97edo: Difference between revisions

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

== JI approximation ==

=== JI approximation ===

97edo has very poor direct approximation for [[superparticular]] intervals among edos up to 200. 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]].

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]].

Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When ~~edos~~ like ~~[[19edo~~|19]], ~~[[29edo|29]], [[31edo|~~31]], ~~[[41edo|~~41]], or ~~[[53edo|~~53]] have ~~mathematically justified~~ harmony, 97edo ~~is essentially "irredeemable" in terms~~ of ~~either modulation or approximation rationales~~. 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.

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@@ Line 11: / Line 11: @@
 edo is the 25th [[prime edo]].
-== JI approximation ==
+=== JI approximation ===
-edo has very poor direct approximation for [[superparticular]] intervals among edos up to 200. 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]].
+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]].
-Since 97edo is a prime edo, it lacks specific modulation circles, symmetrical chords or sub-edos that are present in composite edos. When edos like [[19edo|19]], [[29edo|29]], [[31edo|31]], [[41edo|41]], or [[53edo|53]] have mathematically justified harmony, 97edo is essentially "irredeemable" in terms of either modulation or approximation rationales. 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.
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