742edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro | {{EDO intro}} | ||
== Theory == | == Theory == | ||
742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency|distinctly consistent]] in the [[21-odd-limit]]. | 742edo is a very strong 19-limit system and a [[zeta peak edo]], and is [[consistency|distinctly consistent]] in the [[21-odd-limit]]. It [[tempers out]] the [[vishnuzma]] and the fortune comma in the 5-limit, [[support]]ing [[vishnu]] and [[fortune]]; [[2401/2400]] in the 7-limit, [[9801/9800]] in the 11-limit, [[4096/4095]], [[6656/6655]], [[10648/10647]] in the 13-limit, [[1701/1700]], [[2058/2057]], [[2601/2600]], [[4914/4913]], [[5832/5831]] in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 742 factors into {{factorization|742}}, 742edo has subset edos {{EDOs| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] (224 & 518) with period 14 in the 13-limit, and [[iodine]] (159 & 583f) with period 53 in the 17-limit. | Since 742 factors into {{factorization|742}}, 742edo has subset edos {{EDOs| 2, 7, 14, 53, 106, and 371 }}, of which [[7edo]], [[14edo]] and [[53edo]] are very notable. It supports [[silicon]] ({{nowrap|224 & 518}}) with period 14 in the 13-limit, and [[iodine]] ({{nowrap|159 & 583f}}) with period 53 in the 17-limit. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 26: | Line 26: | ||
| {{monzo| 23 6 -14 }}, {{monzo| -84 53 }} | | {{monzo| 23 6 -14 }}, {{monzo| -84 53 }} | ||
| {{mapping| 742 1176 1723 }} | | {{mapping| 742 1176 1723 }} | ||
| | | −0.0157 | ||
| 0.0555 | | 0.0555 | ||
| 3.43 | | 3.43 | ||
| Line 33: | Line 33: | ||
| 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }} | | 2401/2400, 14348907/14336000, {{monzo| 23 6 -14 }} | ||
| {{mapping| 742 1176 1723 2083 }} | | {{mapping| 742 1176 1723 2083 }} | ||
| | | −0.0035 | ||
| 0.0525 | | 0.0525 | ||
| 3.24 | | 3.24 | ||
| Line 40: | Line 40: | ||
| 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 | | 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 | ||
| {{mapping| 742 1176 1723 2083 2567 }} | | {{mapping| 742 1176 1723 2083 2567 }} | ||
| | | −0.0123 | ||
| 0.0501 | | 0.0501 | ||
| 3.10 | | 3.10 | ||
| Line 47: | Line 47: | ||
| 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 | | 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 | ||
| {{mapping| 742 1176 1723 2083 2567 2746 }} | | {{mapping| 742 1176 1723 2083 2567 2746 }} | ||
| | | −0.0302 | ||
| 0.0608 | | 0.0608 | ||
| 3.76 | | 3.76 | ||
| Line 54: | Line 54: | ||
| 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 | | 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 | ||
| {{mapping| 742 1176 1723 2083 2567 2746 3033 }} | | {{mapping| 742 1176 1723 2083 2567 2746 3033 }} | ||
| | | −0.0317 | ||
| 0.0564 | | 0.0564 | ||
| 3.49 | | 3.49 | ||
| Line 61: | Line 61: | ||
| 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 | | 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 | ||
| {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 }} | | {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 }} | ||
| | | −0.0295 | ||
| 0.0531 | | 0.0531 | ||
| 3.28 | | 3.28 | ||
| Line 68: | Line 68: | ||
| 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 | | 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 | ||
| {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i) | | {{mapping| 742 1176 1723 2083 2567 2746 3033 3152 3357 }} (742i) | ||
| | | −0.0468 | ||
| 0.0699 | | 0.0699 | ||
| 4.32 | | 4.32 | ||
| Line 121: | Line 121: | ||
| [[Iodine]] | | [[Iodine]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Revision as of 14:04, 16 January 2025
| ← 741edo | 742edo | 743edo → |
Theory
742edo is a very strong 19-limit system and a zeta peak edo, and is distinctly consistent in the 21-odd-limit. It tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.068 | +0.209 | -0.093 | +0.165 | +0.443 | +0.166 | +0.061 | -0.781 | +0.611 | -0.022 |
| Relative (%) | +0.0 | -4.2 | +12.9 | -5.7 | +10.2 | +27.4 | +10.3 | +3.8 | -48.3 | +37.8 | -1.4 | |
| Steps (reduced) |
742 (0) |
1176 (434) |
1723 (239) |
2083 (599) |
2567 (341) |
2746 (520) |
3033 (65) |
3152 (184) |
3356 (388) |
3605 (637) |
3676 (708) | |
Subsets and supersets
Since 742 factors into 2 × 7 × 53, 742edo has subset edos 2, 7, 14, 53, 106, and 371, of which 7edo, 14edo and 53edo are very notable. It supports silicon (224 & 518) with period 14 in the 13-limit, and iodine (159 & 583f) with period 53 in the 17-limit.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [23 6 -14⟩, [-84 53⟩ | [⟨742 1176 1723]] | −0.0157 | 0.0555 | 3.43 |
| 2.3.5.7 | 2401/2400, 14348907/14336000, [23 6 -14⟩ | [⟨742 1176 1723 2083]] | −0.0035 | 0.0525 | 3.24 |
| 2.3.5.7.11 | 2401/2400, 9801/9800, 172032/171875, 1240029/1239040 | [⟨742 1176 1723 2083 2567]] | −0.0123 | 0.0501 | 3.10 |
| 2.3.5.7.11.13 | 2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325 | [⟨742 1176 1723 2083 2567 2746]] | −0.0302 | 0.0608 | 3.76 |
| 2.3.5.7.11.13.17 | 1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655 | [⟨742 1176 1723 2083 2567 2746 3033]] | −0.0317 | 0.0564 | 3.49 |
| 2.3.5.7.11.13.17.19 | 1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211 | [⟨742 1176 1723 2083 2567 2746 3033 3152]] | −0.0295 | 0.0531 | 3.28 |
| 2.3.5.7.11.13.17.19.23 | 1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211 | [⟨742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i) | −0.0468 | 0.0699 | 4.32 |
- 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
- 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 137\742 | 221.563 | 8388608/7381125 | Fortune |
| 1 | 303\742 | 490.026 | 896/675 | Surmarvelpyth |
| 2 | 44\742 | 71.159 | 25/24 | Vishnu |
| 14 | 434\742 (10\742) |
701.886 (16.173) |
3/2 (105/104) |
Silicon |
| 53 | 239\742 (1\742) |
386.523 (1.617) |
5/4 (32805/32768) |
Mercator |
| 53 | 565\742 (5\742) |
913.746 (8.086) |
441/260 (196/195) |
Iodine |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct