11358058edo: Difference between revisions
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{{ | {{Mathematical interest}} | ||
{{Infobox ET | |||
{{Infobox ET|Consistency=35|Distinct consistency=35 | | Prime factorization = 2 × 5679029 | ||
| Consistency = 35 | |||
{{ | | Distinct consistency = 35 | ||
}} | |||
{{ED intro}} | |||
Although its step size is far beyond the human melodic [[just-noticeable difference]], it has been noted for its highly accurate approximation of the 31 | Although its step size is far beyond the human melodic [[just-noticeable difference]], it has been noted for its highly accurate approximation of the [[31-limit]], and is [[consistent]] up to the [[Odd prime sum limit|36-OPSL]], where it has a lower maximum error (i.e. the error of the least accurate approximation of any interval in the limit from JI) than any smaller [[edo]], meaning it is very likely a [[The Riemann zeta function and tuning|zeta peak]] edo. | ||
While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts. | While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal | {{Harmonics in equal|11358058|columns=9}} | ||
|11358058 | {{Harmonics in equal|11358058|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 11358058edo (continued)}} | ||
|columns = | |||
}} | |||
Latest revision as of 22:08, 10 August 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. |
| ← 11358057edo | 11358058edo | 11358059edo → |
11358058 equal divisions of the octave (abbreviated 11358058edo or 11358058ed2), also called 11358058-tone equal temperament (11358058tet) or 11358058 equal temperament (11358058et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 11358058 equal parts of about 0.000106 ¢ each. Each step represents a frequency ratio of 21/11358058, or the 11358058th root of 2.
Although its step size is far beyond the human melodic just-noticeable difference, it has been noted for its highly accurate approximation of the 31-limit, and is consistent up to the 36-OPSL, where it has a lower maximum error (i.e. the error of the least accurate approximation of any interval in the limit from JI) than any smaller edo, meaning it is very likely a zeta peak edo.
While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is limited, and the sequence of intervals may eventually start to deviate from their true JI counterparts.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000000 | -0.0000012 | +0.0000028 | -0.0000033 | +0.0000030 | +0.0000059 | -0.0000025 | -0.0000082 | -0.0000067 |
| Relative (%) | +0.0 | -1.1 | +2.6 | -3.1 | +2.8 | +5.6 | -2.4 | -7.7 | -6.3 | |
| Steps (reduced) |
11358058 (0) |
18002096 (6644038) |
26372594 (3656478) |
31886100 (9169984) |
39292425 (5218251) |
42029809 (7955635) |
46425640 (993408) |
48248207 (2815975) |
51378879 (5946647) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000100 | -0.0000039 | -0.0000502 | -0.0000429 | +0.0000204 | -0.0000363 | +0.0000272 | +0.0000055 | +0.0000356 |
| Relative (%) | +9.4 | -3.7 | -47.5 | -40.6 | +19.3 | -34.4 | +25.8 | +5.2 | +33.7 | |
| Steps (reduced) |
55177230 (9744998) |
56270049 (10837817) |
59169273 (2378983) |
60851386 (4061096) |
61631830 (4841540) |
63089342 (6299052) |
65058053 (8267763) |
66815401 (10025111) |
67361659 (10571369) | |