392.61069925394116, 979.41984461023571]
Sensisept 1.3237327072 [703.66969931189112, 390.43247054386023, 963.95801300779738]
Lemba 3.1918932422 [696.25838764909031, 367.91387078363675, 967.91387078363675]
Porcupine 2.5357800965 [711.35714666497972, 385.59524444163281, 977.28570667004158]
Flattone 2.1109027972 [693.77931078251413, 375.11724313005675, 955.98620295737294]
Magic 1.0732854080 [701.75724988867364, 380.35144997773483, 964.21739973281569]
Doublewide 2.4827880237 [702.84525656801804, 377.13394242601373, 977.13394242601373]
Nautilus 3.2417503281 [704.9303328514336, 374.88388808572245, 952.46516642571703]
Beatles 2.3074118267 [711.81070223839708, 396.85183992721278, 976.37859552320595]
Liese 2.2158558643 [697.20834143754701, 388.8333657501887, 956.43058527100618]
Cynder 1.4242319592 [696.57800744750114, 386.31202979000466, 967.80733085083273]
Orwell 0.7479384139 [700.55972555582878, 385.47440333321606, 972.06825777808956]
Garibaldi 0.7255915237 [702.08530745431813, 383.31754036545453, 970.80569563954532]
Myna 0.9521583550 [701.45441394987506, 391.30897255488753, 971.01808976491259]
Miracle 0.5145639219 [700.05110535115693, 383.27371042365002, 966.64963154961401]
Ennealimmal 0.0297232820 [701.95904120105195, 386.2718951349114, 968.62570786771846]

11-limit
Temperament STD Error Optimal Octave-Equivalent Tuning Map
Miracle 0.4840123386 [699.79632416413301, 383.57095514184499, 966.73455861195555, 549.49081041033276]
Diaschismic 0.9053666928 [703.71455972552621, 392.57088054894763, 970.28352219579051, 555.42528329368577]
Orwell 1.1503591124 [699.98048300905839, 385.7226501389743, 971.40626629606641, 542.8515665740166]
Shrutar 1.1466370926 [705.3595735105913, 389.28085297881802, 968.75850728706928, 547.32021324470429]
Schismic 1.3095457353 [702.37200034590228, 381.0239972327837, 966.79199515737184, 557.30399377376352]
Microschismic 0.6723309777 [702.15694727756295, 382.74442177949675, 969.80273811411951, 549.6097873839467]
Magic 1.2257858771 [703.48082510802863, 380.69616502160602, 968.35398025926872, 554.43067982715422]
Meantone 1.4382578689 [696.96764135186027, 387.8705654074418, 969.6764135186047, 545.41754433348933]
Vicentino1 1.4691649744 [696.95310080976435, 387.81240323905735, 966.75794554629556, 542.38275202441082]
Vicentino2 1.3943906114 [696.36341068046954, 385.45364272187931, 963.63410680469781, 540.90852670117465]
Mystery 0.5391768620 [703.44827586206907, 388.45871541806451, 967.76906024565073, 553.97595679737515]
Hemiennialimmal 0.0384487355 [701.91053035867446, 386.19912887134478, 968.57719702534132, 550.95526517933706]

13-limit
Temperament STD Error Optimal Octave-Equivalent Tuning Map
Mystery 0.5125990067 [703.44827586206907, 388.35232641999829, 967.66267124758463, 553.86956779930892, 843.52474021310184]
Diaschismic 0.8269169089 [703.70398327759005, 392.59203344481983, 970.36813377927888, 555.55220066891911, 844.44025083614815]
Cassandra1 0.6963808637 [702.11254505987654, 383.09963952098622, 970.42436916172664, 548.58853637716402, 842.25090119753395]
Cassandra2 1.5170906791 [702.5602978617834, 379.51761710573385, 964.15582993503449, 553.91463848790045, 846.23374490255151]

17-limit
Temperament STD Error Optimal Octave-Equivalent Tuning Map
Mystery 0.8517270149 [703.44827586206907, 388.56265379338703, 967.87299862097279, 554.07989517269766, 843.73506758649057, 98.9074813795941
31]
Diaschismic 0.8903249230 [703.81243587506629, 392.37512824986743, 969.50051299947029, 554.25076949920515, 842.81346187400641, 103.812435875065
74]

To help you deduce the higher limit temperaments, here's the code:

print "11-limit"
showAll([(31,41,'miracle'), (46,58,'diaschismic'), (22,31,'orwell'),
(22,46,'shrutar'),
(12,29,'schismic'), (41,53,'microschismic'), (19,22,'magic'),
(12,31,'meantone'), (24,7,'vicentino1'), (31,38,'vicentino2'),
(29,58,'mystery'), (72,126,'hemiennialimmal')], oe.et.limit11)
print
print "13-limit"
showAll([(29,58,'mystery'), (46,58,'diaschismic'),
(41,53,'cassandra1'), (29,41,'cassandra2'),
], oe.et.limit13)
print
print "17-limit"
showAll([(29,58,'mystery'), (46,58,'diaschismic')], oe.et.limit17)

The two numbers refer to the best (TOP-RMS) mappings for equal temperaments with so many notes to the octave. Except for 7, which is the primes rounded to the nearest integer.

Graham

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

3/26/2007 7:14:13 PM

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

Here is the comparison. Everything is the same except for 3-limit
commas (where I need to modify my routine, I think) and 128/125,
where my definition as stated doesn't apply. I think there can be no
doubt that for the 5-limit we've come across the same thing, bit I
don't think either of us has a clue yet why.

Now I need to think about the 7-limit. While the Tenney integral
would be pretty straightforward, the Kees integral involves an
aggrivating polytope to integrate over, so I may try to check things
numerically.

In some case, where I don't think Paul's names are so well accepted,
I've given alternatives.

> 5-limit
> Temperament STD Error Optimal Octave-Equivalent Tuning Map
> Father 13.4022255182 [743.93459218111104,
456.06540781888947]

16/15
[743.9345921811103768, 456.0654078188896232]

> Bug 11.5723187974 [678.699849851723,
418.04977477758473]

27/25
[678.6998498517232225, 418.0497747775848338]

> Dicot 7.0565596926 [697.01697353731618,
348.50848676865809]

25/24
[697.0169735373157511, 348.5084867686578756]

> Meantone 1.5803785988 [696.23543086425241,
384.94172345700923]

81/80
[696.2354308642524178, 384.9417234570096707]

> Augmented 2.4063634571 [706.62617271307329,
399.99999999999966]
128/125

I would need to modify my definition to cover augmented, and haven't
given much thought at all to how that would work, as the exceptional
cases aren't going to arise much.

> Mavila 6.0227607261 [679.81650695671669,
360.55047912984935]

135/128
[679.8165069567166740, 360.5504791298499779]

> Porcupine 2.6793093003 [708.12266369080533,
380.20443948467585]

250/243
[708.1226636908056110, 380.2044394846760185]

> Blackwood 4.6479585762 [719.99999999999989,
399.53143682162704]
256/243

I think my definition covers Blackwood, but I need to modify my
program.

> Dimipent 3.1072167492 [699.46611416044391,
399.46611416044391]

648/625 = diminished
[699.4661141604437821, 399.4661141604437820]

> Srutal 0.8358526179 [704.89828972433304,
390.20342055133381]

2048/2025 = diaschismic
[704.8982897243330719, 390.2034205513338562]

> Magic 1.1087491460 [700.28912013329887,
380.05782402665994]

3125/3072
[700.2891201332985480, 380.0578240266597097]

> Ripple 2.8179057247 [695.77996059472866,
393.24793695156541]

6561/6250
[695.7799605947286085, 393.2479369515657736]

> Hanson 0.2734646213 [702.03912658922798,
385.03260549102333]

15625/15552
[702.0391265892278257, 385.0326054910231883]

> Negripent 1.6870270975 [696.97883868910765,
377.26587098316935]

16875/16384 = negri
[696.9788386891074433, 377.2658709831694176]

> Tetracot 1.5803785988 [696.23543086425241,
384.94172345700923]

20000/19683
[704.6370868987211190, 385.4334455221225177]

> Superpyth 2.1161729331 [710.07566351907269,
390.68097167165234]

20480/19683
[710.0756635190725176, 390.6809716716526570]

> Helmholtz 0.0570287861 [701.73589554305966,
386.11283565552162]

32805/32768 = schismatic or schismic
[701.7358955430597352, 386.1128356555221188]

> Sensipent 0.3562210560 [701.40221428445079,
387.51713265143655]

78732/78125 = sensi
[701.4022142844504520, 387.5171326514362951]

> Passion 1.5694412716 [706.64844439219667,
394.68124448624258]

262144/253125
[706.6484443921963135, 394.6812444862429493]

> Wuerschmidt 0.2619736591 [702.39424350297281,
387.79928043787191]

393216/390625
[702.3942435029728470, 387.7992804378716060]

> Compton 0.5035612898 [699.99999999999966,
384.88170161444748]

531441/524288

Once again, a 3-limit comma so I need to modify, not my definition,
but my program.

> Amity 0.1404090667 [702.40590781468882,
386.25536031819081]

1600000/1594323
[702.4059078146887962, 386.2553603181908700]

> Orson 0.2152652177 [701.39078167820901,
385.11823642362464]

2109375/2097152 = orwell
[701.3907816782089340, 385.1182364236247428]

> Vishnu 0.0470748042 [701.99757630576642,
386.5703898453292]

6115295232/6103515625
[701.9975763057660616, 386.5703898453283122]

> Luna 0.0154758267 [701.98886879844133,
386.40148416020816]

274877906944/274658203125 = hemithirds
[701.9888687984408810, 386.4014841602078828]

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

3/27/2007 12:22:23 AM

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Now I need to think about the 7-limit. While the Tenney integral
> would be pretty straightforward, the Kees integral involves an
> aggrivating polytope to integrate over, so I may try to check things
> numerically.

Fiddling around numerically, it does seem plausible that 7 limit Kees
integral is once again Graham's optimal OE tuning. But what's the
relationship between that tuning and the Kees metric? It must exist.

🔗Graham Breed <gbreed@gmail.com>

3/27/2007 1:08:36 AM

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" > <genewardsmith@...> wrote:
> > >>Now I need to think about the 7-limit. While the Tenney integral >>would be pretty straightforward, the Kees integral involves an >>aggrivating polytope to integrate over, so I may try to check things >>numerically.
> > > Fiddling around numerically, it does seem plausible that 7 limit Kees > integral is once again Graham's optimal OE tuning. But what's the > relationship between that tuning and the Kees metric? It must exist.

There's an analogy with the TOP-max and Kees-max errors. TOP-max for optimal scale stretch can be written as

[max(W)-min(W)]/[max(W)+min(W)]

where W are the weighted primes. The numerator is a kind of spread and the denominator is double a kind of average. For a good tuning, all W are approximately 1, so the TOP-max is approximately

[max(W) - min(W)]/2

which is also double the Kees-max error.

The TOP-RMS error for optimal scale stretch is

std(W)/rms(W)

where std() is the standard deviation and rms() is the root mean squared. Obviously, once again, the numerator is a spread and the denominator is an average. As the average of W is about 1, the numerator once again approximates the whole thing

std(W)

The analogy suggests that this should have something to do with the Kees weighted RMS.

Graham

🔗Graham Breed <gbreed@gmail.com>

3/27/2007 1:12:01 AM

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > Here is the comparison. Everything is the same except for 3-limit > commas (where I need to modify my routine, I think) and 128/125, > where my definition as stated doesn't apply. I think there can be no > doubt that for the 5-limit we've come across the same thing, bit I > don't think either of us has a clue yet why.
> > Now I need to think about the 7-limit. While the Tenney integral > would be pretty straightforward, the Kees integral involves an > aggrivating polytope to integrate over, so I may try to check things > numerically.

Is this something Maple could solve symbolically? If it's what it looks like, the resulting equation's very simple. Express it in terms of sums over primes.

Graham