Meantone family
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<span style="display: block; text-align: right;">[[toc]] [[xenharmonie/mitteltönig|Deutsch]] </span> 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]]: ~3/2 = 696.239 Mapping generator: ~3 [[Tuning Ranges of Regular Temperaments|valid range]]: [685.714, 720.000] (7 to 5) nice range: [694.786, 701.955] (1/3 comma to Pythagorean) strict range: [694.786, 701.955] [[Map]]: [<1 0 -4|, <0 1 4|] EDOs (patent val edo list is complete): [[5edo|5]], [[7edo|7]], [[12edo|12]], [[19edo|19]], [[24edo|24]], [[26edo|26]], [[31edo|31]], [[36edo|36]], [[38edo|38]], [[43edo|43]], [[45edo|45]], [[50edo|50]], [[55edo|55]], [[57edo|57]], [[62edo|62]], [[67edo|67]], [[69edo|69]], [[74edo|74]], [[76edo|76]], [[81edo|81]], [[86edo|86]], [[88edo|88]], [[93edo|93]], [[98edo|98]], [[100edo|100]], [[105edo|105]], [[117edo|117]], [[129edo|129]], [[212edo|212b]] [[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= <span style="display: block; text-align: right;">[[xenharmonie/septimal-mitteltönig|Deutsch]] </span> [[https://en.wikipedia.org/wiki/Septimal_meantone_temperament|Wikipedia article]] 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|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|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 [[Tuning Ranges of Regular Temperaments|valid range]]: [694.737, 700.000] (19 to 12) nice range: [694.786, 701.955] strict range: [694.786, 700.000] [[POTE tuning|POTE generator]]: 696.495 Mapping generator: ~3 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]], [[143edo|143b]] [[Badness]]: 0.0137 ==Bimeantone== Commas: 81/80, 126/125, 245/242 [[POTE tuning|POTE generator]]: ~3/2 = 696.016 Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|] EDOs: 12, 38d, 50 Badness: 0.0381 ===13-limit=== Commas: 81/80, 105/104, 126/125, 245/242 [[POTE tuning|POTE generator]]: ~3/2 = 695.836 Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|] EDOs: 12f, 50 Badness: 0.0288 ==Unidecimal meantone aka Huygens== See also [[Meantone vs meanpop]] [[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 valid range: [696.774, 700.000] (31 to 12) nice range: [691.202, 701.955] strict range: [696.774, 700.000] [[POTE tuning|POTE generator]]: 696.967 Mapping generator: ~3 [[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|198be]] [[Badness]]: 0.0170 [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3|Twinkle canon – 74 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] ===Tridecimal meantone=== [[Comma]]s: 66/65, 81/80, 99/98, 105/104 valid range: 697.674 (43) nice range: [691.202, 701.955] strict range: 697.674 [[POTE tuning|POTE generator]]: ~3/2 = 696.642 Mapping generator: ~3 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 ===Grosstone=== Commas: 81/80, 99/98, 126/125, 144/143 POTE generator: ~3/2 = 697.264 Mapping generator: ~3 Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|] EDOs: 12, 31, 43, 74 Badness: 0.0259 ===Meridetone=== Commas: 78/77, 81/80, 99/98, 126/125 POTE generator: ~3/2 = 697.529 Mapping generator: ~3 Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|] EDOs: 43, 117df, 160bdf, 203bcdef Badness: 0.0264 ===Hemimeantone=== Commas: 81/80, 99/98, 126/125, 169/168 POTE generator: ~52/45 = 250.304 Mapping generator: ~26/15 Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|] EDOs: 43, 62, 167bef, 229bef Badness: 0.0314 ==Meanpop== See also [[Meantone vs 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 valid range: [694.737, 696.774] (19 to 31) nice range: [691.202, 701.955] strict range: [694.737, 696.774] [[POTE tuning|POTE generator]]: 696.434 Mapping generator: ~3 [[Algebraic generator]]: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. [[@http://soonlabel.com/xenharmonic/archives/607|Scott Joplin's "The Entertainer" tuned into meanpop]] Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] [[Generator]]s: 2, 3 EDOs: [[12edo|12]], [[19edo|19]], [[31edo|31]], [[81edo|81]], [[112edo|112]] [[Badness]]: 0.0215 [[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3|Twinkle canon – 50 edo]] by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]] ===13-limit Meanpop=== [[Comma]]s: 81/80, 105/104, 144/143, 196/195 valid range: [694.737, 696.774] (19 to 31) nice range: [691.202, 701.955] strict range: [694.737, 696.774] POTE generator: ~3/2 = 696.211 Mapping generator: ~3 Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|] EDOS: [[19edo|19]], [[31edo|31]], [[50edo|50]], [[81edo|81]], [[131edo|131bd]], [[212edo|212bdf]] [[Badness]]: 0.0209 ===Meanplop=== Commas: 65/64, 78/77, 81/80, 91/90 POTE generator: ~3/2 = 696.202 Mapping generator: ~3 Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|] EDOs: 12e, 19, 31f, 50f Badness: 0.0277 ==Meanenneadecal== [[Comma]]s: 45/44, 56/55, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.250 Mapping generator: ~3 Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] EDOs: [[7edo|7]], [[12edo|12]], [[19edo|19]], [[31edo|31e]], [[50edo|50e]] [[Badness]]: 0.0214 ===13-limit=== [[Comma]]s: 45/44, 56/55, 78/77, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.146 Mapping generator: ~3 Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] EDOs: [[19edo|19]], [[31edo|31e]], [[50edo|50e]]] [[Badness]]: 0.0212 ===Vincenzo=== Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22 POTE generator: ~3/2 Mapping generator: ~3 Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|] EDOs: 12 Badness: ==Meanundeci== Commas: 33/32, 55/54, 77/75 POTE generator: ~3/2 = 694.689 Mapping generator: ~3 Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|] EDOs: 12e, 19e Badness: 0.0315 ===13-limit=== Commas: 33/32, 55/54, 77/75, 729/728 POTE generator: ~3/2 = 694.764 Mapping generator: ~3 Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|] EDOs: 12e, 19e Badness: 0.0263 ==Meanundec== Commas: 27/26, 40/39, 45/44, 56/55 POTE generator: ~3/2 = 697.254 Mapping generator: ~3 Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|] EDOS: 12f, 19f, 31ef Badness: 0.0242 =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|7/4]] is a diminished seventh interval. Other intervals are [[7_6|7/6]], a diminished third, and [[7_5|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 valid range: [692.308, 694.737] (26 to 19) nice range: [692.353, 701.955] strict range: [692.353, 694.737] [[POTE tuning|POTE generator]]: 693.779 Mapping generator: ~3 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: [[7edo|7]], [[19edo|19]], [[45edo|45]], [[64edo|64]] [[Badness]]: 0.0386 ==11-limit== Commas: 45/44, 81/80, 385/384 valid range: [692.308, 694.737] (26 to 19) nice range: [682.502, 701.955] strict range: [692.308, 694.737] POTE generator: ~3/2 = 693.126 Mapping generator: ~3 Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|] EDOs: 7, 19, 26, 45, 71bc, 116bcde Badness: 0.0338 ==13-limit== 45/44, 65/64, 78/77, 81/80 valid range: [692.308, 694.737] (26 to 19) nice range: [682.502, 701.955] strict range: [692.308, 694.737] POTE generator: ~3/2 = 693.058 Mapping generator: ~3 Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|] EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef Badness: 0.0223 =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|3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]]. valid range: [700.000, 720.000] (12 to 5) nice range: [694.786, 715.587] strict range: [700.000, 715.587] [[POTE tuning|POTE generator]]: 701.573 Mapping generator: ~3 Map: [<1 0 -4 6|, <0 1 4 -2|] [[Wedgie]]: <<1 4 -2 4 -6 -16|| EDOs: [[5edo|5]], [[7edo|7]], [[12edo|12]], [[53edo|53]], [[65edo|65]] [[Badness]]: 0.0207 ==11-limit== Commas: 36/35, 64/63, 56/55 valid range: [700.000, 705.882] (12 to 17) nice range: [691.202, 715.587] strict range: [700.000, 705.882] POTE generator: ~3/2 = 703.254 Mapping generator: ~3 Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|] EDOs: 5, 12, 17c, 29cde Badness: 0.0242 ==13-limit== Commas: 36/35, 56/55, 64/63, 66/65 valid range: 705.882 (17) nice range: [691.202, 715.587] strict range:705.882 POTE generator: ~3/2 = 703.636 Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|] EDOs: 12f, 17c, 29cdef Badness: 0.0241 ==Dominion== Commas: 26/25, 36/35, 56/55, 64/63 POTE generator: ~3/2 = 704.905 Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|] EDOs: 5, 12, 17c, 46cde Badness: 0.0273 ==Domineering== Commas: 36/35, 45/44, 64/63 POTE generator: ~3/2 = 698.776 Mapping generator: ~3 Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|] EDOs: 7, 12, 43de Badness: 0.0220 ==Domination== Commas: 36/35, 64/63, 77/75 POTE generator: ~3/2 = 705.004 Mapping generator: ~3 Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|] EDOs: 17c, 46cd Badness: 0.0366 ===13-limit=== Commas: 26/25, 36/35, 64/63, 66/65 POTE generator: ~3/2 = 705.496 Mapping generator: ~3 Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|] EDOs: 17c Badness: 0.0274 ==Twelve== Commas: 81/80 64/63 45/44 65/64 256/255 153/152 POTE generator: ~3/2 = 696.217 Mapping generator: ~3 Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|] EDOs: 7, 12, 19d, 31def Badness: 0.0204 ==Arnold== Commas: 22/21, 33/32, 36/35 POTE generator: ~3/2 = 698.491 Mapping generator: ~3 Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|] EDOs: 5, 7, 12e Badness: 0.0261 ==13-limit== Commas: 22/21, 27/26, 33/32, 40/39 POTE generator: ~3/2 = 696.743 Mapping generator: ~3 Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|] EDOs: 5, 7, 12ef, 19def, 31def Badness: 0.0233 ==Dominatrix== Commas: 27/26 36/35 45/44 64/63 POTE generator: ~3/2 = 698.544 Mapping generator: ~3 Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|] EDOs: 7, 12f Badness: 0.0183 =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 Mapping generator: ~3 Map: [<1 0 -4 -2|, <0 1 4 3|] [[Wedgie]]: <<1 4 3 4 2 -4|| EDOs: [[5edo|5]], [[12edo|12]] [[Badness]]: 0.0248 =Meansept= Commas: 15/14, 81/80 POTE generator: ~3/2 = 682.895 Mapping generator: ~3 Map: [<1 0 -4 -5|, <0 1 4 5|] Wedgie: <<1 4 5 4 5 0|| EDOs: 7 Badness: 0.0453 ==11-limit== Commas: 15/14, 22/21, 125/121 POTE generator: ~3/2 = 685.234 Mapping generator: ~3 Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|] EDOs: 7 Badness: 0.0325 =Supermean= Commas: 81/80, 672/625 POTE generator: ~3/2 = 704.889 Map: [<1 0 -4 -21|, <0 1 4 15|] EDOs: 17c, 46c Badness: 0.1342 ==11-limit== Commas: 56/55, 81/80, 132/125 POTE generator: ~3/2 = 705.096 Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|] EDOs: 17c, 46c Badness: 0.0633 ==13-limit== Commas: 26/25, 56/55, 66/65, 81/80 POTE generator: ~3/2 = 705.094 Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|] EDOs: 17c, 46c =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 [[19edo]]s, is an excellent tuning for injera. [[http://tech.groups.yahoo.com/group/tuning-math/message/3091|Origin of the name]] valid range: [685.714, 700.000] (14c to 12) nice range: [688.957, 701.955] strict range: [688.957, 700.000] [[POTE tuning|POTE generator]]: 694.375 Mapping generator: ~3 Map: [<2 0 -8 -7|, <0 1 4 4|] [[Wedgie]]: <<2 8 8 8 7 -4|| EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[102edo|102bcd]], [[140edo|140bcd]], [[178edo|178bcd]] [[Badness]]: 0.0311 [[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3|Two Pairs of Socks]] (in [[26edo]]) by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]] [[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3|Injera Jam]] (in [[26edo]]) by [[Zach Curley]] ==11-limit== Commas: 45/44, 50/49, 81/80 valid range: [685.714, 700.000] (14c to 12) nice range: [682.458, 701.955] strict range: [685.714, 700.000] POTE generator: ~3/2 = 692.840 Mapping generator: ~3 Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|] EDOs: 12, 14c, 26. 90bce, 116bce Badness: 0.0231 ==13-limit== Commas: 45/44, 50/49, 81/80, 78/77 valid range: 692.308 (26) nice range: [682.458, 701.955] strict range: 692.308 (26) POTE generator: ~3/2 = 692.673 Mapping generator: ~3 Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|] EDOs: 26, 104bcf Badness: 0.0216 ==Enjera== Commas: 27/26, 40/39, 45/44, 99/98 POTE generator: ~3/2 = 694.121 Mapping generator: ~3 Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|] EDOs: 12f, 26f, 38ef Badness: 0.0265 ==Injerous== Commas: 33/32, 50/49, 55/54 POTE generator: ~3/2 = 690.548 Mapping generator: ~3 Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|] EDOs: 12e, 14c, 26e, 40ce Badness: 0.0386 ==Lahoh== Commas: 50/49, 56/55, 81/77 POTE generator: ~3/2 = 699.001 Mapping generator: ~3 Map: [<2 0 -8 -7 7|, <0 1 4 4 0|] EDOs: 12 Badness: 0.0431 =Godzilla= <span style="display: block; text-align: right;">[[xenharmonie/Semiphor, Semaphor, Godzilla|Deutsch]] </span> Main article: [[Semaphore and 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-quarter intervals these represent give a fourth, and so step-and-a-quarter 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. valid range: [240.000, 257.143] (5 to 14c) nice range: [231.174, 266.871] strict range: [240.000, 257.143] [[POTE tuning|POTE generator]]: ~8/7 = 252.635 Mapping generator: ~7/4 Map: [<1 0 -4 2|, <0 2 8 1|] [[Wedgie]]: <<2 8 1 8 -4 -20|| EDOs: [[5edo|5]], [[9edo|9c]], [[14edo|14c]], [[19edo|19]], [[62edo|62d]], [[81edo|81d]], 143bd [[Badness]]: 0.0267 ==11-limit== Commas: 45/44, 49/48, 81/80 valid range: [252.632, 257.143] (19 to 14c) nice range: [231.174, 266.871] strict range: [252.632, 257.143] POTE generator: ~8/7 = 254.027 Mapping generator: ~7/4 Map: [<1 0 -4 2 -6|, <0 2 8 1 12|] EDOs: 14c, 19, 33cd, 52cd Badness: 0.0290 ==13-limit== Commas: 45/44, 49/48, 78/77, 81/80 valid range: 694.737 (19) nice range: [621.581, 737.652] strict range: 694.737 POTE generator: ~8/7 = 253.603 Mapping generator: ~7/4 Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|] EDOs: 14cf, 19, 33cdf, 52cdf Badness: 0.0225 ==Semafour== Commas: 33/32, 49/48, 55/54 POTE generator: ~8/7 = 254.042 Mapping generator: ~7/4 Map: [<1 0 -4 2 5|, <0 2 8 1 -2|] EDOs: 5, 14c, 19e, 33cde Badness: 0.0285 ==Varan== Commas: 49/48, 77/75, 81/80 POTE generator: ~8/7 = 251.079 Mapping generator: ~7/4 Map: [<1 0 -4 2 -10|, <0 2 8 1 17|] EDOs: 19e, 24, 43de Badness: 0.0396 ===13-limit=== Commas: 49/48, 66/65, 77/75, 81/80 POTE generator: ~8/7 = 251.165 Mapping generator: ~7/4 Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|] EDOs: 19e, 24, 43de Badness: 0.0257 ==Baragon== Commas: 49/48, 56/55, 81/80 POTE generator: ~8/7 = 251.173 Mapping generator: ~7/4 Map: [<1 0 -4 2 9|, <0 2 8 1 -7|] EDOs: 19, 24, 43d Badness: 0.0357 ==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= <span style="display: block; text-align: right;">[[xenharmonie/Mohajira|Deutsch]] </span> [[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. Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4. [[7-limit|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]]: ~128/105 = 348.415 Mapping generator: ~128/105 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]]: ~11/9 = 348.477 Mapping generator: ~11/9 Map: [<1 1 0 6 2|, <0 2 8 -11 5|] [[Generator]]s: 2, 11/9 EDOs: [[7edo|7]], [[24edo|24]], [[31edo|31]] [[Badness]]: 0.0261 ==13-limit== Commas: 81/80, 121/120, 105/104, 66/65 POTE generator: ~11/9 = 348.558 Mapping generator: ~11/9 Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|] EDOs: 7, 24, 31, 117ef, 148bef Badness: 0.0234 =Ptolemy= Commas: 81/80, 121/120, 525/512 POTE generator: ~11/9 = 346.922 Map: [<1 1 0 8 2|, <0 2 8 -18 5|] EDOs: 7, 38d, 45e, 83bcde Badness: 0.0588 ==13-limit== Commas: 65/64, 81/80, 105/104, 121/120 POTE generator: ~11/9 = 346.910 Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|] EDOs: 7, 38df, 45ef, 83bcdef Badness: 0.0343 =Maqamic= <span style="display: block; text-align: right;">[[xenharmonie/maqamisch|Deutsch]] </span> Main article: [[Maqamic]] [[Comma]]s: 81/80, 36/35, 121/120 Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation. [[POTE tuning|POTE generator]]: ~11/9 = 350.934 Mapping generator: ~11/9 Map: [<1 1 0 4 2|, <0 2 8 -4 5|] [[Generator]]s: 2, 11/9 EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]], [[31edo|31d]] ==13-limit== [[Comma]]s: 81/80, 36/35, 121/120, 144/143 [[POTE tuning|POTE generator]]: ~11/9 = 350.816 Mapping generator: ~11/9 Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|] Generators: 2, 11/9 EDOs: [[7edo|7]], [[10edo|10c]], [[17edo|17c]], [[24edo|24d]],[[31edo| 31d]] =Migration= Commas: 81/80, 121/120, 126/125 POTE generator: ~11/9 = 348.182 Mapping generator: ~11/9 Map: [<1 1 0 -3 2|, <0 2 8 20 5|] EDOs: 31, 100de, 131bde, 162bde Badness: 0.0255 =Mohamaq= Commas: 81/80, 392/375 POTE generator: ~25/21 = 350.586 Mapping generator: ~25/21 Map: [<1 1 0 -1|, <0 2 8 13|] EDOs: 17c, 24, 65c, 89cd Badness: 0.0777 ==11-limit== Commas: 56/55, 77/75, 243/242 POTE generator: ~11/9 = 350.565 Mapping generator: ~11/9 Map: [<1 1 0 -1 2|, <0 2 8 13 5|] EDOs: 17c, 24, 65c, 89cd Badness: 0.0362 ==13-limit== Commas: 56/55, 66/65, 77/75, 243/242 POTE generator: ~11/9 = 350.745 Mapping generator: ~11/9 Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|] EDOs: 17c, 24, 41c, 65c Badness: 0.0287 =Orphic= Commas: 81/80, 5898240/5764801 POTE generator: ~7/6 = 275.794 Mapping generator: ~343/288 Map: [<2 1 -4 4|, <0 4 16 3|] Wedgie: <<8 32 6 32 -13 -76|| EDOs: 26, 74, 174bd, 248bd Badness: 0.2588 ==11-limit== Commas: 81/80, 99/98, 73728/73205 POTE generator: ~7/6 = 275.762 Mapping generator: ~77/64 Map: [<2 1 -4 4 8|, <0 4 16 3 -2|] EDOs: 26, 48c, 74, 248bd, 322bd Badness: 0.1015 ==13-limit== Commas: 81/80, 99/98, 144/143, 2200/2197 POTE generator: ~7/6 = 275.774 Mapping generator: ~63/52 Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|] EDOs: 26, 48c, 74, 174bd, 248bd, 322bd Badness: 0.0535 =Mothra= [[Comma]]s: 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. In the 2.3.7-limit, mothra is identical to [[Slendric|slendric]]. Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes. [[7-limit|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>] [[Eigenmonzo]]s: 2, 5 [[POTE tuning|POTE generator]]: ~8/7 = 232.193 Mapping generator: ~8/7 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: [[5edo|5]], [[26edo|26]], [[31edo|31]] [[Badness]]: 0.0371 ==11-limit== [[Comma]]s: 81/80, 99/98, 385/384 POTE generator: ~8/7 = 232.031 Mapping generator: ~8/7 Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] EDOs: [[5edo|5]], [[26edo|26]], [[31edo|31]], [[88edo|88]], [[150edo|150]], [[181edo|181]] [[Badness]]: 0.0256 ==13-limit== Commas: 81/80, 99/98, 105/104, 144/143 POTE generator: ~8/7 = 231.811 Mapping generator: ~8/7 Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|] EDOs: 5, 26, 31, 57, 88 Badness: 0.0240 ==Cynder== Commas: 45/44, 81/80, 1029/1024 POTE generator: ~8/7 = 231.317 Mapping generator: ~8/7 Map: [<1 1 0 3 0|, <0 3 12 -1 18|] EDOs: 26, 57e, 83bce Badness: 0.0557 ===13-limit=== Commas: 45/44, 78/77, 81/80, 640/637 POTE generator: ~8/7 = 231.293 Mapping generator: ~8/7 Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|] EDOs: 26, 57e, 83bce Badness: 0.0341 ==Mosura== Commas: 81/80, 176/175, 1029/1024 POTE generator: ~8/7 = 232.419 Mapping generator: ~8/7 Map: [<1 1 0 3 -1|, <0 3 12 -1 23|] EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce Badness: 0.0313 ===13-limit=== Commas: 81/80, 144/143, 176/175, 1029/1024 POTE generator: ~8/7 = 232.640 Mapping generator: ~8/7 Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|] EDOs: 31, 67, 98 Badness: 0.0369 =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|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]]: ~9/7 = 425.942 Mapping generator: ~9/7 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]], [[31edo|31]], [[262edo|262]], [[293edo|293]] [[Badness]]: 0.0460 Music: By [[Chris Vaisvil]] [[http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3|Square 8]] ==11-limit== Commas: 81/80, 99/98, 121/120 POTE generator: ~9/7 = 425.957 Mapping generator: ~9/7 Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|] EDOs: [[5edo|5]], [[8edo|8]], [[11edo|11]], [[14edo|14]], [[17edo|17]], [[31edo|31]] [[Badness]]: 0.0216 ==13-limit== Commas: 81/80, 99/98, 121/120, 66/65 POTE generator: ~9/7 = 425.550 Mapping generator: ~9/7 Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|] EDOs: 17c, 31, 79cf, 110cef, 141cef [[Badness]]: 0.0255 ==Agora== Commas: 81/80, 99/98, 105/104, 121/120 POTE generator: ~9/7 = 426.276 Mapping generator: ~9/7 Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|] EDOs: 31, 45ef, 76e Badness: 0.0245 =Cuboctahedra= ==11-limit== [[Comma]]s: 81/80, 385/384, 1375/1372 [[POTE tuning|POTE generator]]: ~9/7 = 425.993 Mapping generator: ~9/7 Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] EDOs: [[14edo|14]], [[31edo|31]], [[45edo|45]], [[200edo|200]] [[Badness]]: 0.0568 =Liese= <span style="display: block; text-align: right;">[[xenharmonie/Liese|Deutsch]] </span> [[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]]: ~10/7 = 632.406 Mapping generator: ~10/7 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 ==Liesel== Commas: 56/55, 81/80, 540/539 POTE generator: ~10/7 = 633.073 Mapping generator: ~10/7 Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|] EDOs: 17c, 19, 36, 91ce Badness: 0.0407 ==13-limit== Liesel is a very natural 13-limit tuning, given the generator is so near 13/9. Commas: 56/55, 78/77, 81/80, 91/90 POTE generator: ~10/7 = ~13/9 = 633.042 Mapping generator: ~10/7 Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|] EDOs: 17c, 19, 36, 91cef Badness: 0.0273 ==Elisa== Commas: 77/75, 81/80, 99/98 POTE generator: ~10/7 = 633.061 Mapping generator: ~10/7 Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|] EDOs: 19e, 36e Badness: 0.0416 ==Lisa== Commas: 45/44, 81/80, 343/330 POTE generator: ~10/7 = 631.370 Mapping generator: ~10/7 Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|] EDOs: 19 Badness: 0.0548 ==13-limit== Commas: 45/44, 81/80, 91/88, 147/143 POTE generator: ~10/7 = 631.221 Mapping generator: ~10/7 Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|] EDOs: 19 Badness: 0.0361 =Jerome= Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size. Commas: 81/80, 17280/16807 POTE generator: ~54/49 = 139.343 Mapping generator: ~54/49 Map: [<1 1 0 2|, <0 5 20 7|] Wedgie: <<5 30 7 20 -3 -40|| EDOs: 8, 9, 17, 26, 43, 112 Badness: 0.1087 ==11-limit== Commas: 81/80, 99/98, 864/847 POTE generator: ~12/11 = 139.428 Mapping generator: ~12/11 Map: [<1 1 0 2 3|, <0 5 20 7 4|] EDOs: 8, 9, 17, 26, 43, 241 Badness: 0.0479 ==13-limit== Commas: 77/78, 81/80, 99/98, 144/143 POTE generator: ~13/12 = 139.387 Mapping generator: ~12/11 Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|] EDOs: 8, 9, 17, 26, 43, 155, 198 Badness: 0.0293 ==17-limit== Commas: 78/77, 81/80, 99/98, 144/143, 189/187 POTE generator: ~13/12 = 139.362 Mapping generator: ~12/11 Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|] EDOs: 8, 9, 17, 26, 43, 155 Badness: 0.0209 =Meanmag= Commas: 81/80, 3125/3072 POTE generator: ~8/7 = 238.396 Mapping generator: ~7 Map: [<19 30 44 0|, <0 0 0 1|] Wedgie: <<0 0 19 0 30 44|| EDOs: 19, 57, 76, 171bcd Badness: 0.0770 =Undevigintone= Commas: 49/48, 81/80, 126/125 POTE generator: ~11/8 = 538.047 Mapping generator: ~11 Map: [<19 30 44 53 0|, <0 0 0 0 1|] EDOs: 19, 38d Badness: 0.0364 ==13-limit== Commas: 49/48, 65/64, 81/80, 126/125 POTE generator: ~11/8 = 537.061 Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|] EDOs: 19, 38d Badness: 0.0229
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<html><head><title>Meantone family</title></head><body><span style="display: block; text-align: right;"><!-- ws:start:WikiTextTocRule:184:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:184 --><!-- ws:start:WikiTextTocRule:185: --><div style="margin-left: 2em;"><a href="#x-Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:185 --><!-- ws:start:WikiTextTocRule:186: --><div style="margin-left: 1em;"><a href="#Septimal meantone">Septimal meantone</a></div> <!-- ws:end:WikiTextTocRule:186 --><!-- ws:start:WikiTextTocRule:187: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Bimeantone">Bimeantone</a></div> <!-- ws:end:WikiTextTocRule:187 --><!-- ws:start:WikiTextTocRule:188: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Bimeantone-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:188 --><!-- ws:start:WikiTextTocRule:189: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens">Unidecimal meantone aka Huygens</a></div> <!-- ws:end:WikiTextTocRule:189 --><!-- ws:start:WikiTextTocRule:190: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone">Tridecimal meantone</a></div> <!-- ws:end:WikiTextTocRule:190 --><!-- ws:start:WikiTextTocRule:191: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone">Grosstone</a></div> <!-- ws:end:WikiTextTocRule:191 --><!-- ws:start:WikiTextTocRule:192: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Meridetone">Meridetone</a></div> <!-- ws:end:WikiTextTocRule:192 --><!-- ws:start:WikiTextTocRule:193: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone">Hemimeantone</a></div> <!-- ws:end:WikiTextTocRule:193 --><!-- ws:start:WikiTextTocRule:194: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanpop">Meanpop</a></div> <!-- ws:end:WikiTextTocRule:194 --><!-- ws:start:WikiTextTocRule:195: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanpop-13-limit Meanpop">13-limit Meanpop</a></div> <!-- ws:end:WikiTextTocRule:195 --><!-- ws:start:WikiTextTocRule:196: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanpop-Meanplop">Meanplop</a></div> <!-- ws:end:WikiTextTocRule:196 --><!-- ws:start:WikiTextTocRule:197: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanenneadecal">Meanenneadecal</a></div> <!-- ws:end:WikiTextTocRule:197 --><!-- ws:start:WikiTextTocRule:198: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanenneadecal-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:198 --><!-- ws:start:WikiTextTocRule:199: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanenneadecal-Vincenzo">Vincenzo</a></div> <!-- ws:end:WikiTextTocRule:199 --><!-- ws:start:WikiTextTocRule:200: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanundeci">Meanundeci</a></div> <!-- ws:end:WikiTextTocRule:200 --><!-- ws:start:WikiTextTocRule:201: --><div style="margin-left: 3em;"><a href="#Septimal meantone-Meanundeci-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:201 --><!-- ws:start:WikiTextTocRule:202: --><div style="margin-left: 2em;"><a href="#Septimal meantone-Meanundec">Meanundec</a></div> <!-- ws:end:WikiTextTocRule:202 --><!-- ws:start:WikiTextTocRule:203: --><div style="margin-left: 1em;"><a href="#Flattone">Flattone</a></div> <!-- ws:end:WikiTextTocRule:203 --><!-- ws:start:WikiTextTocRule:204: --><div style="margin-left: 2em;"><a href="#Flattone-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:204 --><!-- ws:start:WikiTextTocRule:205: --><div style="margin-left: 2em;"><a href="#Flattone-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:205 --><!-- ws:start:WikiTextTocRule:206: --><div style="margin-left: 1em;"><a href="#Dominant">Dominant</a></div> <!-- ws:end:WikiTextTocRule:206 --><!-- ws:start:WikiTextTocRule:207: --><div style="margin-left: 2em;"><a href="#Dominant-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:207 --><!-- ws:start:WikiTextTocRule:208: --><div style="margin-left: 2em;"><a href="#Dominant-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:208 --><!-- ws:start:WikiTextTocRule:209: --><div style="margin-left: 2em;"><a href="#Dominant-Dominion">Dominion</a></div> <!-- ws:end:WikiTextTocRule:209 --><!-- ws:start:WikiTextTocRule:210: --><div style="margin-left: 2em;"><a href="#Dominant-Domineering">Domineering</a></div> <!-- ws:end:WikiTextTocRule:210 --><!-- ws:start:WikiTextTocRule:211: --><div style="margin-left: 2em;"><a href="#Dominant-Domination">Domination</a></div> <!-- ws:end:WikiTextTocRule:211 --><!-- ws:start:WikiTextTocRule:212: --><div style="margin-left: 3em;"><a href="#Dominant-Domination-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:212 --><!-- ws:start:WikiTextTocRule:213: --><div style="margin-left: 2em;"><a href="#Dominant-Twelve">Twelve</a></div> <!-- ws:end:WikiTextTocRule:213 --><!-- ws:start:WikiTextTocRule:214: --><div style="margin-left: 2em;"><a href="#Dominant-Arnold">Arnold</a></div> <!-- ws:end:WikiTextTocRule:214 --><!-- ws:start:WikiTextTocRule:215: --><div style="margin-left: 2em;"><a href="#Dominant-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:215 --><!-- ws:start:WikiTextTocRule:216: --><div style="margin-left: 2em;"><a href="#Dominant-Dominatrix">Dominatrix</a></div> <!-- ws:end:WikiTextTocRule:216 --><!-- ws:start:WikiTextTocRule:217: --><div style="margin-left: 1em;"><a href="#Sharptone">Sharptone</a></div> <!-- ws:end:WikiTextTocRule:217 --><!-- ws:start:WikiTextTocRule:218: --><div style="margin-left: 1em;"><a href="#Meansept">Meansept</a></div> <!-- ws:end:WikiTextTocRule:218 --><!-- ws:start:WikiTextTocRule:219: --><div style="margin-left: 2em;"><a href="#Meansept-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:219 --><!-- ws:start:WikiTextTocRule:220: --><div style="margin-left: 1em;"><a href="#Supermean">Supermean</a></div> <!-- ws:end:WikiTextTocRule:220 --><!-- ws:start:WikiTextTocRule:221: --><div style="margin-left: 2em;"><a href="#Supermean-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:221 --><!-- ws:start:WikiTextTocRule:222: --><div style="margin-left: 2em;"><a href="#Supermean-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:222 --><!-- ws:start:WikiTextTocRule:223: --><div style="margin-left: 1em;"><a href="#Injera">Injera</a></div> <!-- ws:end:WikiTextTocRule:223 --><!-- ws:start:WikiTextTocRule:224: --><div style="margin-left: 2em;"><a href="#Injera-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:224 --><!-- ws:start:WikiTextTocRule:225: --><div style="margin-left: 2em;"><a href="#Injera-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:225 --><!-- ws:start:WikiTextTocRule:226: --><div style="margin-left: 2em;"><a href="#Injera-Enjera">Enjera</a></div> <!-- ws:end:WikiTextTocRule:226 --><!-- ws:start:WikiTextTocRule:227: --><div style="margin-left: 2em;"><a href="#Injera-Injerous">Injerous</a></div> <!-- ws:end:WikiTextTocRule:227 --><!-- ws:start:WikiTextTocRule:228: --><div style="margin-left: 2em;"><a href="#Injera-Lahoh">Lahoh</a></div> <!-- ws:end:WikiTextTocRule:228 --><!-- ws:start:WikiTextTocRule:229: --><div style="margin-left: 1em;"><a href="#Godzilla">Godzilla</a></div> <!-- ws:end:WikiTextTocRule:229 --><!-- ws:start:WikiTextTocRule:230: --><div style="margin-left: 2em;"><a href="#Godzilla-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:230 --><!-- ws:start:WikiTextTocRule:231: --><div style="margin-left: 2em;"><a href="#Godzilla-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:231 --><!-- ws:start:WikiTextTocRule:232: --><div style="margin-left: 2em;"><a href="#Godzilla-Semafour">Semafour</a></div> <!-- ws:end:WikiTextTocRule:232 --><!-- ws:start:WikiTextTocRule:233: --><div style="margin-left: 2em;"><a href="#Godzilla-Varan">Varan</a></div> <!-- ws:end:WikiTextTocRule:233 --><!-- ws:start:WikiTextTocRule:234: --><div style="margin-left: 3em;"><a href="#Godzilla-Varan-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:234 --><!-- ws:start:WikiTextTocRule:235: --><div style="margin-left: 2em;"><a href="#Godzilla-Baragon">Baragon</a></div> <!-- ws:end:WikiTextTocRule:235 --><!-- ws:start:WikiTextTocRule:236: --><div style="margin-left: 2em;"><a href="#Godzilla-Music">Music</a></div> <!-- ws:end:WikiTextTocRule:236 --><!-- ws:start:WikiTextTocRule:237: --><div style="margin-left: 1em;"><a href="#Mohajira">Mohajira</a></div> <!-- ws:end:WikiTextTocRule:237 --><!-- ws:start:WikiTextTocRule:238: --><div style="margin-left: 2em;"><a href="#Mohajira-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:238 --><!-- ws:start:WikiTextTocRule:239: --><div style="margin-left: 2em;"><a href="#Mohajira-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:239 --><!-- ws:start:WikiTextTocRule:240: --><div style="margin-left: 1em;"><a href="#Ptolemy">Ptolemy</a></div> <!-- ws:end:WikiTextTocRule:240 --><!-- ws:start:WikiTextTocRule:241: --><div style="margin-left: 2em;"><a href="#Ptolemy-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:241 --><!-- ws:start:WikiTextTocRule:242: --><div style="margin-left: 1em;"><a href="#Maqamic">Maqamic</a></div> <!-- ws:end:WikiTextTocRule:242 --><!-- ws:start:WikiTextTocRule:243: --><div style="margin-left: 2em;"><a href="#Maqamic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:243 --><!-- ws:start:WikiTextTocRule:244: --><div style="margin-left: 1em;"><a href="#Migration">Migration</a></div> <!-- ws:end:WikiTextTocRule:244 --><!-- ws:start:WikiTextTocRule:245: --><div style="margin-left: 1em;"><a href="#Mohamaq">Mohamaq</a></div> <!-- ws:end:WikiTextTocRule:245 --><!-- ws:start:WikiTextTocRule:246: --><div style="margin-left: 2em;"><a href="#Mohamaq-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:246 --><!-- ws:start:WikiTextTocRule:247: --><div style="margin-left: 2em;"><a href="#Mohamaq-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:247 --><!-- ws:start:WikiTextTocRule:248: --><div style="margin-left: 1em;"><a href="#Orphic">Orphic</a></div> <!-- ws:end:WikiTextTocRule:248 --><!-- ws:start:WikiTextTocRule:249: --><div style="margin-left: 2em;"><a href="#Orphic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:249 --><!-- ws:start:WikiTextTocRule:250: --><div style="margin-left: 2em;"><a href="#Orphic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:250 --><!-- ws:start:WikiTextTocRule:251: --><div style="margin-left: 1em;"><a href="#Mothra">Mothra</a></div> <!-- ws:end:WikiTextTocRule:251 --><!-- ws:start:WikiTextTocRule:252: --><div style="margin-left: 2em;"><a href="#Mothra-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:252 --><!-- ws:start:WikiTextTocRule:253: --><div style="margin-left: 2em;"><a href="#Mothra-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:253 --><!-- ws:start:WikiTextTocRule:254: --><div style="margin-left: 2em;"><a href="#Mothra-Cynder">Cynder</a></div> <!-- ws:end:WikiTextTocRule:254 --><!-- ws:start:WikiTextTocRule:255: --><div style="margin-left: 3em;"><a href="#Mothra-Cynder-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:255 --><!-- ws:start:WikiTextTocRule:256: --><div style="margin-left: 2em;"><a href="#Mothra-Mosura">Mosura</a></div> <!-- ws:end:WikiTextTocRule:256 --><!-- ws:start:WikiTextTocRule:257: --><div style="margin-left: 3em;"><a href="#Mothra-Mosura-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:257 --><!-- ws:start:WikiTextTocRule:258: --><div style="margin-left: 1em;"><a href="#Squares">Squares</a></div> <!-- ws:end:WikiTextTocRule:258 --><!-- ws:start:WikiTextTocRule:259: --><div style="margin-left: 2em;"><a href="#Squares-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:259 --><!-- ws:start:WikiTextTocRule:260: --><div style="margin-left: 2em;"><a href="#Squares-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:260 --><!-- ws:start:WikiTextTocRule:261: --><div style="margin-left: 2em;"><a href="#Squares-Agora">Agora</a></div> <!-- ws:end:WikiTextTocRule:261 --><!-- ws:start:WikiTextTocRule:262: --><div style="margin-left: 1em;"><a href="#Cuboctahedra">Cuboctahedra</a></div> <!-- ws:end:WikiTextTocRule:262 --><!-- ws:start:WikiTextTocRule:263: --><div style="margin-left: 2em;"><a href="#Cuboctahedra-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:263 --><!-- ws:start:WikiTextTocRule:264: --><div style="margin-left: 1em;"><a href="#Liese">Liese</a></div> <!-- ws:end:WikiTextTocRule:264 --><!-- ws:start:WikiTextTocRule:265: --><div style="margin-left: 2em;"><a href="#Liese-Liesel">Liesel</a></div> <!-- ws:end:WikiTextTocRule:265 --><!-- ws:start:WikiTextTocRule:266: --><div style="margin-left: 2em;"><a href="#Liese-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:266 --><!-- ws:start:WikiTextTocRule:267: --><div style="margin-left: 2em;"><a href="#Liese-Elisa">Elisa</a></div> <!-- ws:end:WikiTextTocRule:267 --><!-- ws:start:WikiTextTocRule:268: --><div style="margin-left: 2em;"><a href="#Liese-Lisa">Lisa</a></div> <!-- ws:end:WikiTextTocRule:268 --><!-- ws:start:WikiTextTocRule:269: --><div style="margin-left: 2em;"><a href="#Liese-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:269 --><!-- ws:start:WikiTextTocRule:270: --><div style="margin-left: 1em;"><a href="#Jerome">Jerome</a></div> <!-- ws:end:WikiTextTocRule:270 --><!-- ws:start:WikiTextTocRule:271: --><div style="margin-left: 2em;"><a href="#Jerome-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:271 --><!-- ws:start:WikiTextTocRule:272: --><div style="margin-left: 2em;"><a href="#Jerome-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:272 --><!-- ws:start:WikiTextTocRule:273: --><div style="margin-left: 2em;"><a href="#Jerome-17-limit">17-limit</a></div> <!-- ws:end:WikiTextTocRule:273 --><!-- ws:start:WikiTextTocRule:274: --><div style="margin-left: 1em;"><a href="#Meanmag">Meanmag</a></div> <!-- ws:end:WikiTextTocRule:274 --><!-- ws:start:WikiTextTocRule:275: --><div style="margin-left: 1em;"><a href="#Undevigintone">Undevigintone</a></div> <!-- ws:end:WikiTextTocRule:275 --><!-- ws:start:WikiTextTocRule:276: --><div style="margin-left: 2em;"><a href="#Undevigintone-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:276 --><!-- ws:start:WikiTextTocRule:277: --></div> <!-- ws:end:WikiTextTocRule:277 --><br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <a class="wiki_link" href="http://xenharmonie.wikispaces.com/mittelt%C3%B6nig">Deutsch</a><br /> </span><br /> The <a class="wiki_link" href="/5-limit">5-limit</a> 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>: ~3/2 = 696.239<br /> Mapping generator: ~3<br /> <br /> <a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [685.714, 720.000] (7 to 5)<br /> nice range: [694.786, 701.955] (1/3 comma to Pythagorean)<br /> strict range: [694.786, 701.955]<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<1 0 -4|, <0 1 4|]<br /> EDOs (patent val edo list is complete): <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="/24edo">24</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/36edo">36</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/43edo">43</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/50edo">50</a>, <a class="wiki_link" href="/55edo">55</a>, <a class="wiki_link" href="/57edo">57</a>, <a class="wiki_link" href="/62edo">62</a>, <a class="wiki_link" href="/67edo">67</a>, <a class="wiki_link" href="/69edo">69</a>, <a class="wiki_link" href="/74edo">74</a>, <a class="wiki_link" href="/76edo">76</a>, <a class="wiki_link" href="/81edo">81</a>, <a class="wiki_link" href="/86edo">86</a>, <a class="wiki_link" href="/88edo">88</a>, <a class="wiki_link" href="/93edo">93</a>, <a class="wiki_link" href="/98edo">98</a>, <a class="wiki_link" href="/100edo">100</a>, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/117edo">117</a>, <a class="wiki_link" href="/129edo">129</a>, <a class="wiki_link" href="/212edo">212b</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 <a class="wiki_link" href="/7-limit">7-limit</a> 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> <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/septimal-mittelt%C3%B6nig">Deutsch</a><br /> </span><br /> <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Septimal_meantone_temperament" rel="nofollow">Wikipedia article</a><br /> <br /> The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the <a class="wiki_link" href="/7_4">7/4</a> of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and <a class="wiki_link" href="/7_5">7/5</a>, 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="/Tuning%20Ranges%20of%20Regular%20Temperaments">valid range</a>: [694.737, 700.000] (19 to 12)<br /> nice range: [694.786, 701.955]<br /> strict range: [694.786, 700.000]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> Mapping generator: ~3<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 /> <a class="wiki_link" href="/Map">Map</a>: [<1 0 -4 -13|, <0 1 4 10|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 3<br /> <a class="wiki_link" href="/Wedgie">Wedgie</a>: <<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>, <a class="wiki_link" href="/143edo">143b</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-Bimeantone"></a><!-- ws:end:WikiTextHeadingRule:4 -->Bimeantone</h2> Commas: 81/80, 126/125, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.016<br /> <br /> Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|]<br /> EDOs: 12, 38d, 50<br /> Badness: 0.0381<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Septimal meantone-Bimeantone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h3> Commas: 81/80, 105/104, 126/125, 245/242<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 695.836<br /> <br /> Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|]<br /> EDOs: 12f, 50<br /> Badness: 0.0288<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Septimal meantone-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:8 -->Unidecimal meantone aka Huygens</h2> See also <a class="wiki_link" href="/Meantone%20vs%20meanpop">Meantone vs meanpop</a><br /> <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 /> valid range: [696.774, 700.000] (31 to 12)<br /> nice range: [691.202, 701.955]<br /> strict range: [696.774, 700.000]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> Mapping generator: ~3<br /> <br /> <a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> <br /> <a class="wiki_link" href="/Map">Map</a>: [<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">198be</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0170<br /> <br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3" rel="nofollow">Twinkle canon – 74 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:10 -->Tridecimal meantone</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 66/65, 81/80, 99/98, 105/104<br /> <br /> valid range: 697.674 (43)<br /> nice range: [691.202, 701.955]<br /> strict range: 697.674<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.642<br /> Mapping generator: ~3<br /> <br /> 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</a>, <a class="wiki_link" href="/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:12:<h3> --><h3 id="toc6"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Grosstone"></a><!-- ws:end:WikiTextHeadingRule:12 -->Grosstone</h3> Commas: 81/80, 99/98, 126/125, 144/143<br /> <br /> POTE generator: ~3/2 = 697.264<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|]<br /> EDOs: 12, 31, 43, 74<br /> Badness: 0.0259<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Meridetone"></a><!-- ws:end:WikiTextHeadingRule:14 -->Meridetone</h3> Commas: 78/77, 81/80, 99/98, 126/125<br /> <br /> POTE generator: ~3/2 = 697.529<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]<br /> EDOs: 43, 117df, 160bdf, 203bcdef<br /> Badness: 0.0264<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="Septimal meantone-Unidecimal meantone aka Huygens-Hemimeantone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Hemimeantone</h3> Commas: 81/80, 99/98, 126/125, 169/168<br /> <br /> POTE generator: ~52/45 = 250.304<br /> Mapping generator: ~26/15<br /> <br /> Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|]<br /> EDOs: 43, 62, 167bef, 229bef<br /> Badness: 0.0314<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Septimal meantone-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:18 -->Meanpop</h2> See also <a class="wiki_link" href="/Meantone%20vs%20meanpop">Meantone vs meanpop</a><br /> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 126/125, 385/384<br /> <br /> <a class="wiki_link" href="/11-limit">11-limit</a> <a class="wiki_link" href="/minimax">minimax</a> 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 /> valid range: [694.737, 696.774] (19 to 31)<br /> nice range: [691.202, 701.955]<br /> strict range: [694.737, 696.774]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> Mapping generator: ~3<br /> <br /> <a class="wiki_link" href="/Algebraic%20generator">Algebraic generator</a>: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> <br /> <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/607" rel="nofollow" target="_blank">Scott Joplin's "The Entertainer" tuned into meanpop</a><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: <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>, <a class="wiki_link" href="/112edo">112</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0215<br /> <br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3" rel="nofollow">Twinkle canon – 50 edo</a> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h3> --><h3 id="toc10"><a name="Septimal meantone-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit Meanpop</h3> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 105/104, 144/143, 196/195<br /> <br /> valid range: [694.737, 696.774] (19 to 31)<br /> nice range: [691.202, 701.955]<br /> strict range: [694.737, 696.774]<br /> <br /> POTE generator: ~3/2 = 696.211<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]<br /> EDOS: <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>, <a class="wiki_link" href="/131edo">131bd</a>, <a class="wiki_link" href="/212edo">212bdf</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0209<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h3> --><h3 id="toc11"><a name="Septimal meantone-Meanpop-Meanplop"></a><!-- ws:end:WikiTextHeadingRule:22 -->Meanplop</h3> Commas: 65/64, 78/77, 81/80, 91/90<br /> <br /> POTE generator: ~3/2 = 696.202<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|]<br /> EDOs: 12e, 19, 31f, 50f<br /> Badness: 0.0277<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="Septimal meantone-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:24 -->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 /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]<br /> EDOs: <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">31e</a>, <a class="wiki_link" href="/50edo">50e</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0214<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h3> --><h3 id="toc13"><a name="Septimal meantone-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->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 /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]<br /> EDOs: <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/31edo">31e</a>, <a class="wiki_link" href="/50edo">50e</a>]<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0212<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h3> --><h3 id="toc14"><a name="Septimal meantone-Meanenneadecal-Vincenzo"></a><!-- ws:end:WikiTextHeadingRule:28 -->Vincenzo</h3> Commas: 81/80 126/125 45/44 65/64 256/255 153/152 23/22<br /> <br /> POTE generator: ~3/2<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 ... |, <0 1 4 10 6 -4 -5 -3 -6|]<br /> EDOs: 12<br /> Badness:<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="Septimal meantone-Meanundeci"></a><!-- ws:end:WikiTextHeadingRule:30 -->Meanundeci</h2> Commas: 33/32, 55/54, 77/75<br /> <br /> POTE generator: ~3/2 = 694.689<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|]<br /> EDOs: 12e, 19e<br /> Badness: 0.0315<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h3> --><h3 id="toc16"><a name="Septimal meantone-Meanundeci-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h3> Commas: 33/32, 55/54, 77/75, 729/728<br /> <br /> POTE generator: ~3/2 = 694.764<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|]<br /> EDOs: 12e, 19e<br /> Badness: 0.0263<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Septimal meantone-Meanundec"></a><!-- ws:end:WikiTextHeadingRule:34 -->Meanundec</h2> Commas: 27/26, 40/39, 45/44, 56/55<br /> <br /> POTE generator: ~3/2 = 697.254<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|]<br /> EDOS: 12f, 19f, 31ef<br /> Badness: 0.0242<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h1> --><h1 id="toc18"><a name="Flattone"></a><!-- ws:end:WikiTextHeadingRule:36 -->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 <a class="wiki_link" href="/7_4">7/4</a> is a diminished seventh interval. Other intervals are <a class="wiki_link" href="/7_6">7/6</a>, a diminished third, and <a class="wiki_link" href="/7_5">7/5</a>, 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 /> valid range: [692.308, 694.737] (26 to 19)<br /> nice range: [692.353, 701.955]<br /> strict range: [692.353, 694.737]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> Mapping generator: ~3<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: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/19edo">19</a>, <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:38:<h2> --><h2 id="toc19"><a name="Flattone-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2> Commas: 45/44, 81/80, 385/384<br /> <br /> valid range: [692.308, 694.737] (26 to 19)<br /> nice range: [682.502, 701.955]<br /> strict range: [692.308, 694.737]<br /> <br /> POTE generator: ~3/2 = 693.126<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|]<br /> EDOs: 7, 19, 26, 45, 71bc, 116bcde<br /> Badness: 0.0338<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Flattone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2> 45/44, 65/64, 78/77, 81/80<br /> <br /> valid range: [692.308, 694.737] (26 to 19)<br /> nice range: [682.502, 701.955]<br /> strict range: [692.308, 694.737]<br /> <br /> POTE generator: ~3/2 = 693.058<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|]<br /> EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef<br /> Badness: 0.0223<br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h1> --><h1 id="toc21"><a name="Dominant"></a><!-- ws:end:WikiTextHeadingRule:42 -->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 <a class="wiki_link" href="/3_2">3/2</a> 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 /> valid range: [700.000, 720.000] (12 to 5)<br /> nice range: [694.786, 715.587]<br /> strict range: [700.000, 715.587]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br /> Mapping generator: ~3<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: <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="/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:44:<h2> --><h2 id="toc22"><a name="Dominant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:44 -->11-limit</h2> Commas: 36/35, 64/63, 56/55<br /> <br /> valid range: [700.000, 705.882] (12 to 17)<br /> nice range: [691.202, 715.587]<br /> strict range: [700.000, 705.882]<br /> <br /> POTE generator: ~3/2 = 703.254<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|]<br /> EDOs: 5, 12, 17c, 29cde<br /> Badness: 0.0242<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Dominant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->13-limit</h2> Commas: 36/35, 56/55, 64/63, 66/65<br /> <br /> valid range: 705.882 (17)<br /> nice range: [691.202, 715.587]<br /> strict range:705.882<br /> <br /> POTE generator: ~3/2 = 703.636<br /> <br /> Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|]<br /> EDOs: 12f, 17c, 29cdef<br /> Badness: 0.0241<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h2> --><h2 id="toc24"><a name="Dominant-Dominion"></a><!-- ws:end:WikiTextHeadingRule:48 -->Dominion</h2> Commas: 26/25, 36/35, 56/55, 64/63<br /> <br /> POTE generator: ~3/2 = 704.905<br /> <br /> Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|]<br /> EDOs: 5, 12, 17c, 46cde<br /> Badness: 0.0273<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h2> --><h2 id="toc25"><a name="Dominant-Domineering"></a><!-- ws:end:WikiTextHeadingRule:50 -->Domineering</h2> Commas: 36/35, 45/44, 64/63<br /> <br /> POTE generator: ~3/2 = 698.776<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|]<br /> EDOs: 7, 12, 43de<br /> Badness: 0.0220<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h2> --><h2 id="toc26"><a name="Dominant-Domination"></a><!-- ws:end:WikiTextHeadingRule:52 -->Domination</h2> Commas: 36/35, 64/63, 77/75<br /> <br /> POTE generator: ~3/2 = 705.004<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|]<br /> EDOs: 17c, 46cd<br /> Badness: 0.0366<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h3> --><h3 id="toc27"><a name="Dominant-Domination-13-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->13-limit</h3> Commas: 26/25, 36/35, 64/63, 66/65<br /> <br /> POTE generator: ~3/2 = 705.496<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|]<br /> EDOs: 17c<br /> Badness: 0.0274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Dominant-Twelve"></a><!-- ws:end:WikiTextHeadingRule:56 -->Twelve</h2> Commas: 81/80 64/63 45/44 65/64 256/255 153/152<br /> <br /> POTE generator: ~3/2 = 696.217<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|]<br /> EDOs: 7, 12, 19d, 31def<br /> Badness: 0.0204<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h2> --><h2 id="toc29"><a name="Dominant-Arnold"></a><!-- ws:end:WikiTextHeadingRule:58 -->Arnold</h2> Commas: 22/21, 33/32, 36/35<br /> <br /> POTE generator: ~3/2 = 698.491<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|]<br /> EDOs: 5, 7, 12e<br /> Badness: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc30"><a name="Dominant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->13-limit</h2> Commas: 22/21, 27/26, 33/32, 40/39<br /> <br /> POTE generator: ~3/2 = 696.743<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|]<br /> EDOs: 5, 7, 12ef, 19def, 31def<br /> Badness: 0.0233<br /> <br /> <!-- ws:start:WikiTextHeadingRule:62:<h2> --><h2 id="toc31"><a name="Dominant-Dominatrix"></a><!-- ws:end:WikiTextHeadingRule:62 -->Dominatrix</h2> Commas: 27/26 36/35 45/44 64/63<br /> <br /> POTE generator: ~3/2 = 698.544<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|]<br /> EDOs: 7, 12f<br /> Badness: 0.0183<br /> <br /> <!-- ws:start:WikiTextHeadingRule:64:<h1> --><h1 id="toc32"><a name="Sharptone"></a><!-- ws:end:WikiTextHeadingRule:64 -->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 /> Mapping generator: ~3<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: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/12edo">12</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0248<br /> <br /> <!-- ws:start:WikiTextHeadingRule:66:<h1> --><h1 id="toc33"><a name="Meansept"></a><!-- ws:end:WikiTextHeadingRule:66 -->Meansept</h1> Commas: 15/14, 81/80<br /> <br /> POTE generator: ~3/2 = 682.895<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -5|, <0 1 4 5|]<br /> Wedgie: <<1 4 5 4 5 0||<br /> EDOs: 7<br /> Badness: 0.0453<br /> <br /> <!-- ws:start:WikiTextHeadingRule:68:<h2> --><h2 id="toc34"><a name="Meansept-11-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->11-limit</h2> Commas: 15/14, 22/21, 125/121<br /> <br /> POTE generator: ~3/2 = 685.234<br /> Mapping generator: ~3<br /> <br /> Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|]<br /> EDOs: 7<br /> Badness: 0.0325<br /> <br /> <!-- ws:start:WikiTextHeadingRule:70:<h1> --><h1 id="toc35"><a name="Supermean"></a><!-- ws:end:WikiTextHeadingRule:70 -->Supermean</h1> Commas: 81/80, 672/625<br /> <br /> POTE generator: ~3/2 = 704.889<br /> <br /> Map: [<1 0 -4 -21|, <0 1 4 15|]<br /> EDOs: 17c, 46c<br /> Badness: 0.1342<br /> <br /> <!-- ws:start:WikiTextHeadingRule:72:<h2> --><h2 id="toc36"><a name="Supermean-11-limit"></a><!-- ws:end:WikiTextHeadingRule:72 -->11-limit</h2> Commas: 56/55, 81/80, 132/125<br /> <br /> POTE generator: ~3/2 = 705.096<br /> <br /> Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|]<br /> EDOs: 17c, 46c<br /> Badness: 0.0633<br /> <br /> <!-- ws:start:WikiTextHeadingRule:74:<h2> --><h2 id="toc37"><a name="Supermean-13-limit"></a><!-- ws:end:WikiTextHeadingRule:74 -->13-limit</h2> Commas: 26/25, 56/55, 66/65, 81/80<br /> <br /> POTE generator: ~3/2 = 705.094<br /> <br /> Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|]<br /> EDOs: 17c, 46c<br /> <br /> <!-- ws:start:WikiTextHeadingRule:76:<h1> --><h1 id="toc38"><a name="Injera"></a><!-- ws:end:WikiTextHeadingRule:76 -->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 <a class="wiki_link" href="/19edo">19edo</a>s, is an excellent tuning for injera.<br /> <br /> <a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/3091" rel="nofollow">Origin of the name</a><br /> <br /> valid range: [685.714, 700.000] (14c to 12)<br /> nice range: [688.957, 701.955]<br /> strict range: [688.957, 700.000]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br /> Mapping generator: ~3<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="/102edo">102bcd</a>, <a class="wiki_link" href="/140edo">140bcd</a>, <a class="wiki_link" href="/178edo">178bcd</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0311<br /> <br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3" rel="nofollow">Two Pairs of Socks</a> (in <a class="wiki_link" href="/26edo">26edo</a>) by <a class="wiki_link" href="/Igliashon%20Jones">Igliashon Calvin Jones-Coolidge</a><br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20Injera%20Jam.mp3" rel="nofollow">Injera Jam</a> (in <a class="wiki_link" href="/26edo">26edo</a>) by <a class="wiki_link" href="/Zach%20Curley">Zach Curley</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:78:<h2> --><h2 id="toc39"><a name="Injera-11-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->11-limit</h2> Commas: 45/44, 50/49, 81/80<br /> <br /> valid range: [685.714, 700.000] (14c to 12)<br /> nice range: [682.458, 701.955]<br /> strict range: [685.714, 700.000]<br /> <br /> POTE generator: ~3/2 = 692.840<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|]<br /> EDOs: 12, 14c, 26. 90bce, 116bce<br /> Badness: 0.0231<br /> <br /> <!-- ws:start:WikiTextHeadingRule:80:<h2> --><h2 id="toc40"><a name="Injera-13-limit"></a><!-- ws:end:WikiTextHeadingRule:80 -->13-limit</h2> Commas: 45/44, 50/49, 81/80, 78/77<br /> <br /> valid range: 692.308 (26)<br /> nice range: [682.458, 701.955]<br /> strict range: 692.308 (26)<br /> <br /> POTE generator: ~3/2 = 692.673<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|]<br /> EDOs: 26, 104bcf<br /> Badness: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:82:<h2> --><h2 id="toc41"><a name="Injera-Enjera"></a><!-- ws:end:WikiTextHeadingRule:82 -->Enjera</h2> Commas: 27/26, 40/39, 45/44, 99/98<br /> <br /> POTE generator: ~3/2 = 694.121<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|]<br /> EDOs: 12f, 26f, 38ef<br /> Badness: 0.0265<br /> <br /> <!-- ws:start:WikiTextHeadingRule:84:<h2> --><h2 id="toc42"><a name="Injera-Injerous"></a><!-- ws:end:WikiTextHeadingRule:84 -->Injerous</h2> Commas: 33/32, 50/49, 55/54<br /> <br /> POTE generator: ~3/2 = 690.548<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|]<br /> EDOs: 12e, 14c, 26e, 40ce<br /> Badness: 0.0386<br /> <br /> <!-- ws:start:WikiTextHeadingRule:86:<h2> --><h2 id="toc43"><a name="Injera-Lahoh"></a><!-- ws:end:WikiTextHeadingRule:86 -->Lahoh</h2> Commas: 50/49, 56/55, 81/77<br /> <br /> POTE generator: ~3/2 = 699.001<br /> Mapping generator: ~3<br /> <br /> Map: [<2 0 -8 -7 7|, <0 1 4 4 0|]<br /> EDOs: 12<br /> Badness: 0.0431<br /> <br /> <!-- ws:start:WikiTextHeadingRule:88:<h1> --><h1 id="toc44"><a name="Godzilla"></a><!-- ws:end:WikiTextHeadingRule:88 -->Godzilla</h1> <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Semiphor%2C%20Semaphor%2C%20Godzilla">Deutsch</a><br /> </span><br /> Main article: <a class="wiki_link" href="/Semaphore%20and%20Godzilla">Semaphore and Godzilla</a><br /> <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-quarter intervals these represent give a fourth, and so step-and-a-quarter 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 /> valid range: [240.000, 257.143] (5 to 14c)<br /> nice range: [231.174, 266.871]<br /> strict range: [240.000, 257.143]<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 252.635<br /> Mapping generator: ~7/4<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">9c</a>, <a class="wiki_link" href="/14edo">14c</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/62edo">62d</a>, <a class="wiki_link" href="/81edo">81d</a>, 143bd<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0267<br /> <br /> <!-- ws:start:WikiTextHeadingRule:90:<h2> --><h2 id="toc45"><a name="Godzilla-11-limit"></a><!-- ws:end:WikiTextHeadingRule:90 -->11-limit</h2> Commas: 45/44, 49/48, 81/80<br /> <br /> valid range: [252.632, 257.143] (19 to 14c)<br /> nice range: [231.174, 266.871]<br /> strict range: [252.632, 257.143]<br /> <br /> POTE generator: ~8/7 = 254.027<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 -6|, <0 2 8 1 12|]<br /> EDOs: 14c, 19, 33cd, 52cd<br /> Badness: 0.0290<br /> <br /> <!-- ws:start:WikiTextHeadingRule:92:<h2> --><h2 id="toc46"><a name="Godzilla-13-limit"></a><!-- ws:end:WikiTextHeadingRule:92 -->13-limit</h2> Commas: 45/44, 49/48, 78/77, 81/80<br /> <br /> valid range: 694.737 (19)<br /> nice range: [621.581, 737.652]<br /> strict range: 694.737<br /> <br /> POTE generator: ~8/7 = 253.603<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|]<br /> EDOs: 14cf, 19, 33cdf, 52cdf<br /> Badness: 0.0225<br /> <br /> <!-- ws:start:WikiTextHeadingRule:94:<h2> --><h2 id="toc47"><a name="Godzilla-Semafour"></a><!-- ws:end:WikiTextHeadingRule:94 -->Semafour</h2> Commas: 33/32, 49/48, 55/54<br /> <br /> POTE generator: ~8/7 = 254.042<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 5|, <0 2 8 1 -2|]<br /> EDOs: 5, 14c, 19e, 33cde<br /> Badness: 0.0285<br /> <br /> <!-- ws:start:WikiTextHeadingRule:96:<h2> --><h2 id="toc48"><a name="Godzilla-Varan"></a><!-- ws:end:WikiTextHeadingRule:96 -->Varan</h2> Commas: 49/48, 77/75, 81/80<br /> <br /> POTE generator: ~8/7 = 251.079<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 -10|, <0 2 8 1 17|]<br /> EDOs: 19e, 24, 43de<br /> Badness: 0.0396<br /> <br /> <!-- ws:start:WikiTextHeadingRule:98:<h3> --><h3 id="toc49"><a name="Godzilla-Varan-13-limit"></a><!-- ws:end:WikiTextHeadingRule:98 -->13-limit</h3> Commas: 49/48, 66/65, 77/75, 81/80<br /> <br /> POTE generator: ~8/7 = 251.165<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|]<br /> EDOs: 19e, 24, 43de<br /> Badness: 0.0257<br /> <br /> <!-- ws:start:WikiTextHeadingRule:100:<h2> --><h2 id="toc50"><a name="Godzilla-Baragon"></a><!-- ws:end:WikiTextHeadingRule:100 -->Baragon</h2> Commas: 49/48, 56/55, 81/80<br /> <br /> POTE generator: ~8/7 = 251.173<br /> Mapping generator: ~7/4<br /> <br /> Map: [<1 0 -4 2 9|, <0 2 8 1 -7|]<br /> EDOs: 19, 24, 43d<br /> Badness: 0.0357<br /> <br /> <!-- ws:start:WikiTextHeadingRule:102:<h2> --><h2 id="toc51"><a name="Godzilla-Music"></a><!-- ws:end:WikiTextHeadingRule:102 -->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:104:<h1> --><h1 id="toc52"><a name="Mohajira"></a><!-- ws:end:WikiTextHeadingRule:104 -->Mohajira</h1> <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Mohajira">Deutsch</a><br /> </span><br /> <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 /> Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.<br /> <br /> <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> 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>: ~128/105 = 348.415<br /> Mapping generator: ~128/105<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:106:<h2> --><h2 id="toc53"><a name="Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:106 -->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>: ~11/9 = 348.477<br /> Mapping generator: ~11/9<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: <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>: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:108:<h2> --><h2 id="toc54"><a name="Mohajira-13-limit"></a><!-- ws:end:WikiTextHeadingRule:108 -->13-limit</h2> Commas: 81/80, 121/120, 105/104, 66/65<br /> <br /> POTE generator: ~11/9 = 348.558<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]<br /> EDOs: 7, 24, 31, 117ef, 148bef<br /> Badness: 0.0234<br /> <br /> <!-- ws:start:WikiTextHeadingRule:110:<h1> --><h1 id="toc55"><a name="Ptolemy"></a><!-- ws:end:WikiTextHeadingRule:110 -->Ptolemy</h1> Commas: 81/80, 121/120, 525/512<br /> <br /> POTE generator: ~11/9 = 346.922<br /> <br /> Map: [<1 1 0 8 2|, <0 2 8 -18 5|]<br /> EDOs: 7, 38d, 45e, 83bcde<br /> Badness: 0.0588<br /> <br /> <!-- ws:start:WikiTextHeadingRule:112:<h2> --><h2 id="toc56"><a name="Ptolemy-13-limit"></a><!-- ws:end:WikiTextHeadingRule:112 -->13-limit</h2> Commas: 65/64, 81/80, 105/104, 121/120<br /> <br /> POTE generator: ~11/9 = 346.910<br /> <br /> Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|]<br /> EDOs: 7, 38df, 45ef, 83bcdef<br /> Badness: 0.0343<br /> <br /> <!-- ws:start:WikiTextHeadingRule:114:<h1> --><h1 id="toc57"><a name="Maqamic"></a><!-- ws:end:WikiTextHeadingRule:114 -->Maqamic</h1> <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/maqamisch">Deutsch</a><br /> </span><br /> Main article: <a class="wiki_link" href="/Maqamic">Maqamic</a><br /> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120<br /> <br /> Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~11/9 = 350.934<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 4 2|, <0 2 8 -4 5|]<br /> <a class="wiki_link" href="/Generator">Generator</a>s: 2, 11/9<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>, <a class="wiki_link" href="/31edo">31d</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:116:<h2> --><h2 id="toc58"><a name="Maqamic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:116 -->13-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 36/35, 121/120, 144/143<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~11/9 = 350.816<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]<br /> Generators: 2, 11/9<br /> EDOs: <a class="wiki_link" href="/7edo">7</a>, <a class="wiki_link" href="/10edo">10c</a>, <a class="wiki_link" href="/17edo">17c</a>, <a class="wiki_link" href="/24edo">24d</a>,<a class="wiki_link" href="/31edo"> 31d</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:118:<h1> --><h1 id="toc59"><a name="Migration"></a><!-- ws:end:WikiTextHeadingRule:118 -->Migration</h1> Commas: 81/80, 121/120, 126/125<br /> <br /> POTE generator: ~11/9 = 348.182<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 -3 2|, <0 2 8 20 5|]<br /> EDOs: 31, 100de, 131bde, 162bde<br /> Badness: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:120:<h1> --><h1 id="toc60"><a name="Mohamaq"></a><!-- ws:end:WikiTextHeadingRule:120 -->Mohamaq</h1> Commas: 81/80, 392/375<br /> <br /> POTE generator: ~25/21 = 350.586<br /> Mapping generator: ~25/21<br /> <br /> Map: [<1 1 0 -1|, <0 2 8 13|]<br /> EDOs: 17c, 24, 65c, 89cd<br /> Badness: 0.0777<br /> <br /> <!-- ws:start:WikiTextHeadingRule:122:<h2> --><h2 id="toc61"><a name="Mohamaq-11-limit"></a><!-- ws:end:WikiTextHeadingRule:122 -->11-limit</h2> Commas: 56/55, 77/75, 243/242<br /> <br /> POTE generator: ~11/9 = 350.565<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 -1 2|, <0 2 8 13 5|]<br /> EDOs: 17c, 24, 65c, 89cd<br /> Badness: 0.0362<br /> <br /> <!-- ws:start:WikiTextHeadingRule:124:<h2> --><h2 id="toc62"><a name="Mohamaq-13-limit"></a><!-- ws:end:WikiTextHeadingRule:124 -->13-limit</h2> Commas: 56/55, 66/65, 77/75, 243/242<br /> <br /> POTE generator: ~11/9 = 350.745<br /> Mapping generator: ~11/9<br /> <br /> Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|]<br /> EDOs: 17c, 24, 41c, 65c<br /> Badness: 0.0287<br /> <br /> <!-- ws:start:WikiTextHeadingRule:126:<h1> --><h1 id="toc63"><a name="Orphic"></a><!-- ws:end:WikiTextHeadingRule:126 -->Orphic</h1> Commas: 81/80, 5898240/5764801<br /> <br /> POTE generator: ~7/6 = 275.794<br /> Mapping generator: ~343/288<br /> <br /> Map: [<2 1 -4 4|, <0 4 16 3|]<br /> Wedgie: <<8 32 6 32 -13 -76||<br /> EDOs: 26, 74, 174bd, 248bd<br /> Badness: 0.2588<br /> <br /> <!-- ws:start:WikiTextHeadingRule:128:<h2> --><h2 id="toc64"><a name="Orphic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:128 -->11-limit</h2> Commas: 81/80, 99/98, 73728/73205<br /> <br /> POTE generator: ~7/6 = 275.762<br /> Mapping generator: ~77/64<br /> <br /> Map: [<2 1 -4 4 8|, <0 4 16 3 -2|]<br /> EDOs: 26, 48c, 74, 248bd, 322bd<br /> Badness: 0.1015<br /> <br /> <!-- ws:start:WikiTextHeadingRule:130:<h2> --><h2 id="toc65"><a name="Orphic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:130 -->13-limit</h2> Commas: 81/80, 99/98, 144/143, 2200/2197<br /> <br /> POTE generator: ~7/6 = 275.774<br /> Mapping generator: ~63/52<br /> <br /> Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|]<br /> EDOs: 26, 48c, 74, 174bd, 248bd, 322bd<br /> Badness: 0.0535<br /> <br /> <!-- ws:start:WikiTextHeadingRule:132:<h1> --><h1 id="toc66"><a name="Mothra"></a><!-- ws:end:WikiTextHeadingRule:132 -->Mothra</h1> <a class="wiki_link" href="/Comma">Comma</a>s: 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. In the 2.3.7-limit, mothra is identical to <a class="wiki_link" href="/Slendric">slendric</a>.<br /> Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.<br /> <br /> <a class="wiki_link" href="/7-limit">7</a> and <a class="wiki_link" href="/9-limit">9-limit</a> minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 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>: ~8/7 = 232.193<br /> Mapping generator: ~8/7<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: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0371<br /> <br /> <!-- ws:start:WikiTextHeadingRule:134:<h2> --><h2 id="toc67"><a name="Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:134 -->11-limit</h2> <a class="wiki_link" href="/Comma">Comma</a>s: 81/80, 99/98, 385/384<br /> <br /> POTE generator: ~8/7 = 232.031<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/31edo">31</a>, <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:136:<h2> --><h2 id="toc68"><a name="Mothra-13-limit"></a><!-- ws:end:WikiTextHeadingRule:136 -->13-limit</h2> Commas: 81/80, 99/98, 105/104, 144/143<br /> <br /> POTE generator: ~8/7 = 231.811<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|]<br /> EDOs: 5, 26, 31, 57, 88<br /> Badness: 0.0240<br /> <br /> <!-- ws:start:WikiTextHeadingRule:138:<h2> --><h2 id="toc69"><a name="Mothra-Cynder"></a><!-- ws:end:WikiTextHeadingRule:138 -->Cynder</h2> Commas: 45/44, 81/80, 1029/1024<br /> <br /> POTE generator: ~8/7 = 231.317<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 0|, <0 3 12 -1 18|]<br /> EDOs: 26, 57e, 83bce<br /> Badness: 0.0557<br /> <br /> <!-- ws:start:WikiTextHeadingRule:140:<h3> --><h3 id="toc70"><a name="Mothra-Cynder-13-limit"></a><!-- ws:end:WikiTextHeadingRule:140 -->13-limit</h3> Commas: 45/44, 78/77, 81/80, 640/637<br /> <br /> POTE generator: ~8/7 = 231.293<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|]<br /> EDOs: 26, 57e, 83bce<br /> Badness: 0.0341<br /> <br /> <!-- ws:start:WikiTextHeadingRule:142:<h2> --><h2 id="toc71"><a name="Mothra-Mosura"></a><!-- ws:end:WikiTextHeadingRule:142 -->Mosura</h2> Commas: 81/80, 176/175, 1029/1024<br /> <br /> POTE generator: ~8/7 = 232.419<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 -1|, <0 3 12 -1 23|]<br /> EDOs: 31, 129, 136b, 148be, 160be, 191bce, 222bce, 253bce<br /> Badness: 0.0313<br /> <br /> <!-- ws:start:WikiTextHeadingRule:144:<h3> --><h3 id="toc72"><a name="Mothra-Mosura-13-limit"></a><!-- ws:end:WikiTextHeadingRule:144 -->13-limit</h3> Commas: 81/80, 144/143, 176/175, 1029/1024<br /> <br /> POTE generator: ~8/7 = 232.640<br /> Mapping generator: ~8/7<br /> <br /> Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|]<br /> EDOs: 31, 67, 98<br /> Badness: 0.0369<br /> <br /> <!-- ws:start:WikiTextHeadingRule:146:<h1> --><h1 id="toc73"><a name="Squares"></a><!-- ws:end:WikiTextHeadingRule:146 -->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 (<a class="wiki_link" href="/9_7">9/7</a>) 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>: ~9/7 = 425.942<br /> Mapping generator: ~9/7<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>, <a class="wiki_link" href="/31edo">31</a>, <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 /> <a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/tuning-survey/daily20100603-squares8piano.mp3" rel="nofollow">Square 8</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:148:<h2> --><h2 id="toc74"><a name="Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:148 -->11-limit</h2> Commas: 81/80, 99/98, 121/120<br /> <br /> POTE generator: ~9/7 = 425.957<br /> Mapping generator: ~9/7<br /> <br /> Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]<br /> EDOs: <a class="wiki_link" href="/5edo">5</a>, <a class="wiki_link" href="/8edo">8</a>, <a class="wiki_link" href="/11edo">11</a>, <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/17edo">17</a>, <a class="wiki_link" href="/31edo">31</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0216<br /> <br /> <!-- ws:start:WikiTextHeadingRule:150:<h2> --><h2 id="toc75"><a name="Squares-13-limit"></a><!-- ws:end:WikiTextHeadingRule:150 -->13-limit</h2> Commas: 81/80, 99/98, 121/120, 66/65<br /> <br /> POTE generator: ~9/7 = 425.550<br /> Mapping generator: ~9/7<br /> <br /> Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]<br /> EDOs: 17c, 31, 79cf, 110cef, 141cef<br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:152:<h2> --><h2 id="toc76"><a name="Squares-Agora"></a><!-- ws:end:WikiTextHeadingRule:152 -->Agora</h2> Commas: 81/80, 99/98, 105/104, 121/120<br /> <br /> POTE generator: ~9/7 = 426.276<br /> Mapping generator: ~9/7<br /> <br /> Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|]<br /> EDOs: 31, 45ef, 76e<br /> Badness: 0.0245<br /> <br /> <!-- ws:start:WikiTextHeadingRule:154:<h1> --><h1 id="toc77"><a name="Cuboctahedra"></a><!-- ws:end:WikiTextHeadingRule:154 -->Cuboctahedra</h1> <!-- ws:start:WikiTextHeadingRule:156:<h2> --><h2 id="toc78"><a name="Cuboctahedra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:156 -->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>: ~9/7 = 425.993<br /> Mapping generator: ~9/7<br /> <br /> Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]<br /> EDOs: <a class="wiki_link" href="/14edo">14</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/200edo">200</a><br /> <a class="wiki_link" href="/Badness">Badness</a>: 0.0568<br /> <br /> <!-- ws:start:WikiTextHeadingRule:158:<h1> --><h1 id="toc79"><a name="Liese"></a><!-- ws:end:WikiTextHeadingRule:158 -->Liese</h1> <span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/Liese">Deutsch</a><br /> </span><br /> <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>: ~10/7 = 632.406<br /> Mapping generator: ~10/7<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:160:<h2> --><h2 id="toc80"><a name="Liese-Liesel"></a><!-- ws:end:WikiTextHeadingRule:160 -->Liesel</h2> Commas: 56/55, 81/80, 540/539<br /> <br /> POTE generator: ~10/7 = 633.073<br /> Mapping generator: ~10/7<br /> <br /> Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|]<br /> EDOs: 17c, 19, 36, 91ce<br /> Badness: 0.0407<br /> <br /> <!-- ws:start:WikiTextHeadingRule:162:<h2> --><h2 id="toc81"><a name="Liese-13-limit"></a><!-- ws:end:WikiTextHeadingRule:162 -->13-limit</h2> Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.<br /> <br /> Commas: 56/55, 78/77, 81/80, 91/90<br /> <br /> POTE generator: ~10/7 = ~13/9 = 633.042<br /> Mapping generator: ~10/7<br /> <br /> Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|]<br /> EDOs: 17c, 19, 36, 91cef<br /> Badness: 0.0273<br /> <br /> <!-- ws:start:WikiTextHeadingRule:164:<h2> --><h2 id="toc82"><a name="Liese-Elisa"></a><!-- ws:end:WikiTextHeadingRule:164 -->Elisa</h2> Commas: 77/75, 81/80, 99/98<br /> <br /> POTE generator: ~10/7 = 633.061<br /> Mapping generator: ~10/7<br /> <br /> Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|]<br /> EDOs: 19e, 36e<br /> Badness: 0.0416<br /> <br /> <!-- ws:start:WikiTextHeadingRule:166:<h2> --><h2 id="toc83"><a name="Liese-Lisa"></a><!-- ws:end:WikiTextHeadingRule:166 -->Lisa</h2> Commas: 45/44, 81/80, 343/330<br /> <br /> POTE generator: ~10/7 = 631.370<br /> Mapping generator: ~10/7<br /> <br /> Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|]<br /> EDOs: 19<br /> Badness: 0.0548<br /> <br /> <!-- ws:start:WikiTextHeadingRule:168:<h2> --><h2 id="toc84"><a name="Liese-13-limit"></a><!-- ws:end:WikiTextHeadingRule:168 -->13-limit</h2> Commas: 45/44, 81/80, 91/88, 147/143<br /> <br /> POTE generator: ~10/7 = 631.221<br /> Mapping generator: ~10/7<br /> <br /> Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|]<br /> EDOs: 19<br /> Badness: 0.0361<br /> <br /> <!-- ws:start:WikiTextHeadingRule:170:<h1> --><h1 id="toc85"><a name="Jerome"></a><!-- ws:end:WikiTextHeadingRule:170 -->Jerome</h1> Jerome is related to <a class="wiki_link" href="/20ed5">Hieronymus' tuning</a>; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.<br /> <br /> Commas: 81/80, 17280/16807<br /> <br /> POTE generator: ~54/49 = 139.343<br /> Mapping generator: ~54/49<br /> <br /> Map: [<1 1 0 2|, <0 5 20 7|]<br /> Wedgie: <<5 30 7 20 -3 -40||<br /> EDOs: 8, 9, 17, 26, 43, 112<br /> Badness: 0.1087<br /> <br /> <!-- ws:start:WikiTextHeadingRule:172:<h2> --><h2 id="toc86"><a name="Jerome-11-limit"></a><!-- ws:end:WikiTextHeadingRule:172 -->11-limit</h2> Commas: 81/80, 99/98, 864/847<br /> <br /> POTE generator: ~12/11 = 139.428<br /> Mapping generator: ~12/11<br /> <br /> Map: [<1 1 0 2 3|, <0 5 20 7 4|]<br /> EDOs: 8, 9, 17, 26, 43, 241<br /> Badness: 0.0479<br /> <br /> <!-- ws:start:WikiTextHeadingRule:174:<h2> --><h2 id="toc87"><a name="Jerome-13-limit"></a><!-- ws:end:WikiTextHeadingRule:174 -->13-limit</h2> Commas: 77/78, 81/80, 99/98, 144/143<br /> <br /> POTE generator: ~13/12 = 139.387<br /> Mapping generator: ~12/11<br /> <br /> Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|]<br /> EDOs: 8, 9, 17, 26, 43, 155, 198<br /> Badness: 0.0293<br /> <br /> <!-- ws:start:WikiTextHeadingRule:176:<h2> --><h2 id="toc88"><a name="Jerome-17-limit"></a><!-- ws:end:WikiTextHeadingRule:176 -->17-limit</h2> Commas: 78/77, 81/80, 99/98, 144/143, 189/187<br /> <br /> POTE generator: ~13/12 = 139.362<br /> Mapping generator: ~12/11<br /> <br /> Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|]<br /> EDOs: 8, 9, 17, 26, 43, 155<br /> Badness: 0.0209<br /> <br /> <!-- ws:start:WikiTextHeadingRule:178:<h1> --><h1 id="toc89"><a name="Meanmag"></a><!-- ws:end:WikiTextHeadingRule:178 -->Meanmag</h1> Commas: 81/80, 3125/3072<br /> <br /> POTE generator: ~8/7 = 238.396<br /> Mapping generator: ~7<br /> <br /> Map: [<19 30 44 0|, <0 0 0 1|]<br /> Wedgie: <<0 0 19 0 30 44||<br /> EDOs: 19, 57, 76, 171bcd<br /> Badness: 0.0770<br /> <br /> <!-- ws:start:WikiTextHeadingRule:180:<h1> --><h1 id="toc90"><a name="Undevigintone"></a><!-- ws:end:WikiTextHeadingRule:180 -->Undevigintone</h1> Commas: 49/48, 81/80, 126/125<br /> <br /> POTE generator: ~11/8 = 538.047<br /> Mapping generator: ~11<br /> <br /> Map: [<19 30 44 53 0|, <0 0 0 0 1|]<br /> EDOs: 19, 38d<br /> Badness: 0.0364<br /> <br /> <!-- ws:start:WikiTextHeadingRule:182:<h2> --><h2 id="toc91"><a name="Undevigintone-13-limit"></a><!-- ws:end:WikiTextHeadingRule:182 -->13-limit</h2> Commas: 49/48, 65/64, 81/80, 126/125<br /> <br /> POTE generator: ~11/8 = 537.061<br /> <br /> Map: [<19 30 44 53 0 70|, <0 0 0 0 1 0|]<br /> EDOs: 19, 38d<br /> Badness: 0.0229</body></html>