49edo: Difference between revisions
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== Theory == | == Theory == | ||
49edo is very much on the sharp side of things, with sharp tunings of harmonics 3 (it is the first square equal division with a "real" 3 of step coprime to its cardinality), 5, 7, and 11. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7 and 11 limits, [[Archytas family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments and almost identical to the e-based analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]] and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit. | 49edo is very much on the sharp side of things, with sharp tunings of harmonics 3 (it is the first square equal division with a "real" 3 of step coprime to its cardinality), 5, 7, and 11. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7 and 11 limits, [[Archytas family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments and almost identical to the e-based analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]] and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit. | ||
{{primes in edo|49}} | |||
== Intervals == | == Intervals == | ||
| Line 220: | Line 222: | ||
== Just approximation == | == Just approximation == | ||
=== Temperament measures === | === Temperament measures === | ||
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 49et. | The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 49et. | ||
Revision as of 09:18, 26 February 2021
| ← 48edo | 49edo | 50edo → |
49-EDO, or 49 equal temperament divides the octave into 49 equal parts of 24.490 cents each.
Theory
49edo is very much on the sharp side of things, with sharp tunings of harmonics 3 (it is the first square equal division with a "real" 3 of step coprime to its cardinality), 5, 7, and 11. It is the optimal patent val for superpyth temperament in the 7 and 11 limits, archytas (7-limit) and ares (11-limit) planar temperaments and almost identical to the e-based analog of Lucy tuning. It tempers out 64/63, 245/243 and 3125/3087 in the 7-limit, and 100/99 and 1375/1372 in the 11-limit.
Script error: No such module "primes_in_edo".
Intervals
| # | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 24.490 | 50/49 |
| 2 | 48.980 | 81/80, 28/27, 36/35, 49/48 |
| 3 | 73.469 | 25/24, 22/21, 33/32 |
| 4 | 97.959 | 16/15, 21/20 |
| 5 | 122.449 | 15/14 |
| 6 | 146.939 | 12/11 |
| 7 | 171.429 | 10/9, 11/10 |
| 8 | 195.918 | |
| 9 | 220.408 | 9/8, 8/7 |
| 10 | 244.898 | |
| 11 | 269.388 | 7/6 |
| 12 | 293.878 | |
| 13 | 318.367 | 6/5 |
| 14 | 342.857 | 11/9 |
| 15 | 367.347 | 27/22 |
| 16 | 391.837 | 5/4 |
| 17 | 416.327 | 14/11 |
| 18 | 440.816 | 9/7 |
| 19 | 465.306 | |
| 20 | 489.796 | 4/3, 21/16 |
| 21 | 514.286 | |
| 22 | 538.776 | 27/20, 15/11 |
| 23 | 563.265 | 11/8 |
| 24 | 587.755 | 7/5 |
| 25 | 612.245 | 10/7 |
| 26 | 636.735 | 16/11 |
| 27 | 661.244 | 40/27, 22/15 |
| 28 | 685.714 | |
| 29 | 710.204 | 3/2, 32/21 |
| 30 | 734.694 | |
| 31 | 759.184 | 14/9 |
| 32 | 783.673 | 11/7 |
| 33 | 808.163 | 8/5 |
| 34 | 832.653 | 44/27 |
| 35 | 857.143 | 18/11 |
| 36 | 881.633 | 5/3 |
| 37 | 906.122 | |
| 38 | 930.612 | 12/7 |
| 39 | 955.102 | |
| 40 | 979.592 | 16/9, 7/4 |
| 41 | 1004.082 | |
| 42 | 1028.571 | 9/5, 20/11 |
| 43 | 1053.061 | 11/6 |
| 44 | 1077.551 | 28/15 |
| 45 | 1102.041 | 15/8, 40/21 |
| 46 | 1126.531 | 48/25, 21/11, 64/33 |
| 47 | 1151.020 | 160/81, 27/14, 35/18, 96/49 |
| 48 | 1175.510 | 49/25 |
| 49 | 1200.000 | 2/1 |
Just approximation
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) of 49et.
| 3-limit | 5-limit | 7-limit | 11-limit | ||
|---|---|---|---|---|---|
| Octave stretch (¢) | -2.60 | -2.53 | -2.85 | -2.97 | |
| Error | absolute (¢) | 2.60 | 2.12 | 1.92 | 1.74 |
| relative (%) | 10.62 | 8.69 | 7.87 | 7.11 | |
Rank-2 temperaments
| Periods per octave |
Generator | Temperaments |
|---|---|---|
| 1 | 1\49 | Sengagen |
| 1 | 4\49 | Passion |
| 1 | 6\49 | Bohpier |
| 1 | 11\49 | Infraorwell |
| 1 | 13\49 | Catalan |
| 1 | 16\49 | Magus |
| 1 | 18\49 | Clyde |
| 1 | 19\49 | Semisept |
| 1 | 20\49 | Superpyth |
| 7 | 20\49 | Sevond/seville |