190537edo: Difference between revisions
Add list of supersets (since several of these extremely large edos' pages are related) |
m Fix sectioning |
||
| (3 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Mathematical interest}} | |||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} It is the denominator of the next convergent for log<sub>2</sub>3 past [[111202edo|111202]], with another such convergent not occurring until [[10590737edo|10590737]]. | {{ED intro}} It is the denominator of the next convergent for log<sub>2</sub>3 past [[111202edo|111202]], with another such convergent not occurring until [[10590737edo|10590737]]. | ||
== Theory == | |||
190537edo has a [[consistency]] limit of 11, which is rather impressive for a convergent. However, it is strongest in the 2.3.7.17.23 subgroup. Notably, it is the first member of the log<sub>2</sub>3 convergent series with a 3-2 [[Telicity #k-Strong Telicity|telicity ''k''-strength]] greater than 1 since [[665edo]] and it even surpasses 665edo in telicity ''k''-strength. However, the downside is that the step size is many times smaller than the [[JND]]. The 3-limit comma this edo tempers out has been named the [[Archangelic comma]]. | 190537edo has a [[consistency]] limit of 11, which is rather impressive for a convergent. However, it is strongest in the 2.3.7.17.23 subgroup. Notably, it is the first member of the log<sub>2</sub>3 convergent series with a 3-2 [[Telicity #k-Strong Telicity|telicity ''k''-strength]] greater than 1 since [[665edo]] and it even surpasses 665edo in telicity ''k''-strength. However, the downside is that the step size is many times smaller than the [[JND]]. The 3-limit comma this edo tempers out has been named the [[Archangelic comma]]. | ||
| Line 8: | Line 10: | ||
=== Supersets === | === Supersets === | ||
* [[571611edo]] (3×) | * [[User:Aura/571611edo|571611edo]] (3×) | ||
* [[762148edo]] (4×) | * [[User:Aura/762148edo|762148edo]] (4×) | ||
* [[1714833edo]] (9×) | * [[User:Aura/1714833edo|1714833edo]] (9×) | ||
* [[1905370edo]] (10×) | * [[User:Aura/1905370edo|1905370edo]] (10×) | ||
* [[2667518edo]] (14×) | * [[User:Aura/2667518edo|2667518edo]] (14×) | ||
* [[4191814edo]] (22×) | * [[User:Aura/4191814edo|4191814edo]] (22×) | ||
[[Category:3-limit record edos|######]] <!-- 6-digit number --> | [[Category:3-limit record edos|######]] <!-- 6-digit number --> | ||
Latest revision as of 22:50, 14 January 2026
|
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. |
| ← 190536edo | 190537edo | 190538edo → |
(convergent)
190537 equal divisions of the octave (abbreviated 190537edo or 190537ed2), also called 190537-tone equal temperament (190537tet) or 190537 equal temperament (190537et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 190537 equal parts of about 0.0063 ¢ each. Each step represents a frequency ratio of 21/190537, or the 190537th root of 2. It is the denominator of the next convergent for log23 past 111202, with another such convergent not occurring until 10590737.
Theory
190537edo has a consistency limit of 11, which is rather impressive for a convergent. However, it is strongest in the 2.3.7.17.23 subgroup. Notably, it is the first member of the log23 convergent series with a 3-2 telicity k-strength greater than 1 since 665edo and it even surpasses 665edo in telicity k-strength. However, the downside is that the step size is many times smaller than the JND. The 3-limit comma this edo tempers out has been named the Archangelic comma.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | +0.00000 | -0.00134 | +0.00010 | +0.00175 | +0.00200 | +0.00058 | -0.00230 | +0.00048 | -0.00079 | +0.00187 | +0.00242 |
| Relative (%) | +0.0 | +0.0 | -21.3 | +1.5 | +27.8 | +31.7 | +9.3 | -36.5 | +7.6 | -12.5 | +29.8 | +38.4 | |
| Steps (reduced) |
190537 (0) |
301994 (111457) |
442413 (61339) |
534905 (153831) |
659150 (87539) |
705071 (133460) |
778813 (16665) |
809387 (47239) |
861906 (99758) |
925625 (163477) |
943958 (181810) |
992594 (39909) | |
Supersets
- 571611edo (3×)
- 762148edo (4×)
- 1714833edo (9×)
- 1905370edo (10×)
- 2667518edo (14×)
- 4191814edo (22×)