612edo: Difference between revisions
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== Theory == | == Theory == | ||
612edo is a very strong [[5-limit]] system, a fact noted by | 612edo is a very strong [[5-limit]] system, a fact noted by {{w|Robert Holford Macdowall Bosanquet|R. H. M. Bosanquet}} and {{w|James Murray Barbour}}. The equal temperament [[tempering out|tempers out]] the {{monzo| 485 -306 }} ([[sasktel comma]]) in the 3-limit, and in the 5-limit {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -52 -17 34 }} ([[septendecima]]), {{monzo| -53 10 16 }} ([[kwazy comma]]), {{monzo| 54 -37 2 }} ([[monzisma]]), {{monzo| -107 47 14 }} (fortune comma), and {{monzo| 161 -84 -12 }} ([[atom]]). In the 7-limit it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s the [[ennealimmal]] temperament, and in fact provides the [[optimal patent val]] for ennealimmal. The 7-limit val for 612 can be characterized as the ennealimmal commas plus the kwazy comma. In the 11-limit, it tempers out [[3025/3024]] and [[9801/9800]], so that 612 supports the [[hemiennealimmal]] temperament. In the 13-limit, it tempers [[2200/2197]] and [[4096/4095]]. | ||
The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]]. | The 612edo has been proposed as the logarithmic [[interval size measure]] '''skisma''' (or '''sk'''), since one step is nearly the same size as the [[schisma]] (32805/32768), 1/12 of a [[Pythagorean comma]] or 1/11 of a [[syntonic comma]]. Since 612 is divisible by {{EDOs| 2, 3, 4, 6, 9, 12, 17, 18, 34, 36, 51, 68, 102, 153, 204 and 306 }}, it can readily express the step sizes of the 12, 17, 34, and 68 divisions. A table of intervals approximated by 612 can be found under [[Table of 612edo intervals]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|612 | {{Harmonics in equal|612}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 23: | Line 23: | ||
| 2.3.5 | | 2.3.5 | ||
| {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }} | | {{monzo| 1 -27 18 }}, {{monzo| -53 10 16 }} | ||
| | | {{mapping| 612 970 1421 }} | ||
| +0.0044 | | +0.0044 | ||
| 0.0089 | | 0.0089 | ||
| Line 30: | Line 30: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, {{monzo| -53 10 16 }} | | 2401/2400, 4375/4374, {{monzo| -53 10 16 }} | ||
| | | {{mapping| 612 970 1421 1718 }} | ||
| +0.0210 | | +0.0210 | ||
| 0.0297 | | 0.0297 | ||
| Line 37: | Line 37: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }} | | 2401/2400, 3025/3024, 4375/4374, {{monzo| 21 -6 -7 -2 3 }} | ||
| | | {{mapping| 612 970 1421 1718 2117 }} | ||
| +0.0363 | | +0.0363 | ||
| 0.0406 | | 0.0406 | ||
| Line 44: | Line 44: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374 | | 2200/2197, 2401/2400, 3025/3024, 4096/4095, 4375/4374 | ||
| | | {{mapping| 612 970 1421 1718 2117 2265 }} | ||
| +0.0010 | | +0.0010 | ||
| 0.0871 | | 0.0871 | ||
| Line 51: | Line 51: | ||
| 2.3.5.7.11.13.19 | | 2.3.5.7.11.13.19 | ||
| 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095 | | 1331/1330, 1540/1539, 2200/2197, 2376/2375, 2926/2925, 4096/4095 | ||
| | | {{mapping| 612 970 1421 1718 2117 2265 2600 }} | ||
| -0.0168 | | -0.0168 | ||
| 0.0917 | | 0.0917 | ||