Magic family
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The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-10 -1 5>, and flipping that yields <<5 1 -10|| for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)^5 = 3 * 3125/3072. 13/41 is a highly recommendable generator, though 19/60 also makes sense and using [[19edo]] or [[22edo]] is always possible. [[Comma]]: 3125/3072 5-limit minimax [<1 0 0|, <0 1 0|, <2 1/5 0|] [[Eigenmonzo|Eigenmonzos]]: 2, 3 Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 [[Cent|cents]]. Map: [<1 0 2|, <0 5 1|] [[Generator|Generators]]: 2, 5/4 [[Edo|Edos]]: [[6edo|6]], [[16edo|16]], [[19edo|19]], [[22edo|22]], [[41edo|41]], [[60edo|60]] ==Seven limit children== The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator. ===Magic=== Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. [[41edo]] is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1. Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than meantone and simpler than garibaldi. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is <<5 1 12 -10 5 25||. By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, <<5 1 12 -8 ... ||. For this, [[104edo]] provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning. Commas: 225/224, 245/243 7 and 9 limit minimax [|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 3 [[POTE tuning|POTE generator]]: 380.352 Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents. Map: [<1 0 2 -1|, <0 5 1 12|] [[Generator|Generators]]: 2, 5/4 EDOs: 41, 183, 224 ====11-limit==== Commas: 225/224, 245/243, 100/99 [[POTE tuning|POTE generator]]: 380.696 Map: [<1 0 2 -1 6|, <0 1 5 12 -8|] EDOs: 19, 22, 41, 104, 145 Badness: 0.0204 ====13-limit==== Commas: 100/99, 105/104, 144/143, 196/195 POTE generator: ~5/4 = 380.427 Map: [<1 0 2 -1 6 -2|, <1 0 5 12 -8 18|] EDOS: 19, 41, 470 Badness: 0.0215 ===Muggles=== Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is [[19edo]], in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. The muggles wedgie is <<5 1 -7 -10 -25 -19||. Commas: 126/125, 525/512 [[POTE tuning|POTE generator]]: 378.479 Map: [<1 0 2 5|, <0 5 1 -7|] EDOs: 19, 130
Original HTML content:
<html><head><title>Magic family</title></head><body>The 5-limit parent comma for the magic family is 3125/3072, the small diesis or magic comma. Its monzo is |-10 -1 5>, and flipping that yields <<5 1 -10|| for the wedgie. This tells us the generator is a major third, and that to get to the interval class of fifths will require five of these. In fact, (5/4)^5 = 3 * 3125/3072. 13/41 is a highly recommendable generator, though 19/60 also makes sense and using <a class="wiki_link" href="/19edo">19edo</a> or <a class="wiki_link" href="/22edo">22edo</a> is always possible.<br /> <br /> <a class="wiki_link" href="/Comma">Comma</a>: 3125/3072<br /> <br /> 5-limit minimax<br /> [<1 0 0|, <0 1 0|, <2 1/5 0|]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> <br /> Algebraic generator: Terzbirat, the positive root of 9x^2-8x-4 = (4+2*sqrt(13))/9; approximately 380.3175 <a class="wiki_link" href="/Cent">cents</a>.<br /> <br /> Map: [<1 0 2|, <0 5 1|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 5/4<br /> <a class="wiki_link" href="/Edo">Edos</a>: <a class="wiki_link" href="/6edo">6</a>, <a class="wiki_link" href="/16edo">16</a>, <a class="wiki_link" href="/19edo">19</a>, <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/60edo">60</a><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 second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives magic, and 525/512, Avicenna's enharmonic diesis, gives his annoying brother muggles. Both use the major third as a generator.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Seven limit children-Magic"></a><!-- ws:end:WikiTextHeadingRule:2 -->Magic</h3> Magic tempers out not only 3125/3072 and 875/864, but also 225/224, 245/243, and 10976/10935. <a class="wiki_link" href="/41edo">41edo</a> is a good magic tuning, and 19 or 22 note MOS are possible scales. Five major thirds approximate 3/1. Twelve major thirds, less an octave, approximate 7/1.<br /> <br /> Magic, with its accurate fifths, works well with 9-limit harmony. It's more accurate than meantone and simpler than garibaldi. It's a little tricky to work with because in it fifths are a relatively complex interval and it doesn't naturally work with scales of around seven notes to the octave. Its wedgie is <<5 1 12 -10 5 25||.<br /> <br /> By adding 100/99 to the list of commas, magic can be extended to an 11-limit version, <<5 1 12 -8 ... ||. For this, <a class="wiki_link" href="/104edo">104edo</a> provides an excellent tuning, as it does also for the rank three temperaments tempering out 100/99 with 225/224, 245/243 or 875/864. Septimage (see below) is also an excellent 11-limit magic tuning.<br /> <br /> Commas: 225/224, 245/243<br /> <br /> 7 and 9 limit minimax<br /> [|1 0 0 0>, |0 1 0 0>, |2 1/5 0 0>, |-1 12/5 0 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 3<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 380.352<br /> <br /> Algebraic generators: Tirzbirat or Septimage, the real root of 5x^5+4x-20, 380.7604 cents.<br /> <br /> Map: [<1 0 2 -1|, <0 5 1 12|]<br /> <a class="wiki_link" href="/Generator">Generators</a>: 2, 5/4<br /> <br /> EDOs: 41, 183, 224<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h4> --><h4 id="toc2"><a name="x-Seven limit children-Magic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit</h4> Commas: 225/224, 245/243, 100/99<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 380.696<br /> <br /> Map: [<1 0 2 -1 6|, <0 1 5 12 -8|]<br /> EDOs: 19, 22, 41, 104, 145<br /> Badness: 0.0204<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h4> --><h4 id="toc3"><a name="x-Seven limit children-Magic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit</h4> Commas: 100/99, 105/104, 144/143, 196/195<br /> <br /> POTE generator: ~5/4 = 380.427<br /> <br /> Map: [<1 0 2 -1 6 -2|, <1 0 5 12 -8 18|]<br /> EDOS: 19, 41, 470<br /> Badness: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Seven limit children-Muggles"></a><!-- ws:end:WikiTextHeadingRule:8 -->Muggles</h3> Aside from 3125/3072 and 525/512 muggles also tempers out 126/125 and 1323/1280. A good muggles tuning is <a class="wiki_link" href="/19edo">19edo</a>, in which tuning it's the same thing as magic. Muggles works better for small scales than magic in the sense that 7 or 10 note MOS are reasonable choices. The muggles wedgie is <<5 1 -7 -10 -25 -19||.<br /> <br /> Commas: 126/125, 525/512<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 378.479<br /> <br /> Map: [<1 0 2 5|, <0 5 1 -7|]<br /> <br /> EDOs: 19, 130</body></html>