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~~. 97edo is the 25th prime edo~~.

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.

=== Odd harmonics ===

=== Divisors ===

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

== JI approximation ==

97edo has the worst 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/~~15ths~~ equal ~~temperament~~]].

97edo has the worst 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]].

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 towards this tuning through emancipation of the dissonance.

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 towards this tuning through emancipation of the dissonance.

{| class="wikitable center-all"

{| class="wikitable"

|+ Table of errors for superparticular intervals up to 17/16

|-

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Since 97edo has a step of 12.371 cents, it also allows one to use its [[mos]] scales as [[circulating temperament]]s{{clarify}}. It is the first prime edo which does this and the first edo which allows one to use an mos scale with a step 20 degrees or larger as a circulating temperament.

{| class="wikitable mw-collapsible mw-collapsed collapsible"

|+Circulating temperaments in 97edo

|+ style=white-space:nowrap | Circulating temperaments in 97edo

!Tones

!Pattern

@@ 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. 97edo is the 25th prime edo.
+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.
 === Odd harmonics ===
 {{Harmonics in equal|97}}
+=== Divisors ===
+edo is the 25th [[prime edo]].
 == JI approximation ==
-edo has the worst 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 [[16/15ths equal temperament]].
+edo has the worst 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]].
+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 towards this tuning through emancipation of the dissonance.
-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 towards this tuning through emancipation of the dissonance.
+{| class="wikitable center-all"
-{| class="wikitable"
 |+ Table of errors for superparticular intervals up to 17/16
 |-
@@ Line 51: / Line 55: @@
 Since 97edo has a step of 12.371 cents, it also allows one to use its [[mos]] scales as [[circulating temperament]]s{{clarify}}. It is the first prime edo which does this and the first edo which allows one to use an mos scale with a step 20 degrees or larger as a circulating temperament.
 {| class="wikitable mw-collapsible mw-collapsed collapsible"
-|+Circulating temperaments in 97edo
+|+ style=white-space:nowrap | Circulating temperaments in 97edo
 !Tones
 !Pattern