49edo: Difference between revisions
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== Approximation to JI == | == Approximation to JI == | ||
[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]] | |||
=== Zeta peaks === | === Zeta peaks === | ||
The strongest [[The Riemann zeta function and tuning|local zeta peak]] around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for [[Triple BP]]. | The strongest [[The Riemann zeta function and tuning|local zeta peak]] around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for [[Triple BP]]. | ||
Revision as of 05:05, 20 June 2024
| ← 48edo | 49edo | 50edo → |
Theory
49edo is very much on the sharp side of things, with sharp tunings of harmonics 3, 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, being almost exactly equal to 3⁄10-comma superpyth and 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.
Harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.2 | +5.5 | +10.8 | -8.0 | +11.9 | -7.9 | -10.7 | -7.0 | -3.6 | -5.5 | +8.5 |
| Relative (%) | +33.7 | +22.6 | +44.0 | -32.6 | +48.8 | -32.2 | -43.8 | -28.6 | -14.8 | -22.4 | +34.5 | |
| Steps (reduced) |
78 (29) |
114 (16) |
138 (40) |
155 (8) |
170 (23) |
181 (34) |
191 (44) |
200 (4) |
208 (12) |
215 (19) |
222 (26) | |
Miscellany
49edo is the first square equal division with a "real" 3 of step coprime to its cardinality.
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 | 25/21, 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 | 42/25, 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
Approximation to JI
Zeta peaks
The strongest local zeta peak around 49edo is its second closest, 49.141 edo. One step is 24.419 cents, and two steps, 48.839 cents, is a good generator for Triple BP.
Approximation to irrational intervals
Acoustic ϕ and ϕϕ-1
49edo has a very close approximation of both acoustic phi and ϕϕ-1, a kind of logarithmic phi that divides acoustic phi logarithmically by phi (instead of dividing 2/1).
ϕϕ-1 has interesting applications as Metallic MOS, and in particular the fractal-like possibilities of self-similar subdivision of musical scales within acoustic phi.
| Interval | Error (abs, ¢) | #\49 |
|---|---|---|
| ϕ / ϕϕ-1 = ϕ(2-ϕ) | 0.155 | 13 |
| ϕ | -0.437 | 34 |
| ϕϕ-1 | -0.592 | 21 |
Not until 592 do we find a better edo in terms of relative error on these two intervals (but whose edo-steps upon which these intervals are mapped are not based on the Fibonacci sequence, unlike 49edo).
Music
- Sevish - Star Nursery uses a scale based on acoustic phi and ϕϕ-1. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Sstretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [78 -49⟩ | [⟨49 78]] | −2.60 | 2.60 | 10.62 |
| 2.3.5 | 15625/15552, 20480/19683 | [⟨49 78 114]] | −2.53 | 2.12 | 8.69 |
| 2.3.5.7 | 64/63, 245/243, 3125/3087 | [⟨49 78 114 138]] | −2.85 | 1.92 | 7.87 |
| 2.3.5.7.11 | 64/63, 100/99, 245/243, 1331/1323 | [⟨49 78 114 138 170]] | −2.97 | 1.74 | 7.11 |
Rank-2 temperaments
| Periods per 8ve |
Generator | Associated Ratio | Temperaments |
|---|---|---|---|
| 1 | 1\49 | 99/98 | Sengagen |
| 1 | 4\49 | 16/15 | Passion |
| 1 | 6\49 | 12/11 | Bohpier |
| 1 | 8\49 | 28/25 | Didacus |
| 1 | 11\49 | 7/6 | Infraorwell |
| 1 | 12\49 | 25/21 | Kleiboh |
| 1 | 13\49 | 6/5 | Catalan |
| 1 | 16\49 | 5/4 | Magus |
| 1 | 17\49 | 14/11 | Sqrtphi |
| 1 | 18\49 | 9/7 | Clyde |
| 1 | 19\49 | 55/36 | Semisept |
| 1 | 20\49 | 4/3 | Superpyth |
| 7 | 20\49 | 4/3 | Sevond (49) Seville (49c) |
Scales
MOS scales
Instruments
Lumatone
See Lumatone mapping for 49edo
Music
- Deltarune - Man (cover) (2023)