94edo: Difference between revisions
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There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context. | |||
The perfect fifth has three, or perhaps even five, functional options, each differing by one step. Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys. | |||
Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent [[tritone]] (47\94), which divides the octave exactly in half. | |||
The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek [[tetrachord]]al theory, [[Indian]] raga and Turkish [[maqam]], though it has not been used historically as a division in those musical cultures. | |||
While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys. | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|Comma List]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | |||
! colspan="2" | Tuning Error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| 149 -94 }} | |||
| [{{val| 94 149 }}] | |||
| -0.054 | |||
| 0.054 | |||
| 0.43 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, 9765625/9565938 | |||
| [{{val| 94 149 218 }}] | |||
| +0.442 | |||
| 0.704 | |||
| 5.52 | |||
|- | |||
| 2.3.5.7 | |||
| 225/224, 3125/3087, 118098/117649 | |||
| [{{val| 94 149 218 264 }}] | |||
| +0.208 | |||
| 0.732 | |||
| 5.74 | |||
|- | |||
| 2.3.5.7.11 | |||
| 225/224, 385/384, 1331/1323, 2200/2187 | |||
| [{{val| 94 149 218 264 325 }}] | |||
| +0.304 | |||
| 0.683 | |||
| 5.35 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 225/224, 275/273, 325/324, 385/384, 1331/1323 | |||
| [{{val| 94 149 218 264 325 348 }}] | |||
| +0.162 | |||
| 0.699 | |||
| 5.48 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 170/169, 225/224, 275/273, 289/288, 325/324, 385/384 | |||
| [{{val| 94 149 218 264 325 348 384 }}] | |||
| +0.238 | |||
| 0.674 | |||
| 5.28 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384 | |||
| [{{val| 94 149 218 264 325 348 384 399 }}] | |||
| +0.323 | |||
| 0.669 | |||
| 5.24 | |||
|- | |||
| 2.3.5.7.11.13.17.19.23 | |||
| 170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322 | |||
| [{{val| 94 149 218 264 325 348 384 399 425 }}] | |||
| +0.354 | |||
| 0.637 | |||
| 4.99 | |||
|} | |||
94et is lower in relative error than any previous equal temperaments in the 23-limit, and the equal temperament that does better in this subgroup is 193. | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all right-3 left-5" | |||
! Periods<br>per 8ve | |||
! Generator | |||
! Cents | |||
! Associated<br>Ratio | |||
! Temperament | |||
|- | |||
| 1 | |||
| 3\94 | |||
| 38.30 | |||
| 49/48 | |||
| [[Slender]] | |||
|- | |||
| 1 | |||
| 5\94 | |||
| 63.83 | |||
| 25/24 | |||
| [[Sycamore]] / [[betic]] | |||
|- | |||
| 1 | |||
| 11\94 | |||
| 140.43 | |||
| 243/224 | |||
| [[Tsaharuk]] / [[quanic]] | |||
|- | |||
| 1 | |||
| 13\94 | |||
| 165.96 | |||
| 11/10 | |||
| [[Tertiaschis]] | |||
|- | |||
| 1 | |||
| 19\94 | |||
| 242.55 | |||
| 147/128 | |||
| [[Septiquarter]] | |||
|- | |||
| 1 | |||
| 39\94 | |||
| 497.87 | |||
| 4/3 | |||
| [[Helmholtz]] / [[garibaldi]] / [[cassandra]] | |||
|- | |||
| 2 | |||
| 2\94 | |||
| 25.53 | |||
| 64/63 | |||
| [[Ketchup]] | |||
|- | |||
| 2 | |||
| 11\94 | |||
| 140.43 | |||
| 27/25 | |||
| [[Fifive]] | |||
|- | |||
| 2 | |||
| 30\94 | |||
| 382.98 | |||
| 5/4 | |||
| [[Wizard]] / [[gizzard]] | |||
|- | |||
| 2 | |||
| 34\94 | |||
| 434.04 | |||
| 9/7 | |||
| [[Pogo]] / [[supers]] | |||
|- | |||
| 2 | |||
| 43\94 | |||
| 548.94 | |||
| 11/8 | |||
| [[Kleischismic]] | |||
|} | |} | ||