38ed7/3: Difference between revisions
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{{Infobox ET}} | |||
{{ED intro}} | |||
== Theory == | |||
While 38ed7/3 fails to accurately represent low [[prime interval|prime harmonics]], it provides great approximations of the [[13/1|13th]], [[17/1|17th]], [[19/1|19th]], and a multitude of higher primes, and also handles the interval of [[5/3]] well. But 38ed7/3 should, most of all, be noted for the exceptional quality of its approximation to [[11/9]], which is a mere 0.0088 cents off from just. Its natural subgroup in the [[19-limit]] is 5/3.7/3.11/9.13.17.19, but this can extend to include higher primes, especially [[29/1|29]], [[31/1|31]], and [[37/1|37]]. | |||
38ed7/3 possesses a shimmering octave at 31 steps in, therefore making this a potential octave-compressed version of [[31edo]], one that sacrifices its notable accuracy in the [[7-limit]] (though a number of 7-limit intervals are still portrayed passably due to the common flat tendency of harmonics 2, 3, 5, and 7) in favor of a huge number of high primes. | |||
=== Harmonics === | |||
{{Harmonics in equal|38|7|3|columns=11}} | |||
{{Harmonics in equal|38|7|3|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 38ed7/3 (continued)}} | |||
== Intervals == | == Intervals == | ||
{| class="wikitable" | {| class="wikitable center-1 right-2" | ||
! | ! # | ||
! | ! Cents | ||
|- | |- | ||
| 1 | |||
| 38.6 | |||
|- | |- | ||
| | | 2 | ||
| | | 77.2 | ||
|- | |- | ||
| | | 3 | ||
| | | 115.8 | ||
|- | |- | ||
| | | 4 | ||
| 154.4 | |||
| | |||
|- | |- | ||
| | | 5 | ||
| 193.0 | |||
|- | |- | ||
| | | 6 | ||
| | | 231.6 | ||
|- | |- | ||
| | | 7 | ||
| 270.2 | |||
| | |||
|- | |- | ||
| | | 8 | ||
| | | 308.8 | ||
|- | |- | ||
| | | 9 | ||
| | | 347.4 | ||
|- | |- | ||
| | | 10 | ||
| | | 386.0 | ||
|- | |- | ||
| | | 11 | ||
| | | 424.6 | ||
|- | |- | ||
| | | 12 | ||
| | | 463.2 | ||
|- | |- | ||
| | | 13 | ||
| | | 502.7 | ||
|- | |- | ||
| | | 14 | ||
| | | 540.4 | ||
|- | |- | ||
| | | 15 | ||
| | | 579.0 | ||
|- | |- | ||
| | | 16 | ||
| | | 617.6 | ||
|- | |- | ||
| | | 17 | ||
| | | 656.2 | ||
|- | |- | ||
| | | 18 | ||
| | | 694.8 | ||
|- | |- | ||
| | | 19 | ||
| | | 733.4 | ||
|- | |- | ||
| | | 20 | ||
| | | 772.0 | ||
|- | |- | ||
| | | 21 | ||
| | | 810.6 | ||
|- | |- | ||
| | | 22 | ||
| | | 849.2 | ||
|- | |- | ||
| | | 23 | ||
| | | 887.8 | ||
|- | |- | ||
| | | 24 | ||
| | | 926.4 | ||
|- | |- | ||
| | | 25 | ||
| | | 965.0 | ||
|- | |- | ||
| | | 26 | ||
| | | 1003.6 | ||
|- | |- | ||
| | | 27 | ||
| | | 1042.3 | ||
|- | |- | ||
| | | 28 | ||
| 1080.9 | |||
| | |||
|- | |- | ||
| | | 29 | ||
| 1119.5 | |||
|- | |- | ||
| | | 30 | ||
| | | 1158.1 | ||
|- | |- | ||
| | | 31 | ||
| | | 1196.7 | ||
|- | |- | ||
| | | 32 | ||
| | | 1235.3 | ||
|- | |- | ||
| | | 33 | ||
| | | 1273.9 | ||
|- | |- | ||
| | | 34 | ||
| 1312.5 | |||
| | |||
|- | |- | ||
| | | 35 | ||
| 1351.1 | |||
|- | |- | ||
| | | 36 | ||
| | | 1389.7 | ||
|- | |- | ||
| | | 37 | ||
| | | 1428.3 | ||
|- | |- | ||
| 38 | |||
| 1466.9 | |||
|38 | |||
|1466. | |||
|} | |} | ||
Latest revision as of 18:22, 14 May 2026
| ← 37ed7/3 | 38ed7/3 | 39ed7/3 → |
(semiconvergent)
38 equal divisions of 7/3 (abbreviated 38ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 38 equal parts of about 38.6 ¢ each. Each step represents a frequency ratio of (7/3)1/38, or the 38th root of 7/3.
Theory
While 38ed7/3 fails to accurately represent low prime harmonics, it provides great approximations of the 13th, 17th, 19th, and a multitude of higher primes, and also handles the interval of 5/3 well. But 38ed7/3 should, most of all, be noted for the exceptional quality of its approximation to 11/9, which is a mere 0.0088 cents off from just. Its natural subgroup in the 19-limit is 5/3.7/3.11/9.13.17.19, but this can extend to include higher primes, especially 29, 31, and 37.
38ed7/3 possesses a shimmering octave at 31 steps in, therefore making this a potential octave-compressed version of 31edo, one that sacrifices its notable accuracy in the 7-limit (though a number of 7-limit intervals are still portrayed passably due to the common flat tendency of harmonics 2, 3, 5, and 7) in favor of a huge number of high primes.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.3 | -10.5 | -6.7 | -7.0 | -13.8 | -10.5 | -10.0 | +17.7 | -10.3 | +17.7 | -17.1 |
| Relative (%) | -8.7 | -27.1 | -17.3 | -18.1 | -35.8 | -27.1 | -26.0 | +45.8 | -26.7 | +45.8 | -44.4 | |
| Steps (reduced) |
31 (31) |
49 (11) |
62 (24) |
72 (34) |
80 (4) |
87 (11) |
93 (17) |
99 (23) |
103 (27) |
108 (32) |
111 (35) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.3 | -13.8 | -17.4 | -13.4 | -2.5 | +14.3 | -2.1 | -13.7 | +17.7 | +14.3 | +14.6 | +18.1 |
| Relative (%) | -3.4 | -35.8 | -45.2 | -34.6 | -6.5 | +37.1 | -5.4 | -35.4 | +45.8 | +37.2 | +37.8 | +46.9 | |
| Steps (reduced) |
115 (1) |
118 (4) |
121 (7) |
124 (10) |
127 (13) |
130 (16) |
132 (18) |
134 (20) |
137 (23) |
139 (25) |
141 (27) |
143 (29) | |
Intervals
| # | Cents |
|---|---|
| 1 | 38.6 |
| 2 | 77.2 |
| 3 | 115.8 |
| 4 | 154.4 |
| 5 | 193.0 |
| 6 | 231.6 |
| 7 | 270.2 |
| 8 | 308.8 |
| 9 | 347.4 |
| 10 | 386.0 |
| 11 | 424.6 |
| 12 | 463.2 |
| 13 | 502.7 |
| 14 | 540.4 |
| 15 | 579.0 |
| 16 | 617.6 |
| 17 | 656.2 |
| 18 | 694.8 |
| 19 | 733.4 |
| 20 | 772.0 |
| 21 | 810.6 |
| 22 | 849.2 |
| 23 | 887.8 |
| 24 | 926.4 |
| 25 | 965.0 |
| 26 | 1003.6 |
| 27 | 1042.3 |
| 28 | 1080.9 |
| 29 | 1119.5 |
| 30 | 1158.1 |
| 31 | 1196.7 |
| 32 | 1235.3 |
| 33 | 1273.9 |
| 34 | 1312.5 |
| 35 | 1351.1 |
| 36 | 1389.7 |
| 37 | 1428.3 |
| 38 | 1466.9 |