97edo: Difference between revisions

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== Theory ==

~~In the~~ [[patent val]], 97edo tempers out 875/864, 4000/3969 ~~and 1029/1024~~ in the 7-limit, 245/242, ~~100/99,~~ 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 41&97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.

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 41 & 97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.

=== Odd harmonics ===

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97edo is the 25th [[prime edo]].

[[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.

[[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.

=== JI ~~approximation =~~==

== 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%, meaning 97edo can be used as a rough version of [[16/15 equal-step tuning]].

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== Music ==

; [[User:Francium|Francium]]

* [https://www.youtube.com/watch?v=h7bT1oL8T0w Joyous Stellaris]

* [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 (~~Demo version, July 2021~~)]

* [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023)

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 == Theory ==
-In the [[patent val]], 97edo tempers out 875/864, 4000/3969 and 1029/1024 in the 7-limit, 245/242, 100/99, 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 41&amp;97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.
+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 41 &amp; 97 temperament tempering out 100/99, 196/195, 245/242 and 385/384.
 === Odd harmonics ===
@@ Line 11: / Line 11: @@
 edo is the 25th [[prime edo]].
-[[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.
+[[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.
-=== JI approximation ===
+== Approximation to JI ==
 edo 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%, meaning 97edo can be used as a rough version of [[16/15 equal-step tuning]].
@@ Line 24: / Line 24: @@
 == Music ==
 ; [[User:Francium|Francium]]
-* [https://www.youtube.com/watch?v=h7bT1oL8T0w Joyous Stellaris]
+* [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 (Demo version, July 2021)]
+* [https://www.youtube.com/watch?v=3JwH0gZmXHk ''Thanatonautical Tetrapharmacon''] (2023)
 [[Category:Listen]]