49edo
The 49 equal divisions of the octave (49edo), or the 49(-tone) equal temperament (49tet, 49et) when viewed from a regular temperament perspective, divides the octave into 49 equal parts of about 24.5 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-limit, 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.
Prime harmonics
Script error: No such module "primes_in_edo".
Intervals
| # | Cents | Approximate Ratios (*) | Notation |
|---|---|---|---|
| 0 | 0.000 | 1/1 | D |
| 1 | 24.490 | 50/49 | ^D |
| 2 | 48.980 | 81/80, 28/27, 36/35, 49/48 | Eb/^^D |
| 3 | 73.469 | 25/24, 22/21, 33/32 | ^Eb/^^^D |
| 4 | 97.959 | 16/15, 21/20 | ^^Eb/Fb/vvvD# |
| 5 | 122.449 | 15/14 | ^^^Eb/vvD# |
| 6 | 146.939 | 12/11 | vvvE/vD# |
| 7 | 171.429 | 10/9, 11/10 | vvE/D# |
| 8 | 195.918 | 28/25 | vE |
| 9 | 220.408 | 9/8, 8/7 | E |
| 10 | 244.898 | 125/108, 144/125 | ^E/vF |
| 11 | 269.388 | 7/6 | F |
| 12 | 293.878 | 33/28 | ^F |
| 13 | 318.367 | 6/5 | ^^F/Gb |
| 14 | 342.857 | 11/9 | ^^^F/^Gb |
| 15 | 367.347 | 27/22 | vvvF#/^^Gb |
| 16 | 391.837 | 5/4 | vvF#/E# |
| 17 | 416.327 | 14/11 | vF# |
| 18 | 440.816 | 9/7 | F# |
| 19 | 465.306 | 125/96, 162/125 | ^F# |
| 20 | 489.796 | 4/3, 21/16 | G |
| 21 | 514.286 | 75/56 | ^G/vAb |
| 22 | 538.776 | 27/20, 15/11 | Ab/^^G |
| 23 | 563.265 | 11/8 | ^Ab/^^^G |
| 24 | 587.755 | 7/5 | ^^Ab/vvvG# |
| 25 | 612.245 | 10/7 | vvG#/^^^Ab |
| 26 | 636.735 | 16/11 | vG#/vvvA |
| 27 | 661.244 | 40/27, 22/15 | G#/vvA |
| 28 | 685.714 | 112/75 | vA/^G# |
| 29 | 710.204 | 3/2, 32/21 | A |
| 30 | 734.694 | 125/81, 192/125 | ^A/vBb |
| 31 | 759.184 | 14/9 | Bb/^^A |
| 32 | 783.673 | 11/7 | ^Bb/vCb/^^^A |
| 33 | 808.163 | 8/5 | Cb/^^Bb/vvvA# |
| 34 | 832.653 | 44/27 | ^^^Bb/^Cb/vvA# |
| 35 | 857.143 | 18/11 | vvvB/^^Cb/vA# |
| 36 | 881.633 | 5/3 | vvB/^^^Cb/A# |
| 37 | 906.122 | 56/33 | vB/vvvC |
| 38 | 930.612 | 12/7 | B/vvC |
| 39 | 955.102 | 125/72, 216/125 | ^B/vC |
| 40 | 979.592 | 16/9, 7/4 | C/^^B |
| 41 | 1004.082 | 25/14 | ^C/^^^B |
| 42 | 1028.571 | 9/5, 20/11 | ^^C/vvvB#/Db |
| 43 | 1053.061 | 11/6 | ^^^C/vvB#/^Db |
| 44 | 1077.551 | 28/15 | vvvC#/vB#/^^Db |
| 45 | 1102.041 | 15/8, 40/21 | vvC#/B#/^^^Db |
| 46 | 1126.531 | 48/25, 21/11, 64/33 | vC#/vvvD |
| 47 | 1151.020 | 160/81, 27/14, 35/18, 96/49 | C#/vvD |
| 48 | 1175.510 | 49/25 | vD |
| 49 | 1200.000 | 2/1 | D |
(*) Based on 49edo's 11-limit patent val ⟨49 78 114 138 170] mapping
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 | Hanson/Catalan |
| 1 | 16\49 | Magus |
| 1 | 18\49 | Clyde |
| 1 | 19\49 | Semisept |
| 1 | 20\49 | Archy/Superpyth |
| 7 | 20\49 | Sevond/seville |