103169edo: Difference between revisions
this does qualify for "extraordinarily" (basically no relative error, like 4296 does on the 5-limit) |
Tristanbay (talk | contribs) Giving most edo pages over 100000 the mathematical interest note |
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{{Mathematical interest}} | |||
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{{ | 103169edo provides an extraordinarily strong 7-limit system, tempering out {{monzo| 9 -28 37 -18 }}, {{monzo| -92 -17 21 25 }}, and {{monzo| 110 -71 -11 10 }}. It maps the starling comma ([[126/125]]) to 1186 steps, the gamelisma (1029/1024) to 725 steps, the marvel comma ([[225/224]]) to 663 steps, the hemifamity comma (5120/5103) to 495 steps, the breedsma ([[2401/2400]]) to 62 steps, and the ragisma ([[4375/4374]]) to 34 steps. The [[patent val]] tempers out {{monzo| 20 3 -9 -10 7 }}, {{monzo| -29 16 5 -9 5 }}, {{monzo| 3 -26 13 -7 8 }}, and {{monzo| -6 2 -24 11 8 }} in the 11-limit, and 5767168/5767125, 1610510000/1610497161, 12784876137/12784844800, 26796875000/26796587103, and {{monzo| -17 -5 -13 0 17 -1 }} in the 13-limit. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|103169|prec=7}} | |||
=== Subsets and supersets === | |||
Since 103169 factors into 11 × 83 × 113, 103169edo has subset edos 11, 83, 113, 913, 1243, and 9379. | |||
Latest revision as of 17:50, 8 September 2025
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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. |
| ← 103168edo | 103169edo | 103170edo → |
103169 equal divisions of the octave (abbreviated 103169edo or 103169ed2), also called 103169-tone equal temperament (103169tet) or 103169 equal temperament (103169et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 103169 equal parts of about 0.0116 ¢ each. Each step represents a frequency ratio of 21/103169, or the 103169th root of 2.
103169edo provides an extraordinarily strong 7-limit system, tempering out [9 -28 37 -18⟩, [-92 -17 21 25⟩, and [110 -71 -11 10⟩. It maps the starling comma (126/125) to 1186 steps, the gamelisma (1029/1024) to 725 steps, the marvel comma (225/224) to 663 steps, the hemifamity comma (5120/5103) to 495 steps, the breedsma (2401/2400) to 62 steps, and the ragisma (4375/4374) to 34 steps. The patent val tempers out [20 3 -9 -10 7⟩, [-29 16 5 -9 5⟩, [3 -26 13 -7 8⟩, and [-6 2 -24 11 8⟩ in the 11-limit, and 5767168/5767125, 1610510000/1610497161, 12784876137/12784844800, 26796875000/26796587103, and [-17 -5 -13 0 17 -1⟩ in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000000 | +0.0000438 | +0.0000044 | +0.0000005 | -0.0011709 | +0.0038933 | -0.0052791 | -0.0050438 | -0.0042274 | -0.0004802 | -0.0055731 |
| Relative (%) | +0.0 | +0.4 | +0.0 | +0.0 | -10.1 | +33.5 | -45.4 | -43.4 | -36.3 | -4.1 | -47.9 | |
| Steps (reduced) |
103169 (0) |
163519 (60350) |
239551 (33213) |
289632 (83294) |
356906 (47399) |
381771 (72264) |
421699 (9023) |
438254 (25578) |
466691 (54015) |
501193 (88517) |
511119 (98443) | |
Subsets and supersets
Since 103169 factors into 11 × 83 × 113, 103169edo has subset edos 11, 83, 113, 913, 1243, and 9379.