Middle Path table of 5-limit rank-2 temperaments
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This table is reproduced from Table 1 of [[Paul Erlich]]'s [[A Middle Path]], with small modifications to update names and improve readability. The "main sequence" of this table "comprises all possible five-limit 2-D cases where complexity/12.33 + damage/10 < 1. The 'exotemperaments' have larger damage and lend themselves to special measures such as custom inharmonic timbres in order to aid harmoniousness. The 'bonus temperaments' are more complex but have exceedingly small damage and sound like JI." See also [[Middle Path table of seven-limit rank two temperaments]], [[Middle Path table of eleven-limit rank two temperaments]]. || Vanishing Interval's Ratio || V.I. Vector || V.I. cents || Temperament name || TOP period || TOP generator || Mapp. of 2 || Mapp. of 3 || Mapp. of 5 || Cmplx. || TOP Dmg. || ETs || |||||||||||||||||||||||| TWO EXOTEMPERAMENTS: || || 16/15 || [4, -1, -1> || 111.7 || [[Father]] || 1185.9 || 447.4 || 1, 0 || 2, -1 || 2, 1 || 2.15 || 14.13 || || || 27/25 || [0, 3, -2> || 133.2 || [[Bug]] || 1200.0 || 260.3 || 1, 0 || 2, -2 || 3, -3 || 2.55 || 14.18 || || |||||||||||||||||||||||| MAIN SEQUENCE: || || 25/24 || [-3, -1, 2> || 70.7 || [[Dicot]] || 1207.66 || 353.22 || 1, 0 || 1, 2 || 2, 1 || 2.51 || 7.66 || 24c, 41cc, 48cc, 58ccc... || || 81/80 || [-4, 4, -1> || 21.5 || [[Meantone]] || 1201.70 || 504.13 || 1, 0 || 2, -1 || 4, -4 || 3.44 || 1.70 || 12, 19, 24, 26, 29c, 31, 33c, 36, 38... || || 128/125 || [7, 0, -3> || 41.1 || [[Augmented]] || 399.02 || 93.15 || 3, 0 || 5, -1 || 7, 0 || 3.79 || 2.94 || 12, 15, 24, 27, 30, 33, 36, 39... || || 135/128 || [-7, 3, 1> || 92.2 || [[Mavila]] || 1206.55 || 685.03 || 1, 0 || 2, -1 || 1, 3 || 3.83 || 6.55 || 37bc, 44bcc, 67bbccc... || || 250/243 || [1, -5, 3> || 49.2 || [[Porcupine]] || 1196.91 || 1034.59 || 1, 0 || 2, -3 || 3, -5 || 4.32 || 3.09 || 15, 22, 29, 30, 36c, 37, 43c... || || 256/243 || [8, -5, 0> || 90.2 || [[Blackwood]] || 238.87 || 158.78 || 5, 0 || 8, 0 || 11, 1 || 4.33 || 5.67 || 15, 25, 30, 35bc, 40b, 45bc... || || 648/625 || [3, 4, -4> || 62.6 || [[Diminished]] || 299.16 || 197.49 || 4, 0 || 6, 1 || 9, 1 || 5.06 || 3.36 || 12, 24, 28, 32c, 36, 40, 44c... || || 2048/2025 || [11, -4, -2> || 19.6 || [[Srutal]] || 599.56 || 494.86 || 2, 0 || 3, 1 || 5, -2 || 5.97 || 0.89 || 12, 22, 24, 34, 36, 44, 46, 48c, 54, 56... || || 3125/3072 || [-10, -1, 5> || 29.6 || [[Magic]] || 1201.28 || 380.80 || 1, 0 || 0, 5 || 2, 1 || 6.30 || 1.28 || 19, 22, 38, 41, 44, 47b, 54b, 57, 60... || || 6561/6250 || [-1, 8, -5> || 84.1 || [[Ripple]] || 1203.32 || 101.99 || 1, 0 || 2, -5 || 3, -8 || 6.87 || 3.32 || 12, 24, 36, 47, 48c, 59b... || || 15625/15552 || [-6, -5, 6> || 8.11 || [[Hanson]] || 1200.29 || 317.07 || 1, 0 || 0, 6 || 1, 5 || 7.57 || 0.29 || 19, 34, 38, 49, 53, 57, 64b... || || 16875/16384 || [-14, 3, 4> || 51.1 || [[Negri]] || 1201.82 || 1075.68 || 1, 0 || 2, -4 || 2, 3 || 7.62 || 1.82 || 19, 29, 38, 48, 57, 58c, 66b... || || 20000/19683 || [5, -9, 4> || 27.7 || [[Tetracot]] || 1199.03 || 176.11 || 1, 0 || 1, 4 || 1, 9 || 7.76 || 0.97 || 27, 34, 41, 48, 54c, 55c, 61... || || 20480/19683 || [12, -9, 1> || 68.7 || [[Superpyth]] || 1197.60 || 708.17 || 1, 0 || 2, -1 || 6, -9 || 7.77 || 2.40 || 22, 27, 44, 49, 54c, 66, 71... || || 32805/32768 || [-15, 8, 1> || 1.95 || [[Helmholtz]] || 1200.07 || 701.79 || 1, 0 || 2, -1 || -1, 8 || 8.15 || 0.07 || 41, 53, 65, 70c, 77, 82, 89, 94, 99c... || || 78732/78125 || [2, 9, -7> || 13.4 || [[Sensi]] || 1199.59 || 756.60 || 1, 0 || 6, -7 || 8, -9 || 8.84 || 0.41 || 19, 38, 46, 57, 65, 73, 76, 84... || || 262144/253125 || [18, -4, -5> || 60.6 || [[Passion]] || 1198.31 || 98.40 || 1, 0 || 2, -5 || 2, 4 || 9.77 || 1.69 || 61, 73, 85c, 97c, 122bc... || || 273375/262144 || [-18, 7, 3> || 72.6 || [[Triton]] || 1202.01 || 569.09 || 1, 0 || 3, -3 || -1, 7 || 9.80 || 2.01 || 207bbccc, 226bbccc... || || 393216/390625 || [17, 1, -8> || 11.4 || [[Würschmidt]] || 1199.69 || 812.05 || 1, 0 || 7, -8 || 3, -1 || 10.10 || 0.31 || 31, 34, 62, 65, 68, 93, 96, 99... || || 531441/524288 || [-19, 12, 0> || 23.5 || [[Compton]] || 100.05 || 15.13 || 12, 0 || 19, 0 || 28, -1 || 10.33 || 0.62 || 60, 72, 84, 96, 108, 120, 120c, 132c... || || 1594323/1562500 || [-2, 13, -8> || 34.91 || [[Unicorn]] || 1200.85 || 62.64 || 1, 0 || 2, -8 || 3, -13 || 11.19 || 0.85 || 77, 96, 115, 154, 173, 192, 211... || || 1600000/1594323 || [9, -13, 5> || 6.15 || [[Amity]] || 1199.85 || 860.38 || 1, 0 || -2, 5 || -7, 13 || 11.20 || 0.15 || 53, 99, 106, 113, 145, 152... || || 2109375/2097152 || [-21, 3, 7> || 10.1 || [[Orson]] || 1200.24 || 271.65 || 1, 0 || 0, 7 || 3, -3 || 11.41 || 0.24 || 53, 84, 106, 137, 159, 168... || |||||||||||||||||||||||| TWO BONUS TEMPS: || || 6115295232/6103515625 || [23, 6, -14> || 3.34 || [[Vishnu]] || 599.97 || 71.15 || 2, 0 || 4, -7 || 5, -3 || 17.67 || 0.05 || || || 274877906944/274658203125 || [38, -2, -15> || 1.38 || [[Luna]] || 1199.98 || 193.196 || 1, 0 || 4, -15 || 2, 2 || 20.65 || 0.02 || ||
Original HTML content:
<html><head><title>Middle Path table of five-limit rank two temperaments</title></head><body>This table is reproduced from Table 1 of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a>'s <a class="wiki_link" href="/A%20Middle%20Path">A Middle Path</a>, with small modifications to update names and improve readability.<br />
<br />
The "main sequence" of this table "comprises all possible five-limit 2-D cases where complexity/12.33 + damage/10 < 1. The 'exotemperaments' have larger damage and lend themselves to special measures such as custom inharmonic timbres in order to aid harmoniousness. The 'bonus temperaments' are more complex but have exceedingly small damage and sound like JI."<br />
<br />
See also <a class="wiki_link" href="/Middle%20Path%20table%20of%20seven-limit%20rank%20two%20temperaments">Middle Path table of seven-limit rank two temperaments</a>, <a class="wiki_link" href="/Middle%20Path%20table%20of%20eleven-limit%20rank%20two%20temperaments">Middle Path table of eleven-limit rank two temperaments</a>.<br />
<br />
<table class="wiki_table">
<tr>
<td>Vanishing Interval's Ratio<br />
</td>
<td>V.I. Vector<br />
</td>
<td>V.I. cents<br />
</td>
<td>Temperament<br />
name<br />
</td>
<td>TOP<br />
period<br />
</td>
<td>TOP<br />
generator<br />
</td>
<td>Mapp.<br />
of 2<br />
</td>
<td>Mapp.<br />
of 3<br />
</td>
<td>Mapp.<br />
of 5<br />
</td>
<td>Cmplx.<br />
</td>
<td>TOP<br />
Dmg.<br />
</td>
<td>ETs<br />
</td>
</tr>
<tr>
<td colspan="12">TWO EXOTEMPERAMENTS:<br />
</td>
</tr>
<tr>
<td>16/15<br />
</td>
<td>[4, -1, -1><br />
</td>
<td>111.7<br />
</td>
<td><a class="wiki_link" href="/Father">Father</a><br />
</td>
<td>1185.9<br />
</td>
<td>447.4<br />
</td>
<td>1, 0<br />
</td>
<td>2, -1<br />
</td>
<td>2, 1<br />
</td>
<td>2.15<br />
</td>
<td>14.13<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27/25<br />
</td>
<td>[0, 3, -2><br />
</td>
<td>133.2<br />
</td>
<td><a class="wiki_link" href="/Bug">Bug</a><br />
</td>
<td>1200.0<br />
</td>
<td>260.3<br />
</td>
<td>1, 0<br />
</td>
<td>2, -2<br />
</td>
<td>3, -3<br />
</td>
<td>2.55<br />
</td>
<td>14.18<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td colspan="12">MAIN SEQUENCE:<br />
</td>
</tr>
<tr>
<td>25/24<br />
</td>
<td>[-3, -1, 2><br />
</td>
<td>70.7<br />
</td>
<td><a class="wiki_link" href="/Dicot">Dicot</a><br />
</td>
<td>1207.66<br />
</td>
<td>353.22<br />
</td>
<td>1, 0<br />
</td>
<td>1, 2<br />
</td>
<td>2, 1<br />
</td>
<td>2.51<br />
</td>
<td>7.66<br />
</td>
<td>24c, 41cc, 48cc, 58ccc...<br />
</td>
</tr>
<tr>
<td>81/80<br />
</td>
<td>[-4, 4, -1><br />
</td>
<td>21.5<br />
</td>
<td><a class="wiki_link" href="/Meantone">Meantone</a><br />
</td>
<td>1201.70<br />
</td>
<td>504.13<br />
</td>
<td>1, 0<br />
</td>
<td>2, -1<br />
</td>
<td>4, -4<br />
</td>
<td>3.44<br />
</td>
<td>1.70<br />
</td>
<td>12, 19, 24, 26, 29c, 31, 33c, 36, 38...<br />
</td>
</tr>
<tr>
<td>128/125<br />
</td>
<td>[7, 0, -3><br />
</td>
<td>41.1<br />
</td>
<td><a class="wiki_link" href="/Augmented">Augmented</a><br />
</td>
<td>399.02<br />
</td>
<td>93.15<br />
</td>
<td>3, 0<br />
</td>
<td>5, -1<br />
</td>
<td>7, 0<br />
</td>
<td>3.79<br />
</td>
<td>2.94<br />
</td>
<td>12, 15, 24, 27, 30, 33, 36, 39...<br />
</td>
</tr>
<tr>
<td>135/128<br />
</td>
<td>[-7, 3, 1><br />
</td>
<td>92.2<br />
</td>
<td><a class="wiki_link" href="/Mavila">Mavila</a><br />
</td>
<td>1206.55<br />
</td>
<td>685.03<br />
</td>
<td>1, 0<br />
</td>
<td>2, -1<br />
</td>
<td>1, 3<br />
</td>
<td>3.83<br />
</td>
<td>6.55<br />
</td>
<td>37bc, 44bcc, 67bbccc...<br />
</td>
</tr>
<tr>
<td>250/243<br />
</td>
<td>[1, -5, 3><br />
</td>
<td>49.2<br />
</td>
<td><a class="wiki_link" href="/Porcupine">Porcupine</a><br />
</td>
<td>1196.91<br />
</td>
<td>1034.59<br />
</td>
<td>1, 0<br />
</td>
<td>2, -3<br />
</td>
<td>3, -5<br />
</td>
<td>4.32<br />
</td>
<td>3.09<br />
</td>
<td>15, 22, 29, 30, 36c, 37, 43c...<br />
</td>
</tr>
<tr>
<td>256/243<br />
</td>
<td>[8, -5, 0><br />
</td>
<td>90.2<br />
</td>
<td><a class="wiki_link" href="/Blackwood">Blackwood</a><br />
</td>
<td>238.87<br />
</td>
<td>158.78<br />
</td>
<td>5, 0<br />
</td>
<td>8, 0<br />
</td>
<td>11, 1<br />
</td>
<td>4.33<br />
</td>
<td>5.67<br />
</td>
<td>15, 25, 30, 35bc, 40b, 45bc...<br />
</td>
</tr>
<tr>
<td>648/625<br />
</td>
<td>[3, 4, -4><br />
</td>
<td>62.6<br />
</td>
<td><a class="wiki_link" href="/Diminished">Diminished</a><br />
</td>
<td>299.16<br />
</td>
<td>197.49<br />
</td>
<td>4, 0<br />
</td>
<td>6, 1<br />
</td>
<td>9, 1<br />
</td>
<td>5.06<br />
</td>
<td>3.36<br />
</td>
<td>12, 24, 28, 32c, 36, 40, 44c...<br />
</td>
</tr>
<tr>
<td>2048/2025<br />
</td>
<td>[11, -4, -2><br />
</td>
<td>19.6<br />
</td>
<td><a class="wiki_link" href="/Srutal">Srutal</a><br />
</td>
<td>599.56<br />
</td>
<td>494.86<br />
</td>
<td>2, 0<br />
</td>
<td>3, 1<br />
</td>
<td>5, -2<br />
</td>
<td>5.97<br />
</td>
<td>0.89<br />
</td>
<td>12, 22, 24, 34, 36, 44, 46, 48c, 54, 56...<br />
</td>
</tr>
<tr>
<td>3125/3072<br />
</td>
<td>[-10, -1, 5><br />
</td>
<td>29.6<br />
</td>
<td><a class="wiki_link" href="/Magic">Magic</a><br />
</td>
<td>1201.28<br />
</td>
<td>380.80<br />
</td>
<td>1, 0<br />
</td>
<td>0, 5<br />
</td>
<td>2, 1<br />
</td>
<td>6.30<br />
</td>
<td>1.28<br />
</td>
<td>19, 22, 38, 41, 44, 47b, 54b, 57, 60...<br />
</td>
</tr>
<tr>
<td>6561/6250<br />
</td>
<td>[-1, 8, -5><br />
</td>
<td>84.1<br />
</td>
<td><a class="wiki_link" href="/Ripple">Ripple</a><br />
</td>
<td>1203.32<br />
</td>
<td>101.99<br />
</td>
<td>1, 0<br />
</td>
<td>2, -5<br />
</td>
<td>3, -8<br />
</td>
<td>6.87<br />
</td>
<td>3.32<br />
</td>
<td>12, 24, 36, 47, 48c, 59b...<br />
</td>
</tr>
<tr>
<td>15625/15552<br />
</td>
<td>[-6, -5, 6><br />
</td>
<td>8.11<br />
</td>
<td><a class="wiki_link" href="/Hanson">Hanson</a><br />
</td>
<td>1200.29<br />
</td>
<td>317.07<br />
</td>
<td>1, 0<br />
</td>
<td>0, 6<br />
</td>
<td>1, 5<br />
</td>
<td>7.57<br />
</td>
<td>0.29<br />
</td>
<td>19, 34, 38, 49, 53, 57, 64b...<br />
</td>
</tr>
<tr>
<td>16875/16384<br />
</td>
<td>[-14, 3, 4><br />
</td>
<td>51.1<br />
</td>
<td><a class="wiki_link" href="/Negri">Negri</a><br />
</td>
<td>1201.82<br />
</td>
<td>1075.68<br />
</td>
<td>1, 0<br />
</td>
<td>2, -4<br />
</td>
<td>2, 3<br />
</td>
<td>7.62<br />
</td>
<td>1.82<br />
</td>
<td>19, 29, 38, 48, 57, 58c, 66b...<br />
</td>
</tr>
<tr>
<td>20000/19683<br />
</td>
<td>[5, -9, 4><br />
</td>
<td>27.7<br />
</td>
<td><a class="wiki_link" href="/Tetracot">Tetracot</a><br />
</td>
<td>1199.03<br />
</td>
<td>176.11<br />
</td>
<td>1, 0<br />
</td>
<td>1, 4<br />
</td>
<td>1, 9<br />
</td>
<td>7.76<br />
</td>
<td>0.97<br />
</td>
<td>27, 34, 41, 48, 54c, 55c, 61...<br />
</td>
</tr>
<tr>
<td>20480/19683<br />
</td>
<td>[12, -9, 1><br />
</td>
<td>68.7<br />
</td>
<td><a class="wiki_link" href="/Superpyth">Superpyth</a><br />
</td>
<td>1197.60<br />
</td>
<td>708.17<br />
</td>
<td>1, 0<br />
</td>
<td>2, -1<br />
</td>
<td>6, -9<br />
</td>
<td>7.77<br />
</td>
<td>2.40<br />
</td>
<td>22, 27, 44, 49, 54c, 66, 71...<br />
</td>
</tr>
<tr>
<td>32805/32768<br />
</td>
<td>[-15, 8, 1><br />
</td>
<td>1.95<br />
</td>
<td><a class="wiki_link" href="/Helmholtz">Helmholtz</a><br />
</td>
<td>1200.07<br />
</td>
<td>701.79<br />
</td>
<td>1, 0<br />
</td>
<td>2, -1<br />
</td>
<td>-1, 8<br />
</td>
<td>8.15<br />
</td>
<td>0.07<br />
</td>
<td>41, 53, 65, 70c, 77, 82, 89, 94, 99c...<br />
</td>
</tr>
<tr>
<td>78732/78125<br />
</td>
<td>[2, 9, -7><br />
</td>
<td>13.4<br />
</td>
<td><a class="wiki_link" href="/Sensi">Sensi</a><br />
</td>
<td>1199.59<br />
</td>
<td>756.60<br />
</td>
<td>1, 0<br />
</td>
<td>6, -7<br />
</td>
<td>8, -9<br />
</td>
<td>8.84<br />
</td>
<td>0.41<br />
</td>
<td>19, 38, 46, 57, 65, 73, 76, 84...<br />
</td>
</tr>
<tr>
<td>262144/253125<br />
</td>
<td>[18, -4, -5><br />
</td>
<td>60.6<br />
</td>
<td><a class="wiki_link" href="/Passion">Passion</a><br />
</td>
<td>1198.31<br />
</td>
<td>98.40<br />
</td>
<td>1, 0<br />
</td>
<td>2, -5<br />
</td>
<td>2, 4<br />
</td>
<td>9.77<br />
</td>
<td>1.69<br />
</td>
<td>61, 73, 85c, 97c, 122bc...<br />
</td>
</tr>
<tr>
<td>273375/262144<br />
</td>
<td>[-18, 7, 3><br />
</td>
<td>72.6<br />
</td>
<td><a class="wiki_link" href="/Triton">Triton</a><br />
</td>
<td>1202.01<br />
</td>
<td>569.09<br />
</td>
<td>1, 0<br />
</td>
<td>3, -3<br />
</td>
<td>-1, 7<br />
</td>
<td>9.80<br />
</td>
<td>2.01<br />
</td>
<td>207bbccc, 226bbccc...<br />
</td>
</tr>
<tr>
<td>393216/390625<br />
</td>
<td>[17, 1, -8><br />
</td>
<td>11.4<br />
</td>
<td><a class="wiki_link" href="/W%C3%BCrschmidt">Würschmidt</a><br />
</td>
<td>1199.69<br />
</td>
<td>812.05<br />
</td>
<td>1, 0<br />
</td>
<td>7, -8<br />
</td>
<td>3, -1<br />
</td>
<td>10.10<br />
</td>
<td>0.31<br />
</td>
<td>31, 34, 62, 65, 68, 93, 96, 99...<br />
</td>
</tr>
<tr>
<td>531441/524288<br />
</td>
<td>[-19, 12, 0><br />
</td>
<td>23.5<br />
</td>
<td><a class="wiki_link" href="/Compton">Compton</a><br />
</td>
<td>100.05<br />
</td>
<td>15.13<br />
</td>
<td>12, 0<br />
</td>
<td>19, 0<br />
</td>
<td>28, -1<br />
</td>
<td>10.33<br />
</td>
<td>0.62<br />
</td>
<td>60, 72, 84, 96, 108, 120, 120c, 132c...<br />
</td>
</tr>
<tr>
<td>1594323/1562500<br />
</td>
<td>[-2, 13, -8><br />
</td>
<td>34.91<br />
</td>
<td><a class="wiki_link" href="/Unicorn">Unicorn</a><br />
</td>
<td>1200.85<br />
</td>
<td>62.64<br />
</td>
<td>1, 0<br />
</td>
<td>2, -8<br />
</td>
<td>3, -13<br />
</td>
<td>11.19<br />
</td>
<td>0.85<br />
</td>
<td>77, 96, 115, 154, 173, 192, 211...<br />
</td>
</tr>
<tr>
<td>1600000/1594323<br />
</td>
<td>[9, -13, 5><br />
</td>
<td>6.15<br />
</td>
<td><a class="wiki_link" href="/Amity">Amity</a><br />
</td>
<td>1199.85<br />
</td>
<td>860.38<br />
</td>
<td>1, 0<br />
</td>
<td>-2, 5<br />
</td>
<td>-7, 13<br />
</td>
<td>11.20<br />
</td>
<td>0.15<br />
</td>
<td>53, 99, 106, 113, 145, 152...<br />
</td>
</tr>
<tr>
<td>2109375/2097152<br />
</td>
<td>[-21, 3, 7><br />
</td>
<td>10.1<br />
</td>
<td><a class="wiki_link" href="/Orson">Orson</a><br />
</td>
<td>1200.24<br />
</td>
<td>271.65<br />
</td>
<td>1, 0<br />
</td>
<td>0, 7<br />
</td>
<td>3, -3<br />
</td>
<td>11.41<br />
</td>
<td>0.24<br />
</td>
<td>53, 84, 106, 137, 159, 168...<br />
</td>
</tr>
<tr>
<td colspan="12">TWO BONUS TEMPS:<br />
</td>
</tr>
<tr>
<td>6115295232/6103515625<br />
</td>
<td>[23, 6, -14><br />
</td>
<td>3.34<br />
</td>
<td><a class="wiki_link" href="/Vishnu">Vishnu</a><br />
</td>
<td>599.97<br />
</td>
<td>71.15<br />
</td>
<td>2, 0<br />
</td>
<td>4, -7<br />
</td>
<td>5, -3<br />
</td>
<td>17.67<br />
</td>
<td>0.05<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>274877906944/274658203125<br />
</td>
<td>[38, -2, -15><br />
</td>
<td>1.38<br />
</td>
<td><a class="wiki_link" href="/Luna">Luna</a><br />
</td>
<td>1199.98<br />
</td>
<td>193.196<br />
</td>
<td>1, 0<br />
</td>
<td>4, -15<br />
</td>
<td>2, 2<br />
</td>
<td>20.65<br />
</td>
<td>0.02<br />
</td>
<td><br />
</td>
</tr>
</table>
</body></html>