97edo: Difference between revisions

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The 97 equal temperament divides the octave into 97 equal parts of 12.371 cents each. It 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 '''97 equal temperament''' divides the octave into 97 equal parts of 12.371 cents each.

97edo is the 25th prime edo.

== Theory ==

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.

~~== Theory ==~~

Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments{{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.

Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments. 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"

|+Circulating temperaments in 97edo

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=== Dissonance ===

97edo is one of the least harmonic EDOs within double digits or early hundreds, resulting in errors of well over 15%, or alternatively, above one standard deviation of 15.87% in intervals up to 15/14. 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 is one of the least harmonic EDOs within double digits or early hundreds, resulting in errors of well over 15%, or alternatively, above one standard deviation of 15.87% in intervals up to 15/14{{clarify}}. 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]].

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.

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[[Category:Equal divisions of the octave]]

[[Category:Prime EDO]]

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-The 97 equal temperament divides the octave into 97 equal parts of 12.371 cents each. It 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 '''97 equal temperament''' divides the octave into 97 equal parts of 12.371 cents each.
-edo is the 25th prime edo.
+== Theory ==
+{{primes in edo|97}}
+edo 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.
-== Theory ==
+Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments{{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.
-Since 97edo has a step of 12.371 cents, it also allows one to use its MOS scales as circulating temperaments. 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"
 |+Circulating temperaments in 97edo
@@ Line 246: / Line 248: @@
 === Dissonance ===
-edo is one of the least harmonic EDOs within double digits or early hundreds, resulting in errors of well over 15%, or alternatively, above one standard deviation of 15.87% in intervals up to 15/14. 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 is one of the least harmonic EDOs within double digits or early hundreds, resulting in errors of well over 15%, or alternatively, above one standard deviation of 15.87% in intervals up to 15/14{{clarify}}. 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]].
 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.
@@ Line 284: / Line 286: @@
 | 17/16 || 48.3
 |}
 [[Category:Equal divisions of the octave]]
 [[Category:Prime EDO]]