97edo: Difference between revisions
subsectioning the approximation into theory, and as for the "worst" I've actually calculated this with a spreadsheet, maybe not up to 16/15 but I can say for 9/8, rewrite a bit |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
97edo is only [[consistent]] to the [[5-odd-limit]]. The [[patent val]] of 97edo [[tempering out|tempers out]] [[875/864]], [[1029/1024]], and [[4000/3969]] in the 7-limit, [[100/99]], [[245/242]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[352/351]] and [[676/675]] in the 13-limit. It provides the [[optimal patent val]] for the 13-limit {{nowrap|41 & 97}} temperament tempering out 100/99, 196/195, 245/242 and 385/384. | |||
=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|97}} | {{Harmonics in equal|97|columns=14}} | ||
=== | === Subsets and supersets === | ||
97edo is the 25th [[prime edo]]. | 97edo is the 25th [[prime edo]], following [[89edo]] and before [[101edo]]. | ||
== | [[388edo]] and [[2619edo]], which contain 97edo as a subset, have very high consistency limits – 37 and 33 respectively. [[3395edo]], which divides the edostep in 35, is a [[The Riemann zeta function and tuning|zeta edo]]. The [[berkelium]] temperament realizes some relationships between them through a regular temperament perspective. | ||
97edo has very poor direct approximation for [[superparticular]] intervals among edos up to 200, and the worst for intervals up to 9/8 among edos up to 100. It has errors of well above one standard deviation (about 15.87%) in superparticular intervals with denominators up to 14. The first good approximation is the 16/15 semitone using the 9th note, with an error of 3% | |||
== Approximation to JI == | |||
97edo has very poor direct approximation for [[superparticular]] intervals among edos up to 200, and the worst for intervals up to 9/8 among edos up to 100. It has errors of well above one standard deviation (about 15.87%) in superparticular intervals with denominators up to 14. The first good approximation is the 16/15 semitone using the 9th note, with an error of 3%. | |||
Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When notable equal divisions like {{EDOs|19, 31, 41, or 53}} have strong JI-based harmony, 97edo does not have easily representable modulation because of its inability to represent superparticulars. However, this might result in interest in this tuning through JI-agnostic approaches. | Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When notable equal divisions like {{EDOs|19, 31, 41, or 53}} have strong JI-based harmony, 97edo does not have easily representable modulation because of its inability to represent superparticulars. However, this might result in interest in this tuning through JI-agnostic approaches. | ||
{{Q-odd-limit intervals|97}} | |||
== Intervals == | |||
{{Interval table}} | |||
== | == Music == | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/watch?v=hXDdKO3-RL4 ''microtonal improvisation in 97edo''] (2025) | |||
= | ; [[User:Francium|Francium]] | ||
* [https://www.youtube.com/watch?v=3JwH0gZmXHk Thanatonautical Tetrapharmacon ( | * [https://www.youtube.com/watch?v=h7bT1oL8T0w ''Joyous Stellaris''] (2023) – [[semiquartal]] in 97edo tuning | ||
; [[Mercury Amalgam]] | |||
* [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023) | |||
== Instruments == | |||
A [[Lumatone mapping for 97edo]] has now been demonstrated (see the Unnamed high-limit temperament mapping for full gamut coverage). | |||
[[Category:Listen]] | [[Category:Listen]] | ||