742edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{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 32: | Line 24: | ||
| 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 39: | Line 31: | ||
| 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 46: | Line 38: | ||
| 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 53: | Line 45: | ||
| 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 60: | Line 52: | ||
| 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 67: | Line 59: | ||
| 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 | ||
{{comma basis end}} | |||
* 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments. It is only bettered by [[935edo|935]] in terms of absolute error, and by [[1178edo|1178]] in terms of relative error. | * 742et has a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any previous equal temperaments. It is only bettered by [[935edo|935]] in terms of absolute error, and by [[1178edo|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 [[581edo|581]] and is bettered by [[764edo|764]]; in the 23-limit it beats [[718edo|718]] and is bettered by [[814edo|814]]. | * 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 [[581edo|581]] and is bettered by [[764edo|764]]; in the 23-limit it beats [[718edo|718]] and is bettered by [[814edo|814]]. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 101: | Line 88: | ||
|- | |- | ||
| 14 | | 14 | ||
| 434\742<br>(10\742) | | 434\742<br />(10\742) | ||
| 701.886<br>(16.173) | | 701.886<br />(16.173) | ||
| 3/2<br>(105/104) | | 3/2<br />(105/104) | ||
| [[Silicon]] | | [[Silicon]] | ||
|- | |- | ||
| 53 | | 53 | ||
| 239\742<br>(1\742) | | 239\742<br />(1\742) | ||
| 386.523<br>(1.617) | | 386.523<br />(1.617) | ||
| 5/4<br>(32805/32768) | | 5/4<br />(32805/32768) | ||
| [[Mercator]] | | [[Mercator]] | ||
|- | |- | ||
| 53 | | 53 | ||
| 565\742<br>(5\742) | | 565\742<br />(5\742) | ||
| 913.746<br>(8.086) | | 913.746<br />(8.086) | ||
| 441/260<br>(196/195) | | 441/260<br />(196/195) | ||
| [[Iodine]] | | [[Iodine]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
Revision as of 05:11, 16 November 2024
| ← 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. The equal temperament 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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf