Kees integral tuning vs poptimal

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.

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

--- 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.

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

--- 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]

--- 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.

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

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