Tuning-Math Archive Section 6: 5675

Message: 5675 - Contents - Hide Contents Date: Sat, 30 Nov 2002 00:16:21 Subject: Re: Even more ridiculous 5-comma list From: wallyesterpaulrus --- In tuning-math@y..., "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> > Thanks. Now just one more question: How do you calculate the second > generator (e.g. 162.9960265 in "porcupine")? I get slightly different > values, ones that still work, but I am wondering why they are different > from your's. Makes me wonder if I am simplifying something incorrectly. For > example, in "kleismic" I just use ln(6/5)/ln(2) * 1200, which is 315.641287 > instead of 317.079753. > > Paul

looks like you're just using the pure minor third for kleismic. maybe you're using minimax optimization instead of rms?

Message: 5677 - Contents - Hide Contents Date: Sun, 1 Dec 2002 22:25:29 Subject: Some 17-limit basis stuff From: Gene W Smith I was looking at what turns up when we take six successive 17-limit superparticular commas, and it turns out that 72 is an outlier, in the sense that it gets created ten times in this way; its nearest competition is 764, which appears four times. We get 72 from any successive six commas in this list [385/384, 441/440, 442/441, 540/539, 561/560, 595/594, 625/624, 676/675] or this list [625/624, 676/675, 715/714, 729/728, 833/832, 936/935, 1001/1000, 1089/1088] or this list: [1089/1088, 1156/1155, 1225/1224, 1275/1274, 1701/1700, 1716/1715, 2058/2057] The TM basis is this: [169/168, 221/220, 225/224, 243/242, 273/272, 325/324] We also have an interesting situation with 46, where two versions of the 46 et appear. We have h46+v17: [46, 73, 107, 129, 159, 170, 189] This is defined by six contiguous commas in the list [325/324, 351/350, 352/351, 364/363, 375/374, 385/384, 441/440] and has basis [52/51, 91/90, 121/119, 126/125, 169/168, 176/175] We also have h46: [46, 73, 107, 129, 159, 170, 188] This is defined by six contiguous commas of the list [256/255, 273/272, 289/288, 325/324, 351/350, 352/351, 364/363] or by the six commas [2601/2600, 3025/3024, 4096/4095, 4225/4224, 4375/4374, 4914/4913] It has basis [91/90, 121/120, 126/125, 136/135, 154/153, 169/168]

Message: 5678 - Contents - Hide Contents Date: Wed, 04 Dec 2002 22:05:03 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> Not that any list is really ultimate, but with rms error < 40, >geometric complexity < 500,

i'll stick with <100; apologies are still due pierre :)

> and badness < 3500, it covers a lot of >ground. > > 27/25 3.739252 35.60924 1861.731473

if i recall correctly, this gave some kind of 8-equal monster . . . bug

> 135/128 4.132031 18.077734 1275.36536 pelogic > 256/243 5.493061 12.759741 2114.877638 blackwood > 25/24 3.025593 28.851897 799.108711 dicot > 648/625 6.437752 11.06006 2950.938432 diminished > 16875/16384 8.17255 5.942563 3243.743713 negri > 250/243 5.948286 7.975801 1678.609846 porcupine > 128/125 4.828314 9.677666 1089.323984 augmented > 3125/3072 7.741412 4.569472 2119.95499 magic > 20000/19683 9.785568 2.504205 2346.540676 tetracot > 531441/524288 13.183347 1.382394 3167.444999 aristoxenean > 81/80 4.132031 4.217731 297.556531 meantone > 2048/2025 6.271199 2.612822 644.408867 diaschismic > 67108864/66430125 15.510107 .905187 3377.402314

carl seems to have missed this before. misty.

> 78732/78125 12.192182 1.157498 2097.802867 semisixths > 393216/390625 12.543123 1.07195 2115.395301 wuerschmidt > 2109375/2097152 12.772341 .80041 1667.723301 orwell > 4294967296/4271484375 18.573955 .483108 3095.692488

this one also seems to have escaped carl's notice. escapade.

> 15625/15552 9.338935 1.029625 838.631548 kleismic > 1600000/1594323 13.7942 .383104 1005.555381 AMT > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 parakleismic > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 vulture > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 semisuper > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 enneadecal > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 vavoom > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 tricot > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 schismic > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 turkey > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 semithirds > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 minortone > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 ennealimmal > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 astro > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 crazy > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 whoosh > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 egads > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 monzismic > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 fortune > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 gross > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 senior > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 pirate

i'll produce the next set of dualzoomers in accordance with this list.

Message: 5679 - Contents - Hide Contents Date: Wed, 04 Dec 2002 23:18:08 Subject: Re: Ultimate 5-limit comma list From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

>> 27/25 3.739252 35.60924 1861.731473

> if i recall correctly, this gave some kind of 8-equal monster . . . > bug

I called it limmal. Why bug?

>> 1600000/1594323 13.7942 .383104 1005.555381 > AMT Amity?

>> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 > vulture

>> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 > turkey

I suppose this makes 53-et the turkey vulture, though putting these two together actually leads to 3-torsion, so maybe not. Turkey is interesting in that it says 5 ~ 2^(-69) 3^45 = 2^(-24) (3/2)^45. This means that like meantone and schismic, it has a generator of a fifth.

>> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 > crazy Kwasi.

>> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 > pirate

When *I* tried to name it everyone booed. :( This is one of the two 4296-et power commas; the other is viking: 2^161 * 3^(-15) * 5^49 Also of note is raider = pirate * viking = 2^71 * 3^(-99) * 5^33 The TM basis for 4296 is <pirate, raider>.

Message: 5680 - Contents - Hide Contents Date: Wed, 04 Dec 2002 23:33:14 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote: >

>>> 27/25 3.739252 35.60924 1861.731473

>> if i recall correctly, this gave some kind of 8-equal

monster . . .

>> bug >

> I called it limmal.

too many definitions of limma -- see The Proxomitron Reveals... * [with cont.] (Wayb.) arts.org/dict/limma.htm

> Why bug?

a simple, undistinguished animal :)

>>> 1600000/1594323 13.7942 .383104 1005.555381 >> AMT > > Amity? OK.

>>> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 >> vulture >

>>> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 >> turkey >

> I suppose this makes 53-et the turkey vulture, though putting these >two together actually leads to 3-torsion, so maybe not.

a 159-tone periodicity block? for the birds! :)

> Turkey is interesting in that it says 5 ~ 2^(-69) 3^45 = > 2^(-24) (3/2)^45. This means that like meantone and schismic, it

has a generator of a fifth.

>>> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 >> crazy > > Kwasi.

oh yeah, kwazy.

>>> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 >> pirate >

> When *I* tried to name it everyone booed. :(

so sorry :( :( :(

> This is one of the two 4296-et power commas; the other is viking: > > 2^161 * 3^(-15) * 5^49 > > Also of note is raider = pirate * viking = > > 2^71 * 3^(-99) * 5^33 > > The TM basis for 4296 is <pirate, raider>.

u got it. i will up the complexity limit, if that gets these in there . . .

Message: 5681 - Contents - Hide Contents Date: Wed, 04 Dec 2002 23:37:44 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

if i raise my complexity limit anywhere from 5 to 43 points, i'd be adding these two and only these two to my list. seems like a sensible place to stop, at least until our next fit of mathematical irrelevance! :)

Message: 5682 - Contents - Hide Contents Date: Wed, 04 Dec 2002 23:50:58 Subject: Re: A Property of MOS/DE Scales From: Kalle Aho Paul wrote on tuning-list: "sorry i got confused on this matter! but the main thrust of my reply was to suggest to you that we have more sophisticated tools at our disposal now than i did when i wrote that paper. we should not be restricted to an ET conception. each MOS/DE scale can be defined independently in terms of its own generator, in cents; and period of repetition, in 1/octaves. by restricting ourselves to ETs and their best approximations to consonant intervals, we may miss some interesting and wonderful possibilities. it might be a good idea to follow up to tuning-math, where the "searchers" may be more willing to publically help." I am aware of these more sophisticated (and also more elegant) tools and possibilities. So if anyone is interested please read the thread in tuning list and continue on tuning-math. Kalle

Message: 5683 - Contents - Hide Contents Date: Thu, 05 Dec 2002 18:49:30 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> Viking [161, -84, -12] .015361 cents

this is the difference between 11 pythagorean commas and 12 syntonic commas. i'm going to call it "atomic" instead, unless someone comes up with a better name . . .

Message: 5684 - Contents - Hide Contents Date: Thu, 05 Dec 2002 20:45:49 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus what if we pushed up the badness limit until ampersand made it in? the 5-limit comma does have a name, so it must be of some use . . . --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> Not that any list is really ultimate, but with rms error < 40,

geometric complexity < 500, and badness < 3500, it covers a lot of ground.

> > 27/25 3.739252 35.60924 1861.731473 > > 135/128 4.132031 18.077734 1275.36536 > > 256/243 5.493061 12.759741 2114.877638 > > 25/24 3.025593 28.851897 799.108711 > > 648/625 6.437752 11.06006 2950.938432 > > 16875/16384 8.17255 5.942563 3243.743713 > > 250/243 5.948286 7.975801 1678.609846 > > 128/125 4.828314 9.677666 1089.323984 > > 3125/3072 7.741412 4.569472 2119.95499 > > 20000/19683 9.785568 2.504205 2346.540676 > > 531441/524288 13.183347 1.382394 3167.444999 > > 81/80 4.132031 4.217731 297.556531 > > 2048/2025 6.271199 2.612822 644.408867 > > 67108864/66430125 15.510107 .905187 3377.402314 > > 78732/78125 12.192182 1.157498 2097.802867 > > 393216/390625 12.543123 1.07195 2115.395301 > > 2109375/2097152 12.772341 .80041 1667.723301 > > 4294967296/4271484375 18.573955 .483108 3095.692488 > > 15625/15552 9.338935 1.029625 838.631548 > > 1600000/1594323 13.7942 .383104 1005.555381 > > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 > > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 > > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 > > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 > > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 > > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 > > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 > > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 > > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 > > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 > > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 > > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 > > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 > > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 > > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 > > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 > > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 > > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 > > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 > > (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721 > > (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642 > > (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514 > > (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003 > > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 > > (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668 > > (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625 > > (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527 > > (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421 > > (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113 > > (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823 > > (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

Message: 5685 - Contents - Hide Contents Date: Thu, 05 Dec 2002 21:51:19 Subject: Re: Ultimate 5-limit comma list From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

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

>> Viking [161, -84, -12] .015361 cents

> this is the difference between 11 pythagorean commas and 12 syntonic > commas. i'm going to call it "atomic" instead, unless someone comes > up with a better name . . .

Then shouldn't pirate and raider be electron and proton? Good spot, by the way.

Message: 5686 - Contents - Hide Contents Date: Thu, 05 Dec 2002 22:22:56 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

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

>>> Viking [161, -84, -12] .015361 cents >

>> this is the difference between 11 pythagorean commas and 12 syntonic >> commas. i'm going to call it "atomic" instead, unless someone comes >> up with a better name . . . >

> Then shouldn't pirate and raider be electron and proton?

if you say so, but which is which? then again, speaking of "electronic music" might get confusing . . .

Message: 5688 - Contents - Hide Contents Date: Thu, 05 Dec 2002 00:22:07 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

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

>> This is one of the two 4296-et power commas; the other is viking: >> >> 2^161 * 3^(-15) * 5^49 >> >> Also of note is raider = pirate * viking = >> >> 2^71 * 3^(-99) * 5^33 >> >> The TM basis for 4296 is <pirate, raider>. >

> if i raise my complexity limit anywhere from 5 to 43 points, i'd be > adding these two and only these two to my list. seems like a sensible > place to stop, at least until our next fit of mathematical > irrelevance! :)

wait a minute. i went back to your list with only the 2s exponent and looked up the complexity based on that. but something's wrong. 2^161 * 3^(-15) * 5^49 = 301200.046966396 cents 2^71 * 3^(-99) * 5^33 = -11145.1925281338 cents what are they really supposed to be, and what is the geometric complexity of them?

Message: 5690 - Contents - Hide Contents Date: Thu, 05 Dec 2002 04:24:09 Subject: Re: Ultimate 5-limit comma list From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> u got it. i will up the complexity limit, if that gets these in > there . . .

4296 is certainly a logical stopping point; all you need to do is up the limit to 105. Viking is 100.53, and raider is 104.66.

Message: 5692 - Contents - Hide Contents Date: Thu, 05 Dec 2002 04:31:52 Subject: Re: Ultimate 5-limit comma list From: Gene Ward Smith --- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> 2^161 * 3^(-15) * 5^49 = 301200.046966396 cents > 2^71 * 3^(-99) * 5^33 = -11145.1925281338 cents

Pirate [-90, -15, 49] .046966 cents Raider [71, -99, 37] .062327 cents Viking [161, -84, -12] .015361 cents

Message: 5696 - Contents - Hide Contents Date: Fri, 06 Dec 2002 05:14:59 Subject: 25/24, 49/48, 50/49 From: Gene Ward Smith "Standard" ets for which h(25/24)=0, h(49/48)=1 7,17 h(25/24)=1 h(49/48)=0 5,9,15,19,25,29 h(50/49)=0 h(25/24)=1 2,12,16,18,22,26,32 Linear temperaments with 25/24 a comma and geometric badness below 5000: 25/24 [0, 0, 4, 9, -6, 0] [[4, 6, 9, 0], [0, 0, 0, 1]] complexity 6.647907 rms 63.402645 badness 2802.058941 generators [300.0000000, 3306.069668] [0, 0, 7, 16, -11, 0] [[7, 11, 16, 0], [0, 0, 0, 1]] complexity 11.633838 rms 25.354978 badness 3431.699321 generators [171.4285715, 3348.926812] [2, 1, -4, -12, 12, -3] [[1, 1, 2, 4], [0, 2, 1, -4]] complexity 9.849244 rms 25.068246 badness 2431.810367 generators [1200., 358.0334333] [0, 0, 3, 7, -5, 0] [[3, 5, 7, 0], [0, 0, 0, 1]] complexity 4.985930 rms 61.312549 badness 1524.199406 generators [400.0000000, 3406.069668] [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] complexity 10.574200 rms 23.945252 badness 2677.407574 generators [600.0000000, 950.9775006] [2, 1, 6, 11, -4, -3] [[1, 1, 2, 1], [0, 2, 1, 6]] complexity 9.849244 rms 26.099830 badness 2531.881840 generators [1200., 360.2272895] [2, 1, 3, 4, 1, -3] [[1, 1, 2, 2], [0, 2, 1, 3]] complexity 6.245166 rms 48.926006 badness 1908.216791 generators [1200., 322.2119006] [2, 1, -1, -5, 7, -3] [[1, 1, 2, 3], [0, 2, 1, -1]] complexity 6.245166 rms 53.747748 badness 2096.274853 generators [1200., 318.5700997] Ditto with 49/48: [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] complexity 10.574200 rms 23.945252 badness 2677.407574 generators [600.0000000, 950.9775006] [6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]] complexity 16.383068 rms 12.273810 badness 3294.350648 generators [1200., 316.6640534] [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]] complexity 14.729697 rms 12.188571 badness 2644.480844 generators [1200., -125.4687958] [2, 3, 1, -6, 4, 0] [[1, 0, 0, 2], [0, 2, 3, 1]] complexity 6.691597 rms 34.566097 badness 1547.782446 generators [1200., 929.2070233] [6, 0, 3, 7, 12, -14] [[3, 0, 7, 6], [0, 2, 0, 1]] complexity 17.012788 rms 16.786584 badness 4858.624067 generators [400.0000000, 956.3327071] [0, 5, 0, -14, 0, 8] [[5, 8, 0, 14], [0, 0, 1, 0]] complexity 10.254281 rms 15.815352 badness 1662.988582 generators [240.0000000, 2789.386744] [2, 8, 1, -20, 4, 8] [[1, 0, -4, 2], [0, 2, 8, 1]] complexity 15.298626 rms 12.690078 badness 2970.086938 generators [1200., 947.2576877] [2, -6, 1, 19, 4, -14] [[1, 0, 7, 2], [0, 2, -6, 1]] complexity 15.298626 rms 18.261110 badness 4273.975544 generators [1200., 939.2948284] Ditto with 50/49: [4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]] complexity 11.405897 rms 19.136993 badness 2489.617178 generators [300.0000000, 1885.698206] [0, 2, 2, -1, -3, 3] [[2, 3, 0, 1], [0, 0, 1, 1]] complexity 4.275602 rms 59.723378 badness 1091.789278 generators [600.0000000, 2726.592310] [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]] complexity 19.126831 rms 11.798337 badness 4316.252447 generators [600.0000000, 231.2978354] [2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]] complexity 11.925109 rms 10.903178 badness 1550.521640 generators [600.0000000, 1908.814331] [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]] complexity 11.925109 rms 18.863889 badness 2682.600333 generators [600.0000000, 1928.512337] [2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]] complexity 15.871133 rms 11.218941 badness 2825.971103 generators [600.0000000, 1893.651026] [8, 6, 6, -3, 13, -9] [[2, 1, 3, 4], [0, 4, 3, 3]] complexity 21.576275 rms 10.132266 badness 4716.930929 generators [600.0000000, 325.6113679] [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] complexity 10.574200 rms 23.945252 badness 2677.407574 generators [600.0000000, 950.9775006]

Message: 5697 - Contents - Hide Contents Date: Sat, 07 Dec 2002 03:44:07 Subject: Re: 25/24, 49/48, 50/49 From: wallyesterpaulrus i don't know if the usual badness measure was appropriate in this case -- see my results and then you might get a better feel for it. the first key to interpreting this is to note that the chromatic unison vector's size in generators will equal the cardinality of the scale (per period) . . . let's cap the rms error at 20-22 cents . . .

> Ditto with 49/48: > > [6, 5, 3, -7, 12, -6] [[1, 0, 1, 2], [0, 6, 5, 3]] > complexity 16.383068 rms 12.273810 badness 3294.350648 > generators [1200., 316.6640534]

25:24 chroma = 10 - 4 = 7 generators -> 4 note scale graham complexity = 6 X

> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]] > complexity 14.729697 rms 12.188571 badness 2644.480844 > generators [1200., -125.4687958]

25:24 chroma = -6 - 4 = -10 generators -> 10 note scale graham complexity = 7 -> 6 tetrads

> [6, 0, 3, 7, 12, -14] [[3, 0, 7, 6], [0, 2, 0, 1]] > complexity 17.012788 rms 16.786584 badness 4858.624067 > generators [400.0000000, 956.3327071]

25:24 chroma = 0 - 2 = -2 generators -> 6 note scale graham complexity = 2*3=6 X

> [0, 5, 0, -14, 0, 8] [[5, 8, 0, 14], [0, 0, 1, 0]] > complexity 10.254281 rms 15.815352 badness 1662.988582 > generators [240.0000000, 2789.386744] blackwood-10

25:24 chroma = 2 - 0 = 2 generators -> 10 note scale graham complexity = 1*5=5 -> 10 tetrads

> [2, 8, 1, -20, 4, 8] [[1, 0, -4, 2], [0, 2, 8, 1]] > complexity 15.298626 rms 12.690078 badness 2970.086938 > generators [1200., 947.2576877]

25:24 chroma = 16 - 2 = 14 generators -> 14 note scale graham complexity = 8 -> 12 tetrads

> [2, -6, 1, 19, 4, -14] [[1, 0, 7, 2], [0, 2, -6, 1]] > complexity 15.298626 rms 18.261110 badness 4273.975544 > generators [1200., 939.2948284]

25:24 chroma = -12 - 2 = -14 generators -> 14 note scale graham complexity = 8 -> 12 tetrads

> Ditto with 50/49: > > [4, 4, 4, -2, 5, -3] [[4, 0, 3, 5], [0, 1, 1, 1]] > complexity 11.405897 rms 19.136993 badness 2489.617178 > generators [300.0000000, 1885.698206]

25:24 chroma = 2 - 1 = 1 generator -> 4 note scale graham complexity = 1*4 = 4 X

> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]] > complexity 19.126831 rms 11.798337 badness 4316.252447 > generators [600.0000000, 231.2978354]

25:24 chroma = -2 - 3 = 5 generators -> 10 note scale graham complexity = 8 -> 4 tetrads

> [2, -4, -4, 2, 12, -11] [[2, 0, 11, 12], [0, 1, -2, -2]] > complexity 11.925109 rms 10.903178 badness 1550.521640 > generators [600.0000000, 1908.814331] pajara-10

25:24 chroma = -4 - 1 = 5 generators -> 10 note scale graham complexity = 3*2 = 6 -> 8 tetrads

> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]] > complexity 11.925109 rms 18.863889 badness 2682.600333 > generators [600.0000000, 1928.512337]

25:24 chroma = 6 - 1 = 5 generators -> 10 note scale graham complexity = 3*2 = 6 -> 8 tetrads

> [2, 8, 8, -4, -7, 8] [[2, 0, -8, -7], [0, 1, 4, 4]] > complexity 15.871133 rms 11.218941 badness 2825.971103 > generators [600.0000000, 1893.651026] injera-14

25:24 chroma = 8 - 1 = 7 generators -> 14 note scale graham complexity = 4*2 = 8 -> 12 tetrads

> [8, 6, 6, -3, 13, -9] [[2, 1, 3, 4], [0, 4, 3, 3]] > complexity 21.576275 rms 10.132266 badness 4716.930929 > generators [600.0000000, 325.6113679]

25:24 chroma = 6 - 4 = 2 generators -> 4 note scale graham complexity = 4*2 = 8 X p.s. note that this one:

> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]] > complexity 10.574200 rms 23.945252 badness 2677.407574 > generators [600.0000000, 950.9775006]

appeared in all three lists . . .

Message: 5698 - Contents - Hide Contents Date: Sat, 07 Dec 2002 11:23:21 Subject: Re: 25/24, 49/48, 50/49 From: Kalle Aho Thank you, Gene and Paul! Great job! How is complexity calculated and why? Badness is somewhere around complexity^2*rms, what's the idea behind this? I guess everything is in the archives, any old messages you'd like me to read? Kalle

Message: 5699 - Contents - Hide Contents Date: Sat, 07 Dec 2002 20:04:26 Subject: Re: 25/24, 49/48, 50/49 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Kalle Aho <kalleaho@m...>" <kalleaho@m...> wrote:

> How is complexity calculated and why?

I calculated complexity using a rather complicated formula, but Paul tells me that Graham complexity would have been better; evidentally you are interested only in complete tetrads. It is easy to compute: take the range in generator steps of the representations of {1,3,5,7,5/3,7/3,7/5} and multiply by the number of periods in an octave. Badness is somewhere around

> complexity^2*rms, what's the idea behind this?

It makes the badness measure "log-flat", roughly meaning about as many temperaments appear in one size range as in another.