241200edo: Difference between revisions
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{{Mathematical interest}} | |||
{{Infobox ET}} | {{Infobox ET}} | ||
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241200edo is the 60th [[zeta peak edo]] and the first one that 1200 divides, making it compatible with [[cent]]s. It is also a [[zeta peak integer edo]]. It is a strong 37-limit system, distinctly [[consistent]] in the 39-odd-limit, with a lower 37-limit [[relative error]] than any previous equal temperaments. | |||
=== Prime harmonics === | |||
{{Harmonics in equal|241200|columns=9}} | |||
{{Harmonics in equal|241200|columns=9|start=10|collapsed=1|title=Approximation of prime harmonics in 241200edo (continued)}} | |||
{{Harmonics in equal|241200|columns=9|start=19|collapsed=1|title=Approximation of prime harmonics in 241200edo (continued)}} | |||
Latest revision as of 11:15, 15 January 2026
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 241199edo | 241200edo | 241201edo → |
241200 equal divisions of the octave (abbreviated 241200edo or 241200ed2), also called 241200-tone equal temperament (241200tet) or 241200 equal temperament (241200et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 241200 equal parts of about 0.00498 ¢ each. Each step represents a frequency ratio of 21/241200, or the 241200th root of 2.
241200edo is the 60th zeta peak edo and the first one that 1200 divides, making it compatible with cents. It is also a zeta peak integer edo. It is a strong 37-limit system, distinctly consistent in the 39-odd-limit, with a lower 37-limit relative error than any previous equal temperaments.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | +0.00022 | -0.00028 | -0.00004 | +0.00047 | -0.00030 | -0.00019 | -0.00058 | -0.00072 |
| Relative (%) | +0.0 | +4.5 | -5.6 | -0.7 | +9.4 | -6.0 | -3.7 | -11.6 | -14.4 | |
| Steps (reduced) |
241200 (0) |
382293 (141093) |
560049 (77649) |
677134 (194734) |
834415 (110815) |
892546 (168946) |
985896 (21096) |
1024600 (59800) |
1091083 (126283) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.00008 | -0.00075 | -0.00076 | +0.00227 | -0.00029 | +0.00084 | -0.00206 | +0.00247 | +0.00077 |
| Relative (%) | -1.6 | -15.0 | -15.2 | +45.6 | -5.9 | +16.9 | -41.4 | +49.6 | +15.4 | |
| Steps (reduced) |
1171745 (206945) |
1194952 (230152) |
1256520 (50520) |
1292242 (86242) |
1308815 (102815) |
1339767 (133767) |
1381574 (175574) |
1418894 (212894) |
1430494 (224494) | |
| Harmonic | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00143 | -0.00003 | +0.00157 | +0.00042 | +0.00243 | +0.00048 | +0.00011 | +0.00194 | +0.00036 |
| Relative (%) | +28.7 | -0.5 | +31.6 | +8.4 | +48.9 | +9.6 | +2.2 | +39.0 | +7.3 | |
| Steps (reduced) |
1463141 (15941) |
1483319 (36119) |
1492986 (45786) |
1520472 (73272) |
1537660 (90460) |
1561947 (114747) |
1591899 (144699) |
1605961 (158761) |
1612784 (165584) | |