146edo: Difference between revisions
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Subgroup interpretation |
Clarify (major third -> harmonic 5) |
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{{EDO intro}} | {{EDO intro}} | ||
146edo has an accurate | 146edo has an accurate [[harmonic]] [[5/1|5]], only 0.012344 cents compressed from just. 146 is the denominator of a convergent to log<sub>2</sub>5, after [[3edo|3]], [[28edo|28]] and [[59edo|59]], and before [[643edo|643]]. Combined with fairly accurate approximations of [[7/1|7]], [[9/1|9]], [[11/1|11]], [[17/1|17]], and [[19/1|19]], it commends itself as a 2.9.5.7.11.13.17.19 [[subgroup]] system. | ||
However, it also provides the [[optimal patent val]] for the 11-limit [[newspeak]] temperament. Using the [[patent val]], it [[tempering out|tempers out]] the 2109375/2097152 ([[semicomma]]), and {{monzo| -6 17 -9 }} in the 5-limit; [[225/224]], [[1728/1715]], and 100442349/97656250 in the 7-limit; [[441/440]], 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; [[1001/1000]], [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit. | However, it also provides the [[optimal patent val]] for the 11-limit [[newspeak]] temperament. Using the [[patent val]], it [[tempering out|tempers out]] the 2109375/2097152 ([[semicomma]]), and {{monzo| -6 17 -9 }} in the 5-limit; [[225/224]], [[1728/1715]], and 100442349/97656250 in the 7-limit; [[441/440]], 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; [[1001/1000]], [[1188/1183]], [[1287/1280]], and [[1573/1568]] in the 13-limit. | ||
Revision as of 15:05, 18 May 2024
| ← 145edo | 146edo | 147edo → |
146edo has an accurate harmonic 5, only 0.012344 cents compressed from just. 146 is the denominator of a convergent to log25, after 3, 28 and 59, and before 643. Combined with fairly accurate approximations of 7, 9, 11, 17, and 19, it commends itself as a 2.9.5.7.11.13.17.19 subgroup system.
However, it also provides the optimal patent val for the 11-limit newspeak temperament. Using the patent val, it tempers out the 2109375/2097152 (semicomma), and [-6 17 -9⟩ in the 5-limit; 225/224, 1728/1715, and 100442349/97656250 in the 7-limit; 441/440, 1375/1372, 1944/1925, and 43923/43750 in the 11-limit; 1001/1000, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.32 | -0.01 | +1.04 | +1.57 | -0.63 | -2.17 | -3.34 | +1.89 | -1.62 | -2.29 | -3.62 |
| Relative (%) | -40.5 | -0.2 | +12.6 | +19.1 | -7.7 | -26.4 | -40.6 | +23.0 | -19.7 | -27.8 | -44.0 | |
| Steps (reduced) |
231 (85) |
339 (47) |
410 (118) |
463 (25) |
505 (67) |
540 (102) |
570 (132) |
597 (13) |
620 (36) |
641 (57) |
660 (76) | |
Subsets and supersets
Since 146 factors into 2 × 73, 146edo contains 2edo and 73edo as its subsets.