97edo: Difference between revisions
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97edo is the 25th prime edo. | 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. 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" | {| class="wikitable" | ||
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|77 | |77 | ||
|20L 57s | |20L 57s | ||
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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]]. | |||
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" | |||
|+ Table of errors for superparticular intervals up to 17/16 | |||
|- | |||
! Interval (JI) !! Error ([[Relative cent|r¢]]) | |||
|- | |||
| 3/2 || 25.9 | |||
|- | |||
| 4/3 || 25.8 | |||
|- | |||
| 5/4 || 22.7 | |||
|- | |||
| 6/5 || 48.6 | |||
|- | |||
| 7/6 || 42.8 | |||
|- | |||
| 8/7 || 31.4 | |||
|- | |||
| 9/8 || 48.2 | |||
|- | |||
| 10/9 || 25.6 | |||
|- | |||
| 11/10 || 33.7 | |||
|- | |||
| 12/11 || 17.6 | |||
|- | |||
| 13/12 || 20.1 | |||
|- | |||
| 14/13 || 37.0 | |||
|- | |||
| 15/14 || 34.6 | |||
|- | |||
| 16/15 || 3.1 | |||
|- | |||
| 17/16 || 48.3 | |||
|} | |} | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||