Meantone family
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2011-06-07 17:56:35 UTC.
- The original revision id was 234978070.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[toc|flat]] The 5-limit parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. [[POTE tuning|POTE generator]]: 696.239 Map: [<1 0 -4|, <0 1 4|] EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]] [[Badness]]: 0.00736 ==Seven limit children== The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>]. =Septimal meantone= The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it. [[Comma]]s: 81/80, 126/125 7 and [[9-limit]] minimax [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 696.495 Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. Map: [<1 0 -4 -13|, <0 1 4 10|] [[Generator]]s: 2, 3 Wedgie: <<1 4 10 4 13 12|| EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]] [[Badness]]: 0.0137 ==Unidecimal meantone aka Huygens== [[Comma]]s: 81/80, 126/125, 99/98 [[11-limit]] minimax [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] [[Eigenmonzo]]s: 2, 11/9 [[POTE tuning|POTE generator]]: 696.967 Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|] [[Generator]]s: 2, 3 EDOs: [[7edo|7]], [[12edo|12]], [[31edo|31]], [[105edo|105]], [[198edo|198]] [[Badness]]: 0.0170 ===Tridecimal meantone=== [[Comma]]s: 66/65, 81/80, 99/98, 105/104 POTE generator: ~3/2 = 696.642 Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] EDOs: [[12edo|12]], [[19edo|19], [[31edo|31]], [[267edo|267]], [[298edo|298]] [[Badness]]: 0.0180 ==Meanpop== [[Comma]]s: 81/80, 126/125, 385/384 11-limit minimax 1/4 comma [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 696.434 Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] [[Generator]]s: 2, 3 EDOs: 12, 19, 31, 81, [[112edo|112]] [[Badness]]: 0.0215 ===13-limit Meanpop=== [[Comma]]s: 81/80, 105/104, 144/143, 196/195 POTE generator: ~3/2 = 696.211 Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] EDOS: 7, 12, 19, 31, 50, 81, [[131edo|131]] [[Badness]]: 0.0209 ==Meanenneadecal== [[Comma]]s: 45/44, 56/55, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.250 Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] EDOs: 7, 12, 19, 31, 50, 81 [[Badness]]: 0.0214 ===13-limit=== [[Comma]]s: 45/44, 56/55, 78/77, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.146 Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] EDOs: 7, 12, 19, 31, 50, [[131edo|131]], [[181edo|181]] [[Badness]]: 0.0212 =Flattone= [[Comma]]s: 81/80, 525/512 The [[wedgie]] for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. [[7-limit]] minimax [|1 0 0 0>, |21/13 0 1/13 -1/13>, |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>] [[Eigenmonzo]]s: 2, 7/5 [[9-limit]] minimax [|1 0 0 0>, |17/11 2/11 0 -1/11>, |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] [[Eigenmonzo]]s: 2, 9/7 [[POTE tuning|POTE generator]]: 693.779 Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4. Map: [<1 0 -4 17|, <0 1 4 -9|] [[Wedgie]]: <<1 4 -9 4 -17 -32|| [[Generator]]s: 2, 3 EDOs: 7, 19, [[45edo|45]], [[64edo|64]] [[Badness]]: 0.0386 =Dominant= [[Comma]]s: 36/35, 64/63 The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]]. [[POTE tuning|POTE generator]]: 701.573 Map: [<1 0 -4 6|, <0 1 4 -2|] [[Wedgie]]: <<1 4 -2 4 -6 -16|| EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]] [[Badness]]: 0.0207 =Sharptone= [[Comma]]s: 21/20, 28/27 Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done. [[POTE tuning|POTE generator]]: 700.140 Map: [<1 0 -4 -2|, <0 1 4 3|] [[Wedgie]]: <<1 4 3 4 2 -4|| EDOs: 5, 12 [[Badness]]: 0.0248 =Injera= [[Comma]]s: 50/49, 81/80 The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel 19edos, is an excellent tuning for injera. [[POTE tuning|POTE generator]]: 694.375 Map: [<2 0 -8 -7|, <0 1 4 4|] [[Wedgie]]: <<2 8 8 8 7 -4|| EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]] [[Badness]]: 0.0311 =Godzilla= [[Comma]]s: 49/48, 81/80 Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes. [[POTE tuning|POTE generator]]: 252.635 Map: [<1 0 -4 2|, <0 2 8 1|] [[Wedgie]]: <<2 8 1 8 -4 -20|| EDOs: [[5edo|5]], [[9edo|9]], [[14edo|14]], 19 [[Badness]]: 0.0267 ==Music== [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3|Godzilla Example]] by [[Cameron Bobro]] [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]] in Godzilla[9] by [[Igliashon Jones]] =Mohajira= [[Comma]]s: 81/80, 6144/6125 Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. 7 and 9-limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 348.415 Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly. Map: [<1 1 0 6|, <0 2 8 -11|] [[Generator]]s: 2, 128/105 [[Wedgie]]: <<2 8 -11 8 -23 -48|| EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] [[Badness]]: 0.0557 ==11-limit== [[Comma]]s: 81/80, 121/120, 176/175 [[11-limit]] minimax 1/4 comma [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |6 0 -11/8 0 0>, |2 0 5/8 0 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 348.477 Map: [<1 1 0 6 2|, <0 2 8 -11 5|] [[Generator]]s: 2, 11/9 EDOs: 7, 24, 31 [[Badness]]: 0.0261 =Mothra= Commas: 81/80, 1029/1024 Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 232.193 Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents. Map: [<1 1 0 3|, <0 3 12 -1|] [[Generator]]s: 2, 8/7 [[Wedgie]]: <<3 12 -1 12 -10 -36|| EDOs: 5, [[26edo|26]], 31 [[Badness]]: 0.0371 ==11-limit== [[Comma]]s: 81/80, 99/98, 385/384 POTE generator: ~63/55 = 232.031 Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] EDOs: 5, [[26edo|26]], 31, [[88edo|88]], [[150edo|150]], [[181edo|181]] [[Badness]]: 0.0256 =Squares= [[Comma]]s: 81/80, 2401/2400 Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 425.942 Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents. Map: [<1 3 8 6|, <0 -4 -16 -9|] [[Generator]]s: 2, 9/7 EDOs: [[14edo|14]], 31, [[262edo|262]], [[293edo|293]] [[Badness]]: 0.0460 Music: By [[Chris Vaisvil]] http://tinyurl.com/25kv7cq http://tinyurl.com/24cbxse ==11-limit== [[Comma]]s: 81/80, 385/384, 1375/1372 [[POTE tuning|POTE generator]]: 425.993 Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] EDOs: [[14edo|14]], 31, [[200edo|200]] [[Badness]]: 0.0568 =Liese= [[Comma]]s: 81/80, 686/675 Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: 632.406 Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges. Map: [<1 0 -4 -3|, <0 3 12 11|] [[Generator]]s: 2, 10/7 EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]] [[Badness]]: 0.0467 =Squares= Commas: 81/80, 2401/2400 POTE generator: ~9/7 = 425.942 Map: [<1 3 8 6|, <0 -4 -16 -9|] Wedgie: <<4 16 9 16 3 -24|| EDOs: 5, 8, 11, 14, 17, 31 Badness: 0.0460 ==11-limit== Commas: 81/80, 99/98, 121/120 POTE generator: ~9/7 = 425.957 Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|] EDOs: 5, 8, 11, 14, 17, 31 Badness: 0.0216 ==13-limit== Commas: 81/80, 99/98, 121/120, 66/65 POTE generator: ~9/7 = 425.550 Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|] EDOs: 5, 8, 11, 14, 17, 31 Badness: 0.0255
Original HTML content:
<html><head><title>Meantone family</title></head><body><!-- ws:start:WikiTextTocRule:48:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --> | <a href="#Septimal meantone">Septimal meantone</a><!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --> | <a href="#Flattone">Flattone</a><!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --> | <a href="#Dominant">Dominant</a><!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --> | <a href="#Sharptone">Sharptone</a><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --> | <a href="#Injera">Injera</a><!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --> | <a href="#Godzilla">Godzilla</a><!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --> | <a href="#Mohajira">Mohajira</a><!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --> | <a href="#Mothra">Mothra</a><!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --> | <a href="#Liese">Liese</a><!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --> | <a href="#Squares">Squares</a><!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --> <!-- ws:end:WikiTextTocRule:73 --><br /> The 5-limit parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> <br /> Map: [<1 0 -4|, <0 1 4|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/81edo">81</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.00736<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>].<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h1> The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br /> <br /> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125<br /> <br /> 7 and <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> <br /> Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br /> <br /> Map: [<1 0 -4 -13|, <0 1 4 10|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> Wedgie: <<1 4 10 4 13 12||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/81edo">81</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0137<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal meantone aka Huygens</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 99/98<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> minimax<br /> [|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, <br /> |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 11/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> <br /> Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> <br /> Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/198edo">198</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0170<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tridecimal meantone</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br /> <br /> POTE generator: ~3/2 = 696.642<br /> <br /> Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/19edo">19], [[31edo|31</a>, <a class="wiki_link" href="/267edo">267</a>, <a class="wiki_link" href="/298edo">298</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0180<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:8 -->Meanpop</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> <br /> 11-limit minimax 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |-3 0 5/2 0 0>, |11 0 -13/4 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> <br /> Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: 12, 19, 31, 81, <a class="wiki_link" href="/112edo">112</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit Meanpop</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> <br /> POTE generator: ~3/2 = 696.211<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> EDOS: 7, 12, 19, 31, 50, 81, <a class="wiki_link" href="/131edo">131</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0209<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:12 -->Meanenneadecal</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br /> <br /> Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]<br /> EDOs: 7, 12, 19, 31, 50, 81<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0214<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 45/44, 56/55, 78/77, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br /> <br /> Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]<br /> EDOs: 7, 12, 19, 31, 50, <a class="wiki_link" href="/131edo">131</a>, <a class="wiki_link" href="/181edo">181</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0212<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h1> --><h1 id="toc8"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Flattone</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 525/512<br /> <br /> The <a class="wiki_link" href="/wedgie">wedgie</a> for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> <br /> <a class="wiki_link" href="/7-limit">7-limit</a> minimax<br /> [|1 0 0 0>, |21/13 0 1/13 -1/13>, <br /> |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/5<br /> <br /> <a class="wiki_link" href="/9-limit">9-limit</a> minimax<br /> [|1 0 0 0>, |17/11 2/11 0 -1/11>, <br /> |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> <br /> Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br /> <br /> Map: [<1 0 -4 17|, <0 1 4 -9|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 -9 4 -17 -32||<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> EDOs: 7, 19, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0386<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h1> --><h1 id="toc9"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:18 -->Dominant</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 36/35, 64/63<br /> <br /> The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br /> <br /> Map: [<1 0 -4 6|, <0 1 4 -2|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 -2 4 -6 -16||<br /> EDOs: 5, 7, 12, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Sharptone</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 21/20, 28/27<br /> <br /> Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 700.140<br /> <br /> Map: [<1 0 -4 -2|, <0 1 4 3|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<1 4 3 4 2 -4||<br /> EDOs: 5, 12<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0248<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h1> --><h1 id="toc11"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:22 -->Injera</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 50/49, 81/80<br /> <br /> The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel 19edos, is an excellent tuning for injera.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br /> <br /> Map: [<2 0 -8 -7|, <0 1 4 4|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 8 8 7 -4||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/178edo">178</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0311<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:24 -->Godzilla</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 49/48, 81/80<br /> <br /> Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 252.635<br /> <br /> Map: [<1 0 -4 2|, <0 2 8 1|]<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 1 8 -4 -20||<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/9edo">9</a>, <a class="wiki_link" href="/14edo">14</a>, 19<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0267<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="Godzilla-Music"></a><!-- ws:end:WikiTextHeadingRule:26 -->Music</h2> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3" rel="nofollow">Godzilla Example</a> by <a class="wiki_link" href="/Cameron%20Bobro">Cameron Bobro</a><br /> <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">"Change is on the Wind"</a> in Godzilla[9] by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Jones</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h1> --><h1 id="toc14"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:28 -->Mohajira</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 6144/6125<br /> <br /> Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> <br /> 7 and 9-limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> <br /> Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br /> <br /> Map: [<1 1 0 6|, <0 2 8 -11|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 128/105<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<2 8 -11 8 -23 -48||<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/24edo">24</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0557<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 121/120, 176/175<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> minimax 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |6 0 -11/8 0 0>, |2 0 5/8 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> <br /> Map: [<1 1 0 6 2|, <0 2 8 -11 5|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br /> EDOs: 7, 24, 31<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h1> --><h1 id="toc16"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:32 -->Mothra</h1> Commas: 81/80, 1029/1024<br /> <br /> Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> <br /> 7 and 9 limit minimax 1/4 comma <br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 232.193<br /> <br /> Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br /> <br /> Map: [<1 1 0 3|, <0 3 12 -1|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 8/7<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<3 12 -1 12 -10 -36||<br /> EDOs: 5, <a class="wiki_link" href="/26edo">26</a>, 31<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0371<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> <br /> POTE generator: ~63/55 = 232.031<br /> <br /> Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]<br /> EDOs: 5, <a class="wiki_link" href="/26edo">26</a>, 31, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/150edo">150</a>, <a class="wiki_link" href="/181edo">181</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0256<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h1> --><h1 id="toc18"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:36 -->Squares</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 2401/2400<br /> <br /> Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> <br /> Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br /> <br /> Map: [<1 3 8 6|, <0 -4 -16 -9|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 9/7<br /> EDOs: <a class="wiki_link" href="/14edo">14</a>, 31, <a class="wiki_link" href="/262edo">262</a>, <a class="wiki_link" href="/293edo">293</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0460<br /> <br /> Music:<br /> By <a class="wiki_link" href="/Chris%20Vaisvil">Chris Vaisvil</a><br /> <!-- ws:start:WikiTextUrlRule:532:http://tinyurl.com/25kv7cq --><a class="wiki_link_ext" href="http://tinyurl.com/25kv7cq" rel="nofollow">http://tinyurl.com/25kv7cq</a><!-- ws:end:WikiTextUrlRule:532 --><br /> <!-- ws:start:WikiTextUrlRule:533:http://tinyurl.com/24cbxse --><a class="wiki_link_ext" href="http://tinyurl.com/24cbxse" rel="nofollow">http://tinyurl.com/24cbxse</a><!-- ws:end:WikiTextUrlRule:533 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 385/384, 1375/1372<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.993<br /> <br /> Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]<br /> EDOs: <a class="wiki_link" href="/14edo">14</a>, 31, <a class="wiki_link" href="/200edo">200</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0568<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h1> --><h1 id="toc20"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:40 -->Liese</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 686/675<br /> <br /> Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> <br /> Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br /> <br /> Map: [<1 0 -4 -3|, <0 3 12 11|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 10/7<br /> EDOs: <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/74edo">74</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0467<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:42 -->Squares</h1> Commas: 81/80, 2401/2400<br /> <br /> POTE generator: ~9/7 = 425.942<br /> <br /> Map: [<1 3 8 6|, <0 -4 -16 -9|]<br /> Wedgie: <<4 16 9 16 3 -24||<br /> EDOs: 5, 8, 11, 14, 17, 31<br /> Badness: 0.0460<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc22"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> Commas: 81/80, 99/98, 121/120<br /> <br /> POTE generator: ~9/7 = 425.957<br /> <br /> Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]<br /> EDOs: 5, 8, 11, 14, 17, 31<br /> Badness: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> Commas: 81/80, 99/98, 121/120, 66/65<br /> <br /> POTE generator: ~9/7 = 425.550<br /> <br /> Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]<br /> EDOs: 5, 8, 11, 14, 17, 31<br /> Badness: 0.0255</body></html>