38ed7/3: Difference between revisions
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{{ED intro}} | {{ED intro}} | ||
While 38ed7/3 fails to accurately represent low | == 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. | 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 == | ||
| Line 241: | Line 246: | ||
|1466.8709 | |1466.8709 | ||
|} | |} | ||
Revision as of 18:17, 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
| Degrees | Enneatonic | ed11\9~ed7/3 | ||
|---|---|---|---|---|
| 1 | G^ | 38.5965 | 38.6019 | |
| Jbv | Abv | |||
| 2 | Jb | Ab | 77.193 | 77.2037 |
| 3 | Jb^ | Ab^ | 115.7895 | 115.8056 |
| G#v | ||||
| 4 | G# | 154.386 | 154.4075 | |
| 5 | G#^ | 192.98245 | 193.0093 | |
| Jv | Av | |||
| 6 | J | A | 231.57895 | 231.6112 |
| 7 | J^/Av | A^/Bv | 270.1754 | 270.2131 |
| 8 | A | B | 308.7719 | 308.8149 |
| 9 | A^/Bbv | B^/Cbv | 347.3684 | 347.4168 |
| 10 | Bb | Cb | 385.9649 | 386.0187 |
| 11 | Bb^/A#v | Cb^/B#v | 424.5614 | 424.6205 |
| 12 | A# | B# | 463.1579 | 463.2224 |
| 13 | A#^/Bv | B#^/Cv | 501.7544 | 502.6667 |
| 14 | B | C | 540.3509 | 540.4261 |
| 15 | B^/Cv | C^/Qv | 578.9474 | 579.028 |
| 16 | C | Q | 617.5439 | 617.6299 |
| 17 | C^/Qbv | Q^/Dbv | 656.14035 | 656.2317 |
| 18 | Qb | Db | 694.7368 | 694.8336 |
| 19 | Qb^/C#v | Db^/Q#v | 733.3 | 733.43545 |
| 20 | C# | Q# | 771.9298 | 772.0373 |
| 21 | C#^/Qv | Q#/Dv | 810.5263 | 810.6392 |
| 22 | Q | D | 849.1228 | 849.24105 |
| 23 | Q^/Dv | D^/Sv | 887.7193 | 887.8429 |
| 24 | D | S | 926.3158 | 926.4448 |
| 25 | D^ | S^ | 964.9123 | 965.04665 |
| Ebv | ||||
| 26 | Eb | 1003.5088 | 1003.6485 | |
| 27 | Eb^ | 1042.1053 | 1042.2504 | |
| D#v | S#v | |||
| 28 | D# | S# | 1080.70175 | 1080.85225 |
| 29 | D#^ | S#^ | 1119.29825 | 1119.4541 |
| Ev | ||||
| 30 | E | 1157.8947 | 1158.0559 | |
| 31 | E^/Fbv | 1196.4912 | 1196.6578 | |
| 32 | Fb | 1235.0877 | 1235.2567 | |
| 33 | Fb^/E#v | 1273.68425 | 1273.8616 | |
| 34 | E# | 1312.2807 | 1312.4634 | |
| 35 | E#^/Fv | 1350.8772 | 1351.0654 | |
| 36 | F | 1389.4737 | 1389.6672 | |
| 37 | F^/Gv | 1428.0702 | 1428.269 | |
| 38 | G | 1466.6 | 1466.8709 | |