Meantone family
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The 5-limit parent [[Comma|comma]] of the [[meantone]] family is the Didymus or [[http://en.wikipedia.org/wiki/Syntonic_comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. [[POTE tuning|POTE generator]]: 696.239 Map: [<1 0 -4|, <0 1 4|] EDOs: 5, 7, 12, 19, 31, 50, 81 Badness: 0.00736 ==Seven limit children== The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>]. ==Septimal meantone== The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The [[Wedgies and Multivals|wedgie]] for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and [[31edo]] is a good tuning for it. Commas: 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>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 696.495 Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. Map: [<1 0 -4 -13|, <0 1 4 10|] Generators: 2, 3 Wedgie: <<1 4 10 4 13 12|| EDOs: 12, 19, 31, 81 Badness: 0.0137 ==Unidecimal meantone aka Huygens== Commas: 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>] Eigenmonzos: 2, 11/9 [[POTE tuning|POTE generator]]: 696.967 Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|] Generators: 2, 3 EDOs: 7, 12, 31, [[105edo|105]], [[198edo|198]] Badness: 0.0170 ===Tridecimal meantone=== Commas: 66/65, 81/80, 99/98, 105/104 POTE generator: ~3/2 = 696.642 Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|] EDOs: 12, 19, 31, 267, 298 Badness: 0.0180 ==Meanpop== Commas: 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>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 696.434 Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] Generators: 2, 3 EDOs: 12, 19, 31, 81, 112 Badness: 0.0215 ===13-limit Meanpop=== Commas: 81/80, 105/104, 144/143, 196/195 POTE generator: ~3/2 = 696.211 Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|] EDOS: 7, 12, 19, 31, 50, 81, 131 Badness: 0.0209 ==Meanenneadecal== Commas: 45/44, 56/55, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.250 Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|] EDOs: 7, 12, 19, 31, 50, 81 Badness: 0.0214 ===13-limit=== Commas: 45/44, 56/55, 78/77, 81/80 [[POTE tuning|POTE generator]]: ~3/2 = 696.146 Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|] EDOs: 7, 12, 19, 31, 50, 131, 181 Badness: 0.0212 ==Flattone== Commas: 81/80, 525/512 The wedgie for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. 7-limit minimax [|1 0 0 0>, |21/13 0 1/13 -1/13>, |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>] Eigenmonzos: 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>] Eigenmonzos: 2, 9/7 [[POTE tuning|POTE generator]]: 693.779 Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4. Map: [<1 0 -4 17|, <0 1 4 -9|] Wedgie: <<1 4 -9 4 -17 -32|| Generators: 2, 3 EDOs: 7, 19, [[45edo|45]], [[64edo|64]] Badness: 0.0386 ==Dominant== Commas: 36/35, 64/63 The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]]. [[POTE tuning|POTE generator]]: 701.573 Map: [<1 0 -4 6|, <0 1 4 -2|] Wedgie: <<1 4 -2 4 -6 -16|| EDOs: 5, 7, 12, [[53edo|53]], [[65edo|65]] Badness: 0.0207 ==Sharptone== Commas: 21/20, 28/27 Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done. [[POTE tuning|POTE generator]]: 700.140 Map: [<1 0 -4 -2|, <0 1 4 3|] Wedgie: <<1 4 3 4 2 -4|| EDOs: 5, 12 Badness: 0.0248 ==Injera== Commas: 50/49, 81/80 The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel 19edos, is an excellent tuning for injera. [[POTE tuning|POTE generator]]: 694.375 Map: [<2 0 -8 -7|, <0 1 4 4|] Wedgie: <<2 8 8 8 7 -4|| EDOs: [[12edo|12]], [[26edo|26]], [[38edo|38]], [[140edo|140]], [[178edo|178]] Badness: 0.0311 ==Godzilla== Commas: 49/48, 81/80 Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. MOS are of 5, 9, or 14 notes. [[POTE tuning|POTE generator]]: 252.635 Map: [<1 0 -4 2|, <0 2 8 1|] Wedgie: <<2 8 1 8 -4 -20|| EDOs: 5, 9, 14, 19 Badness: 0.0267 Music: Igliashon Jones, [[http://tinyurl.com/4uyumk9|"Change is on the Wind"]], in Godzilla[9] ==Mohajira== Commas: 81/80, 6144/6125 Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 348.415 Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly. Map: [<1 1 0 6|, <0 2 8 -11|] Generators: 2, 128/105 Wedgie: <<2 8 -11 8 -23 -48|| EDOs: 7, 24, 31 Badness: 0.0557 ===11-limit=== Commas: 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>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 348.477 Map: [<1 1 0 6 2|, <0 2 8 -11 5|] Generators: 2, 11/9 EDOs: 7, 24, 31 Badness: 0.0261 ==Mothra== Commas: 81/80, 1029/1024 Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 232.193 Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents. Map: [<1 1 0 3|, <0 3 12 -1|] Generators: 2, 8/7 Wedgie: <<3 12 -1 12 -10 -36|| EDOs: 5, 26, 31 Badness: 0.0371 ===11-limit=== Commas: 81/80, 99/98, 385/384 POTE generator: ~63/55 = 232.031 Map: [<1 1 0 3 5|, <0 3 12 -1 -8|] EDOs: 5, 26, 31, 88, 150, 181 Badness: 0.0256 ==Squares== Commas: 81/80, 2401/2400 Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. 7 and 9 limit minimax 1/4 comma [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 425.942 Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents. Map: [<1 3 8 6|, <0 -4 -16 -9|] Generators: 2, 9/7 EDOs: 14, 31, 262, 293 Badness: 0.0460 Music: By Chris Vaisvil http://tinyurl.com/25kv7cq http://tinyurl.com/24cbxse ===11-limit=== Commas: 81/80, 385/384, 1375/1372 [[POTE tuning|POTE generator]]: 425.993 Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|] EDOs: 14, 31, 200 Badness: 0.0568 ==Liese== Commas: 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>] Eigenmonzos: 2, 5 [[POTE tuning|POTE generator]]: 632.406 Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges. Map: [<1 0 -4 -3|, <0 3 12 11|] Generators: 2, 10/7 EDOs: [[17edo|17]], [[19edo|19]], [[55edo|55]], [[74edo|74]] Badness: 0.0467
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<html><head><title>Meantone family</title></head><body>The 5-limit parent <a class="wiki_link" href="/Comma">comma</a> of the <a class="wiki_link" href="/meantone">meantone</a> family is the Didymus or <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Syntonic_comma" rel="nofollow">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> <br /> Map: [<1 0 -4|, <0 1 4|]<br /> EDOs: 5, 7, 12, 19, 31, 50, 81<br /> Badness: 0.00736<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>].<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-Septimal meantone"></a><!-- ws:end:WikiTextHeadingRule:2 -->Septimal meantone</h2> The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and <a class="wiki_link" href="/31edo">31edo</a> is a good tuning for it.<br /> <br /> Commas: 81/80, 126/125<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> <br /> Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br /> <br /> Map: [<1 0 -4 -13|, <0 1 4 10|]<br /> Generators: 2, 3<br /> Wedgie: <<1 4 10 4 13 12||<br /> EDOs: 12, 19, 31, 81<br /> Badness: 0.0137<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-Unidecimal meantone aka Huygens"></a><!-- ws:end:WikiTextHeadingRule:4 -->Unidecimal meantone aka Huygens</h2> Commas: 81/80, 126/125, 99/98<br /> <br /> 11-limit 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 /> Eigenmonzos: 2, 11/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> <br /> Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> <br /> Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|]<br /> Generators: 2, 3<br /> EDOs: 7, 12, 31, <a class="wiki_link" href="/105edo">105</a>, <a class="wiki_link" href="/198edo">198</a><br /> Badness: 0.0170<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Unidecimal meantone aka Huygens-Tridecimal meantone"></a><!-- ws:end:WikiTextHeadingRule:6 -->Tridecimal meantone</h3> Commas: 66/65, 81/80, 99/98, 105/104<br /> <br /> POTE generator: ~3/2 = 696.642<br /> <br /> Map: Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]<br /> EDOs: 12, 19, 31, 267, 298<br /> Badness: 0.0180<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="x-Meanpop"></a><!-- ws:end:WikiTextHeadingRule:8 -->Meanpop</h2> Commas: 81/80, 126/125, 385/384<br /> <br /> 11-limit minimax 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |-3 0 5/2 0 0>, |11 0 -13/4 0 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> <br /> Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> Generators: 2, 3<br /> EDOs: 12, 19, 31, 81, 112<br /> Badness: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="x-Meanpop-13-limit Meanpop"></a><!-- ws:end:WikiTextHeadingRule:10 -->13-limit Meanpop</h3> Commas: 81/80, 105/104, 144/143, 196/195<br /> <br /> POTE generator: ~3/2 = 696.211<br /> <br /> Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]<br /> EDOS: 7, 12, 19, 31, 50, 81, 131<br /> Badness: 0.0209<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="x-Meanenneadecal"></a><!-- ws:end:WikiTextHeadingRule:12 -->Meanenneadecal</h2> Commas: 45/44, 56/55, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.250<br /> <br /> Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]<br /> EDOs: 7, 12, 19, 31, 50, 81<br /> Badness: 0.0214<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h3> --><h3 id="toc7"><a name="x-Meanenneadecal-13-limit"></a><!-- ws:end:WikiTextHeadingRule:14 -->13-limit</h3> Commas: 45/44, 56/55, 78/77, 81/80<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 696.146<br /> <br /> Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]<br /> EDOs: 7, 12, 19, 31, 50, 131, 181<br /> Badness: 0.0212<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="x-Flattone"></a><!-- ws:end:WikiTextHeadingRule:16 -->Flattone</h2> Commas: 81/80, 525/512<br /> <br /> The wedgie for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are <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 /> 7-limit 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 /> Eigenmonzos: 2, 7/5<br /> <br /> 9-limit 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 /> Eigenmonzos: 2, 9/7<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> <br /> Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br /> <br /> Map: [<1 0 -4 17|, <0 1 4 -9|]<br /> Wedgie: <<1 4 -9 4 -17 -32||<br /> Generators: 2, 3<br /> EDOs: 7, 19, <a class="wiki_link" href="/45edo">45</a>, <a class="wiki_link" href="/64edo">64</a><br /> Badness: 0.0386<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="x-Dominant"></a><!-- ws:end:WikiTextHeadingRule:18 -->Dominant</h2> Commas: 36/35, 64/63<br /> <br /> The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br /> <br /> Map: [<1 0 -4 6|, <0 1 4 -2|]<br /> Wedgie: <<1 4 -2 4 -6 -16||<br /> EDOs: 5, 7, 12, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/65edo">65</a><br /> Badness: 0.0207<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h2> --><h2 id="toc10"><a name="x-Sharptone"></a><!-- ws:end:WikiTextHeadingRule:20 -->Sharptone</h2> Commas: 21/20, 28/27<br /> <br /> Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 700.140<br /> <br /> Map: [<1 0 -4 -2|, <0 1 4 3|]<br /> Wedgie: <<1 4 3 4 2 -4||<br /> EDOs: 5, 12<br /> Badness: 0.0248<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="x-Injera"></a><!-- ws:end:WikiTextHeadingRule:22 -->Injera</h2> Commas: 50/49, 81/80<br /> <br /> The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel 19edos, is an excellent tuning for injera.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br /> <br /> Map: [<2 0 -8 -7|, <0 1 4 4|]<br /> Wedgie: <<2 8 8 8 7 -4||<br /> EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/26edo">26</a>, <a class="wiki_link" href="/38edo">38</a>, <a class="wiki_link" href="/140edo">140</a>, <a class="wiki_link" href="/178edo">178</a><br /> Badness: 0.0311<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h2> --><h2 id="toc12"><a name="x-Godzilla"></a><!-- ws:end:WikiTextHeadingRule:24 -->Godzilla</h2> Commas: 49/48, 81/80<br /> <br /> Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. MOS are of 5, 9, or 14 notes.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 252.635<br /> <br /> Map: [<1 0 -4 2|, <0 2 8 1|]<br /> Wedgie: <<2 8 1 8 -4 -20||<br /> EDOs: 5, 9, 14, 19<br /> Badness: 0.0267<br /> <br /> Music: Igliashon Jones, <a class="wiki_link_ext" href="http://tinyurl.com/4uyumk9" rel="nofollow">"Change is on the Wind"</a>, in Godzilla[9]<br /> <!-- ws:start:WikiTextHeadingRule:26:<h2> --><h2 id="toc13"><a name="x-Mohajira"></a><!-- ws:end:WikiTextHeadingRule:26 -->Mohajira</h2> Commas: 81/80, 6144/6125<br /> <br /> Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> <br /> Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br /> <br /> Map: [<1 1 0 6|, <0 2 8 -11|]<br /> Generators: 2, 128/105<br /> Wedgie: <<2 8 -11 8 -23 -48||<br /> EDOs: 7, 24, 31<br /> Badness: 0.0557<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h3> --><h3 id="toc14"><a name="x-Mohajira-11-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->11-limit</h3> Commas: 81/80, 121/120, 176/175<br /> <br /> 11-limit minimax 1/4 comma<br /> [|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, <br /> |6 0 -11/8 0 0>, |2 0 5/8 0 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> <br /> Map: [<1 1 0 6 2|, <0 2 8 -11 5|]<br /> Generators: 2, 11/9<br /> EDOs: 7, 24, 31<br /> Badness: 0.0261<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc15"><a name="x-Mothra"></a><!-- ws:end:WikiTextHeadingRule:30 -->Mothra</h2> Commas: 81/80, 1029/1024<br /> <br /> Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> <br /> 7 and 9 limit minimax 1/4 comma <br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 232.193<br /> <br /> Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br /> <br /> Map: [<1 1 0 3|, <0 3 12 -1|]<br /> Generators: 2, 8/7<br /> Wedgie: <<3 12 -1 12 -10 -36||<br /> EDOs: 5, 26, 31<br /> Badness: 0.0371<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h3> --><h3 id="toc16"><a name="x-Mothra-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h3> Commas: 81/80, 99/98, 385/384<br /> <br /> POTE generator: ~63/55 = 232.031<br /> <br /> Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]<br /> EDOs: 5, 26, 31, 88, 150, 181<br /> Badness: 0.0256<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="x-Squares"></a><!-- ws:end:WikiTextHeadingRule:34 -->Squares</h2> Commas: 81/80, 2401/2400<br /> <br /> Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> <br /> 7 and 9 limit minimax 1/4 comma<br /> [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>]<br /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> <br /> Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br /> <br /> Map: [<1 3 8 6|, <0 -4 -16 -9|]<br /> Generators: 2, 9/7<br /> EDOs: 14, 31, 262, 293<br /> Badness: 0.0460<br /> <br /> Music:<br /> By Chris Vaisvil<br /> <!-- ws:start:WikiTextUrlRule:359:http://tinyurl.com/25kv7cq --><a class="wiki_link_ext" href="http://tinyurl.com/25kv7cq" rel="nofollow">http://tinyurl.com/25kv7cq</a><!-- ws:end:WikiTextUrlRule:359 --><br /> <!-- ws:start:WikiTextUrlRule:360:http://tinyurl.com/24cbxse --><a class="wiki_link_ext" href="http://tinyurl.com/24cbxse" rel="nofollow">http://tinyurl.com/24cbxse</a><!-- ws:end:WikiTextUrlRule:360 --><br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h3> --><h3 id="toc18"><a name="x-Squares-11-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->11-limit</h3> Commas: 81/80, 385/384, 1375/1372<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.993<br /> <br /> Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]<br /> EDOs: 14, 31, 200<br /> Badness: 0.0568<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="x-Liese"></a><!-- ws:end:WikiTextHeadingRule:38 -->Liese</h2> Commas: 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 /> Eigenmonzos: 2, 5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> <br /> Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br /> <br /> Map: [<1 0 -4 -3|, <0 3 12 11|]<br /> Generators: 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 /> Badness: 0.0467</body></html>