Magic family
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[[toc]] A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than [[Meantone family|meantone]] and is also closer to just intonation. It is the simplest rank 2 temperament that tunes every 9-limit interval better than is possible in [[12edo]]. Properties may depend on tuning and extension. The most prominent deficiency of magic temperaments is that they lack [[Rothenberg propriety|proper]] or nearly-proper MOS scales in the 5 to 10 note "diatonic" region. =Five limit magic= 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, the optimal patent val generator, also makes a lot of 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√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]], [[221edo|221c]], [[281edo|281c]] ==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 family|meantone]] and simpler than [[Garibaldi temperament|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||. 225/224 is the [[Marvel+temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament. 243/242 leads to another essentially tempered 9-limit triad with two thirds approximating 9/7 and the other 6/5. (This "octarod comma" is shared with [[Sensi|sensi]], [[Semaphore and Godzilla|godzilla]], [[Superpyth|superpyth]], [[Tetracot family|octacot]], [[Gamelismic clan|rodan]], [[Shrutar|shrutar]], [[Porcupine family|hedgehog]], [[Clyde node|clyde]], and [[Sensamagic clan|bohpier]]. See [[http://x31eq.com/cgi-bin/uv.cgi?uvs=245:243|temperament finder]].) 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, 142cd, 183cd, 224cd ==11-limit== Commas: 225/224, 245/243, 100/99 [[POTE tuning|POTE generator]]: 380.696 Map: [<1 0 2 -1 6|, <0 5 1 12 -8|] EDOs: 19, 22, 41, 104, 145c 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|, <0 5 1 12 -8 18|] EDOS: 19, 41, 265cdef Badness: 0.0215 ===Sorcery=== Commas: 65/64, 78/77, 91/90, 100/99 POTE generator: ~5/4 = 380.477 Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|] EDOs: 19, 22, 31f, 41f Badness: 0.0258 ===Necromancy=== Commas: 100/99, 225/224, 245/243, 275/273 POTE generator: ~5/4 = 380.787 Map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|] EDOs: 19, 22, 41, 63, 104 Badness: 0.0253 ==Telepathy== Commas: 55/54, 99/98, 176/175 POTE generator: ~5/4 = 381.019 Map: [<1 0 2 -1 -1|, <0 5 1 12 14|] EDOs: 19, 22, 41e, 63e Badness: 0.0271 ===13-limit telepathy=== Commas: 55/54, 65/64, 91/90, 99/98 POTE generator: ~5/4 = 380.520 Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|] EDOs: 19, 22, 31f, 34ef, 41ef Badness: 0.0255 ==Horcrux== Commas: 45/44, 56/55, 245/243 POTE generator: ~5/4 = 379.642 Map: [<1 0 2 -1 0|, <0 5 1 12 11|] EDOs: 19, 60e Badness: 0.0393 =Divination= Commas: 121/120, 225/224, 245/243 POTE generator: ~5/4 = 380.233 Map: [<2 0 4 -2 5|, <0 5 1 12 3|] EDOs: 22, 38d, 60e, 142cde Badness: 0.0359 ==13-limit== Commas: 105/104, 121/120, 196/195, 245/243 POTE generator: ~5/4 = 379.920 Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|] EDOs: 22f, 60e Badness: 0.0346 =Soothsaying= Commas: 100/99, 225/224, 245/243, 1352/1331 POTE generator: ~5/4 = 380.508 Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|] EDOs: 22, 60, 82 Badness: 0.0554 =Witchcraft= Commas: 225/224, 245/243, 441/440 POTE generator: ~5/4 = 380.232 Map: [<1 0 2 -1 -7|, <0 5 1 12 33|] EDOs: 41, 60e, 101cd, 243cde Badness: 0.0307 ==13-limit== Commas: 105/104, 196/195, 245/243, 275/273 POTE generator: ~5/4 = 380.189 Map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|] EDOs: 41, 60e, 101cd Badness: 0.0235 =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]]: ~5/4 = 378.479 Map: [<1 0 2 5|, <0 5 1 -7|] EDOs: 19, 73bcd, 92bcd Badness: 0.0562 ==11-limit== Commas: 45/44, 126/125, 385/384 [[POTE tuning|POTE generator]]: ~5/4 = 377.724 Map: [<1 0 2 5 0|, <0 5 1 -7 11|] EDOs: 16, 19, 35, 54bd Badness: 0.0480 ==13-limit== Commas: 45/44, 65/64, 78/77, 126/125 [[POTE tuning|POTE generator]]: ~5/4 = 377.724 Map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|] EDOs: 16, 19, 35f, 54bdf Badness: 0.0309 =Astrology= Commas: 50/49, 3125/3072 POTE generator: ~5/4 = 380.578 Map: [<2 0 4 5|, <0 5 1 1|] Wedgie: <<10 2 2 -20 -25 -1|| EDOs: 6, 16, 22, 60d, 82d Badness: 0.0827 ==11-limit== Commas: 50/49, 121/120, 176/175 POTE generator: ~5/4 = 380.530 Map: [<2 0 4 5 5|, <0 5 1 1 3|] EDOs: 6, 16, 22, 60de, 82de Badness: 0.0392 ==13-limit== Commas: 50/49, 65/64, 78/77, 121/120 POTE generator: ~5/4 = 379.787 Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|] EDOs: 6, 16, 22, 38f Badness: 0.0344 [[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3|Astrology Percussion Quintet No 1]] by [[https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet|Joel Taylor]] ==Horoscope== Commas: 50/49, 66/65, 105/104, 121/120 POTE generator: ~5/4 = 379.837 Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|] EDOs: 16, 22f, 38 Badness: 0.0353 =Spell= Commas: 49/48, 3125/3072 POTE generator: ~28/25 = 189.927 Map: [<1 0 2 2|, <0 10 2 5|] Wedgie: <<10 2 5 -20 -20 6|| EDOs: 6, 19, 82d Badness: 0.0810 ==11-limit== Commas: 49/48, 56/55, 125/121 POTE generator: ~11/10 = 190.285 Map: [<1 0 2 2 3|, <0 10 2 5 3|] EDOs: 6, 19, 44de, 63de, 82de Badness: 0.0598 ==13-limit== Commas: 49/48, 56/55, 78/77, 125/121 POTE generator: ~11/10 = 189.928 Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|] EDOs: 6, 19, 82def Badness: 0.0456 ==Cantrip== Commas: 49/48, 56/55, 91/90, 125/121 POTE generator: ~11/10 = 190.360 Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|] EDOs: 19, 44de, 63de, 82de Badness: 0.0416 =Hocum= Commas: 3125/3072, 4000/3969 POTE generator: ~63/50 = 400.108 Map: [<1 5 3 -3|, <0 -10 -2 17|] Wedgie: <<10 2 -17 -20 -55 -45|| EDOs: 38, 41, 161c, 202c, 243c, 284c Badness: 0.1071 =Hocus= Commas: 225/224, 243/242, 245/242 POTE generator: ~14/11 = 409.910 Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|] EDOs: 38d, 41, 120cd, 161cd, 202cd Badness: 0.0385 ==13-limit== Commas: 105/104, 196/195, 243/242, 245/242 POTE generator: ~14/11 = 410.004 Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|] EDOs: 41, 79d, 120cd Badness: 0.0303 =Trismegistus= Commas: 3125/3072, 1029/1024 POTE generator: ~147/100 = 673.290 Map: [<1 10 4 0|, <0 -15 -3 5|] Wedgie: <<15 3 -5 -30 -50 -20|| EDOs: 16, 25, 41, 139c, 180c, 221c, 262c Badness: 0.0983 ==11-limit== Commas: 385/384, 441/440, 625/616 POTE generator: ~22/15 = 673.340 Map: [<1 10 4 0 13|, <0 -15 -3 5 -17|] EDOs: 16, 25e, 41, 98c, 139c, 180c Badness: 0.0456 ==13-limit== Commas: 105/104, 144/143, 275/273, 625/616 POTE generator: ~22/15 = 673.359 Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|] EDOs: 16, 25e, 41, 98c, 139cf Badness: 0.0331 =Quadrimage= Commas: 2401/2400, 3125/3072 POTE generator: ~28/25 = 204.987 Map: [<1 5 3 4|, <0 -20 -4 -7|] Wedgie: <<20 4 7 -40 -45 5|| EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd Badness: 0.1274 ==11-limit== Commas: 245/242, 385/384, 625/616 POTE generator: ~28/25 = 204.956 Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|] EDOs: 6, 35, 41, 199cde, 240cde, 281cde Badness: 0.0616 ==13-limit== Commas: 105/104, 144/143, 245/242, 625/616 POTE generator: ~28/25 = 205.028 Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|] EDOs: 41, 117c, 158cd, 199cdef Badness: 0.0440
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<html><head><title>Magic family</title></head><body><!-- ws:start:WikiTextTocRule:70:<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:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 1em;"><a href="#Five limit magic">Five limit magic</a></div> <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Five limit magic-Seven limit children">Seven limit children</a></div> <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 1em;"><a href="#Magic">Magic</a></div> <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Magic-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Magic-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Sorcery">Sorcery</a></div> <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 3em;"><a href="#Magic-13-limit-Necromancy">Necromancy</a></div> <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Magic-Telepathy">Telepathy</a></div> <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 3em;"><a href="#Magic-Telepathy-13-limit telepathy">13-limit telepathy</a></div> <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Magic-Horcrux">Horcrux</a></div> <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 1em;"><a href="#Divination">Divination</a></div> <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Divination-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 1em;"><a href="#Soothsaying">Soothsaying</a></div> <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 1em;"><a href="#Witchcraft">Witchcraft</a></div> <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 2em;"><a href="#Witchcraft-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Muggles">Muggles</a></div> <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><div style="margin-left: 2em;"><a href="#Muggles-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><div style="margin-left: 2em;"><a href="#Muggles-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><div style="margin-left: 1em;"><a href="#Astrology">Astrology</a></div> <!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><div style="margin-left: 2em;"><a href="#Astrology-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><div style="margin-left: 2em;"><a href="#Astrology-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><div style="margin-left: 2em;"><a href="#Astrology-Horoscope">Horoscope</a></div> <!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><div style="margin-left: 1em;"><a href="#Spell">Spell</a></div> <!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><div style="margin-left: 2em;"><a href="#Spell-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 2em;"><a href="#Spell-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 2em;"><a href="#Spell-Cantrip">Cantrip</a></div> <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Hocum">Hocum</a></div> <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 1em;"><a href="#Hocus">Hocus</a></div> <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 2em;"><a href="#Hocus-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 1em;"><a href="#Trismegistus">Trismegistus</a></div> <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 2em;"><a href="#Trismegistus-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Trismegistus-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 1em;"><a href="#Quadrimage">Quadrimage</a></div> <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Quadrimage-11-limit">11-limit</a></div> <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Quadrimage-13-limit">13-limit</a></div> <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --></div> <!-- ws:end:WikiTextTocRule:106 -->A magic temperament is optimal, for some searches, in the 9-limit. It has slightly higher complexity than <a class="wiki_link" href="/Meantone%20family">meantone</a> and is also closer to just intonation. It is the simplest rank 2 temperament that tunes every 9-limit interval better than is possible in <a class="wiki_link" href="/12edo">12edo</a>. Properties may depend on tuning and extension.<br /> <br /> The most prominent deficiency of magic temperaments is that they lack <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> or nearly-proper MOS scales in the 5 to 10 note "diatonic" region.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Five limit magic"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit magic</h1> 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, the optimal patent val generator, also makes a lot of 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√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>, <a class="wiki_link" href="/221edo">221c</a>, <a class="wiki_link" href="/281edo">281c</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="Five limit magic-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->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:4:<h1> --><h1 id="toc2"><a name="Magic"></a><!-- ws:end:WikiTextHeadingRule:4 -->Magic</h1> 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 <a class="wiki_link" href="/Meantone%20family">meantone</a> and simpler than <a class="wiki_link" href="/Garibaldi%20temperament">garibaldi</a>. 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 /> 225/224 is the [[Marvel+temperaments|marvel]] comma. Because the augmented triad is the simplest triad in magic temperaments, it is especially significant in magic temperament.<br /> <br /> 243/242 leads to another essentially tempered 9-limit triad with two thirds approximating 9/7 and the other 6/5. (This "octarod comma" is shared with <a class="wiki_link" href="/Sensi">sensi</a>, <a class="wiki_link" href="/Semaphore%20and%20Godzilla">godzilla</a>, <a class="wiki_link" href="/Superpyth">superpyth</a>, <a class="wiki_link" href="/Tetracot%20family">octacot</a>, <a class="wiki_link" href="/Gamelismic%20clan">rodan</a>, <a class="wiki_link" href="/Shrutar">shrutar</a>, <a class="wiki_link" href="/Porcupine%20family">hedgehog</a>, <a class="wiki_link" href="/Clyde%20node">clyde</a>, and <a class="wiki_link" href="/Sensamagic%20clan">bohpier</a>. See <a class="wiki_link_ext" href="http://x31eq.com/cgi-bin/uv.cgi?uvs=245:243" rel="nofollow">temperament finder</a>.) <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, 142cd, 183cd, 224cd<br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="Magic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2> 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 5 1 12 -8|]<br /> EDOs: 19, 22, 41, 104, 145c<br /> Badness: 0.0204<br /> <br /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><h2 id="toc4"><a name="Magic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h2> 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|, <0 5 1 12 -8 18|]<br /> EDOS: 19, 41, 265cdef<br /> Badness: 0.0215<br /> <br /> <!-- ws:start:WikiTextHeadingRule:10:<h3> --><h3 id="toc5"><a name="Magic-13-limit-Sorcery"></a><!-- ws:end:WikiTextHeadingRule:10 -->Sorcery</h3> Commas: 65/64, 78/77, 91/90, 100/99<br /> <br /> POTE generator: ~5/4 = 380.477<br /> <br /> Map: [<1 0 2 -1 6 4|, <0 5 1 12 -8 -1|]<br /> EDOs: 19, 22, 31f, 41f<br /> Badness: 0.0258<br /> <br /> <!-- ws:start:WikiTextHeadingRule:12:<h3> --><h3 id="toc6"><a name="Magic-13-limit-Necromancy"></a><!-- ws:end:WikiTextHeadingRule:12 -->Necromancy</h3> Commas: 100/99, 225/224, 245/243, 275/273<br /> <br /> POTE generator: ~5/4 = 380.787<br /> <br /> Map: [<1 0 2 -1 6 11|, <0 5 1 12 -8 -23|]<br /> EDOs: 19, 22, 41, 63, 104<br /> Badness: 0.0253<br /> <br /> <!-- ws:start:WikiTextHeadingRule:14:<h2> --><h2 id="toc7"><a name="Magic-Telepathy"></a><!-- ws:end:WikiTextHeadingRule:14 -->Telepathy</h2> Commas: 55/54, 99/98, 176/175<br /> <br /> POTE generator: ~5/4 = 381.019<br /> <br /> Map: [<1 0 2 -1 -1|, <0 5 1 12 14|]<br /> EDOs: 19, 22, 41e, 63e<br /> Badness: 0.0271<br /> <br /> <!-- ws:start:WikiTextHeadingRule:16:<h3> --><h3 id="toc8"><a name="Magic-Telepathy-13-limit telepathy"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit telepathy</h3> Commas: 55/54, 65/64, 91/90, 99/98<br /> <br /> POTE generator: ~5/4 = 380.520<br /> <br /> Map: [<1 0 2 -1 -1 4|, <0 5 1 12 14 -1|]<br /> EDOs: 19, 22, 31f, 34ef, 41ef<br /> Badness: 0.0255<br /> <br /> <!-- ws:start:WikiTextHeadingRule:18:<h2> --><h2 id="toc9"><a name="Magic-Horcrux"></a><!-- ws:end:WikiTextHeadingRule:18 -->Horcrux</h2> Commas: 45/44, 56/55, 245/243<br /> <br /> POTE generator: ~5/4 = 379.642<br /> <br /> Map: [<1 0 2 -1 0|, <0 5 1 12 11|]<br /> EDOs: 19, 60e<br /> Badness: 0.0393<br /> <br /> <!-- ws:start:WikiTextHeadingRule:20:<h1> --><h1 id="toc10"><a name="Divination"></a><!-- ws:end:WikiTextHeadingRule:20 -->Divination</h1> Commas: 121/120, 225/224, 245/243<br /> <br /> POTE generator: ~5/4 = 380.233<br /> <br /> Map: [<2 0 4 -2 5|, <0 5 1 12 3|]<br /> EDOs: 22, 38d, 60e, 142cde<br /> Badness: 0.0359<br /> <br /> <!-- ws:start:WikiTextHeadingRule:22:<h2> --><h2 id="toc11"><a name="Divination-13-limit"></a><!-- ws:end:WikiTextHeadingRule:22 -->13-limit</h2> Commas: 105/104, 121/120, 196/195, 245/243<br /> <br /> POTE generator: ~5/4 = 379.920<br /> <br /> Map: [<2 0 4 -2 5 -4|, <0 5 1 12 3 18|]<br /> EDOs: 22f, 60e<br /> Badness: 0.0346<br /> <br /> <!-- ws:start:WikiTextHeadingRule:24:<h1> --><h1 id="toc12"><a name="Soothsaying"></a><!-- ws:end:WikiTextHeadingRule:24 -->Soothsaying</h1> Commas: 100/99, 225/224, 245/243, 1352/1331<br /> <br /> POTE generator: ~5/4 = 380.508<br /> <br /> Map: [<2 0 4 -2 12 15|, <0 5 1 12 -8 -12|]<br /> EDOs: 22, 60, 82<br /> Badness: 0.0554<br /> <br /> <!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc13"><a name="Witchcraft"></a><!-- ws:end:WikiTextHeadingRule:26 -->Witchcraft</h1> Commas: 225/224, 245/243, 441/440<br /> <br /> POTE generator: ~5/4 = 380.232<br /> <br /> Map: [<1 0 2 -1 -7|, <0 5 1 12 33|]<br /> EDOs: 41, 60e, 101cd, 243cde<br /> Badness: 0.0307<br /> <br /> <!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc14"><a name="Witchcraft-13-limit"></a><!-- ws:end:WikiTextHeadingRule:28 -->13-limit</h2> Commas: 105/104, 196/195, 245/243, 275/273<br /> <br /> POTE generator: ~5/4 = 380.189<br /> <br /> Map: [<1 0 2 -1 -7 -2|, <0 5 1 12 33 18|]<br /> EDOs: 41, 60e, 101cd<br /> Badness: 0.0235<br /> <br /> <!-- ws:start:WikiTextHeadingRule:30:<h1> --><h1 id="toc15"><a name="Muggles"></a><!-- ws:end:WikiTextHeadingRule:30 -->Muggles</h1> 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>: ~5/4 = 378.479<br /> <br /> Map: [<1 0 2 5|, <0 5 1 -7|]<br /> EDOs: 19, 73bcd, 92bcd<br /> Badness: 0.0562<br /> <br /> <!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc16"><a name="Muggles-11-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->11-limit</h2> Commas: 45/44, 126/125, 385/384<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> <br /> Map: [<1 0 2 5 0|, <0 5 1 -7 11|]<br /> EDOs: 16, 19, 35, 54bd<br /> Badness: 0.0480<br /> <br /> <!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc17"><a name="Muggles-13-limit"></a><!-- ws:end:WikiTextHeadingRule:34 -->13-limit</h2> Commas: 45/44, 65/64, 78/77, 126/125<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~5/4 = 377.724<br /> <br /> Map: [<1 0 2 5 0 4|, <0 5 1 -7 11 -1|]<br /> EDOs: 16, 19, 35f, 54bdf<br /> Badness: 0.0309<br /> <br /> <!-- ws:start:WikiTextHeadingRule:36:<h1> --><h1 id="toc18"><a name="Astrology"></a><!-- ws:end:WikiTextHeadingRule:36 -->Astrology</h1> Commas: 50/49, 3125/3072<br /> <br /> POTE generator: ~5/4 = 380.578<br /> <br /> Map: [<2 0 4 5|, <0 5 1 1|]<br /> Wedgie: <<10 2 2 -20 -25 -1||<br /> EDOs: 6, 16, 22, 60d, 82d<br /> Badness: 0.0827<br /> <br /> <!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc19"><a name="Astrology-11-limit"></a><!-- ws:end:WikiTextHeadingRule:38 -->11-limit</h2> Commas: 50/49, 121/120, 176/175<br /> <br /> POTE generator: ~5/4 = 380.530<br /> <br /> Map: [<2 0 4 5 5|, <0 5 1 1 3|]<br /> EDOs: 6, 16, 22, 60de, 82de<br /> Badness: 0.0392<br /> <br /> <!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc20"><a name="Astrology-13-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->13-limit</h2> Commas: 50/49, 65/64, 78/77, 121/120<br /> <br /> POTE generator: ~5/4 = 379.787<br /> <br /> Map: [<2 0 4 5 5 8|, <0 5 1 1 3 -1|]<br /> EDOs: 6, 16, 22, 38f<br /> Badness: 0.0344<br /> <br /> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/AstrologyPercQuintet1_c.mp3" rel="nofollow">Astrology Percussion Quintet No 1</a> by <a class="wiki_link_ext" href="https://soundcloud.com/joelgranttaylor/astrology-percussion-quintet" rel="nofollow">Joel Taylor</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:42:<h2> --><h2 id="toc21"><a name="Astrology-Horoscope"></a><!-- ws:end:WikiTextHeadingRule:42 -->Horoscope</h2> Commas: 50/49, 66/65, 105/104, 121/120<br /> <br /> POTE generator: ~5/4 = 379.837<br /> <br /> Map: [<2 0 4 5 5 3|, <0 5 1 1 3 7|]<br /> EDOs: 16, 22f, 38<br /> Badness: 0.0353<br /> <br /> <!-- ws:start:WikiTextHeadingRule:44:<h1> --><h1 id="toc22"><a name="Spell"></a><!-- ws:end:WikiTextHeadingRule:44 -->Spell</h1> Commas: 49/48, 3125/3072<br /> <br /> POTE generator: ~28/25 = 189.927<br /> <br /> Map: [<1 0 2 2|, <0 10 2 5|]<br /> Wedgie: <<10 2 5 -20 -20 6||<br /> EDOs: 6, 19, 82d<br /> Badness: 0.0810<br /> <br /> <!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc23"><a name="Spell-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2> Commas: 49/48, 56/55, 125/121<br /> <br /> POTE generator: ~11/10 = 190.285<br /> <br /> Map: [<1 0 2 2 3|, <0 10 2 5 3|]<br /> EDOs: 6, 19, 44de, 63de, 82de<br /> Badness: 0.0598<br /> <br /> <!-- ws:start:WikiTextHeadingRule:48:<h2> --><h2 id="toc24"><a name="Spell-13-limit"></a><!-- ws:end:WikiTextHeadingRule:48 -->13-limit</h2> Commas: 49/48, 56/55, 78/77, 125/121<br /> <br /> POTE generator: ~11/10 = 189.928<br /> <br /> Map: [<1 0 2 2 3 4|, <0 10 2 5 3 -2|]<br /> EDOs: 6, 19, 82def<br /> Badness: 0.0456<br /> <br /> <!-- ws:start:WikiTextHeadingRule:50:<h2> --><h2 id="toc25"><a name="Spell-Cantrip"></a><!-- ws:end:WikiTextHeadingRule:50 -->Cantrip</h2> Commas: 49/48, 56/55, 91/90, 125/121<br /> <br /> POTE generator: ~11/10 = 190.360<br /> <br /> Map: [<1 0 2 2 3 1|, <0 10 2 5 3 17|]<br /> EDOs: 19, 44de, 63de, 82de<br /> Badness: 0.0416<br /> <br /> <!-- ws:start:WikiTextHeadingRule:52:<h1> --><h1 id="toc26"><a name="Hocum"></a><!-- ws:end:WikiTextHeadingRule:52 -->Hocum</h1> Commas: 3125/3072, 4000/3969<br /> <br /> POTE generator: ~63/50 = 400.108<br /> <br /> Map: [<1 5 3 -3|, <0 -10 -2 17|]<br /> Wedgie: <<10 2 -17 -20 -55 -45||<br /> EDOs: 38, 41, 161c, 202c, 243c, 284c<br /> Badness: 0.1071<br /> <br /> <!-- ws:start:WikiTextHeadingRule:54:<h1> --><h1 id="toc27"><a name="Hocus"></a><!-- ws:end:WikiTextHeadingRule:54 -->Hocus</h1> Commas: 225/224, 243/242, 245/242<br /> <br /> POTE generator: ~14/11 = 409.910<br /> <br /> Map: [<1 5 3 11 12|, <0 -10 -2 -24 -25|]<br /> EDOs: 38d, 41, 120cd, 161cd, 202cd<br /> Badness: 0.0385<br /> <br /> <!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc28"><a name="Hocus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2> Commas: 105/104, 196/195, 243/242, 245/242<br /> <br /> POTE generator: ~14/11 = 410.004<br /> <br /> Map: [<1 5 3 11 12 16|, <0 -10 -2 -24 -25 -36|]<br /> EDOs: 41, 79d, 120cd<br /> Badness: 0.0303<br /> <br /> <!-- ws:start:WikiTextHeadingRule:58:<h1> --><h1 id="toc29"><a name="Trismegistus"></a><!-- ws:end:WikiTextHeadingRule:58 -->Trismegistus</h1> Commas: 3125/3072, 1029/1024<br /> <br /> POTE generator: ~147/100 = 673.290<br /> <br /> Map: [<1 10 4 0|, <0 -15 -3 5|]<br /> Wedgie: <<15 3 -5 -30 -50 -20||<br /> EDOs: 16, 25, 41, 139c, 180c, 221c, 262c<br /> Badness: 0.0983<br /> <br /> <!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc30"><a name="Trismegistus-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2> Commas: 385/384, 441/440, 625/616<br /> <br /> POTE generator: ~22/15 = 673.340<br /> <br /> Map: [<1 10 4 0 13|, <0 -15 -3 5 -17|]<br /> EDOs: 16, 25e, 41, 98c, 139c, 180c<br /> Badness: 0.0456<br /> <br /> <!-- ws:start:WikiTextHeadingRule:62:<h2> --><h2 id="toc31"><a name="Trismegistus-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2> Commas: 105/104, 144/143, 275/273, 625/616<br /> <br /> POTE generator: ~22/15 = 673.359<br /> <br /> Map: [<1 10 4 0 13 11|, <0 -15 -3 5 -17 -13|]<br /> EDOs: 16, 25e, 41, 98c, 139cf<br /> Badness: 0.0331<br /> <br /> <!-- ws:start:WikiTextHeadingRule:64:<h1> --><h1 id="toc32"><a name="Quadrimage"></a><!-- ws:end:WikiTextHeadingRule:64 -->Quadrimage</h1> Commas: 2401/2400, 3125/3072<br /> <br /> POTE generator: ~28/25 = 204.987<br /> <br /> Map: [<1 5 3 4|, <0 -20 -4 -7|]<br /> Wedgie: <<20 4 7 -40 -45 5||<br /> EDOs: 6, 35, 41, 158cd, 199cd, 240cd, 281cd<br /> Badness: 0.1274<br /> <br /> <!-- ws:start:WikiTextHeadingRule:66:<h2> --><h2 id="toc33"><a name="Quadrimage-11-limit"></a><!-- ws:end:WikiTextHeadingRule:66 -->11-limit</h2> Commas: 245/242, 385/384, 625/616<br /> <br /> POTE generator: ~28/25 = 204.956<br /> <br /> Map: [<1 5 3 4 5|, <0 -20 -4 -7 -9|]<br /> EDOs: 6, 35, 41, 199cde, 240cde, 281cde<br /> Badness: 0.0616<br /> <br /> <!-- ws:start:WikiTextHeadingRule:68:<h2> --><h2 id="toc34"><a name="Quadrimage-13-limit"></a><!-- ws:end:WikiTextHeadingRule:68 -->13-limit</h2> Commas: 105/104, 144/143, 245/242, 625/616<br /> <br /> POTE generator: ~28/25 = 205.028<br /> <br /> Map: [<1 5 3 4 5 9|, <0 -20 -4 -7 -9 -31|]<br /> EDOs: 41, 117c, 158cd, 199cdef<br /> Badness: 0.0440</body></html>