49edo: Difference between revisions

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→Intervals: mark inconsistencies italic
Line 35: Line 35:
| 2
| 2
| 48.980
| 48.980
| [[81/80]], [[28/27]], [[36/35]], [[49/48]]
| ''[[81/80]]'', ''[[28/27]]'', [[36/35]], ''[[49/48]]''
| Eb/^^D
| Eb/^^D
|-
|-
| 3
| 3
| 73.469
| 73.469
| [[25/24]], [[22/21]], [[33/32]]
| [[25/24]], [[22/21]], ''[[33/32]]''
| ^Eb/^^^D
| ^Eb/^^^D
|-
|-
| 4
| 4
| 97.959
| 97.959
| [[16/15]], [[21/20]]
| ''[[16/15]]'', [[21/20]]
| ^^Eb/Fb/vvvD#
| ^^Eb/Fb/vvvD#
|-
|-
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| 9
| 9
| 220.408
| 220.408
| [[9/8]], [[8/7]]
| ''[[9/8]]'', [[8/7]]
| E
| E
|-
|-
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| 12
| 12
| 293.878
| 293.878
| [[33/28]]
| [[25/21]], [[33/28]]
| ^F
| ^F
|-
|-
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| 19
| 19
| 465.306
| 465.306
| 125/96, 162/125
| 125/96, ''162/125''
| ^F#
| ^F#
|-
|-
| 20
| 20
| 489.796
| 489.796
| [[4/3]], [[21/16]]
| [[4/3]], ''[[21/16]]''
| G
| G
|-
|-
Line 135: Line 135:
| 22
| 22
| 538.776
| 538.776
| [[27/20]], [[15/11]]
| ''[[27/20]]'', [[15/11]]
| Ab/^^G
| Ab/^^G
|-
|-
Line 160: Line 160:
| 27
| 27
| 661.244
| 661.244
| [[40/27]], [[22/15]]
| ''[[40/27]]'', [[22/15]]
| G#/vvA
| G#/vvA
|-
|-
Line 170: Line 170:
| 29
| 29
| 710.204
| 710.204
| [[3/2]], [[32/21]]
| [[3/2]], ''[[32/21]]''
| A
| A
|-
|-
| 30
| 30
| 734.694
| 734.694
| 125/81, 192/125
| ''125/81'', 192/125
| ^A/vBb
| ^A/vBb
|-
|-
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| 37
| 37
| 906.122
| 906.122
| [[56/33]]
| [[42/25]], [[56/33]]
| vB/vvvC
| vB/vvvC
|-
|-
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| 40
| 40
| 979.592
| 979.592
| [[16/9]], [[7/4]]
| ''[[16/9]]'', [[7/4]]
| C/^^B
| C/^^B
|-
|-
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| 45
| 45
| 1102.041
| 1102.041
| [[15/8]], [[40/21]]
| ''[[15/8]]'', [[40/21]]
| vvC#/B#/^^^Db
| vvC#/B#/^^^Db
|-
|-
| 46
| 46
| 1126.531
| 1126.531
| [[48/25]], [[21/11]], [[64/33]]
| [[48/25]], [[21/11]], ''[[64/33]]''
| vC#/vvvD
| vC#/vvvD
|-
|-
| 47
| 47
| 1151.020
| 1151.020
| [[160/81]], [[27/14]], [[35/18]], [[96/49]]
| ''[[160/81]]'', ''[[27/14]]'', [[35/18]], ''[[96/49]]''
| C#/vvD
| C#/vvD
|-
|-

Revision as of 19:28, 17 November 2021

← 48edo 49edo 50edo →
Prime factorization 72
Step size 24.4898 ¢ 
Fifth 29\49 (710.204 ¢)
Semitones (A1:m2) 7:2 (171.4 ¢ : 48.98 ¢)
Dual sharp fifth 29\49 (710.204 ¢)
Dual flat fifth 28\49 (685.714 ¢) (→ 4\7)
Dual major 2nd 8\49 (195.918 ¢)
Consistency limit 7
Distinct consistency limit 7

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 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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		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

Rank-2 temperaments by generators
Periods
per octave 		Generator Temperaments
1 1\49 Sengagen
1 4\49 Passion
1 6\49 Bohpier
1 8\49 Didacus
1 11\49 Infraorwell
1 12\49 Kleiboh
1 13\49 Hanson / catalan
1 16\49 Magus
1 17\49 Sqrtphi
1 18\49 Clyde
1 19\49 Semisept
1 20\49 Archy / superpyth
7 20\49 Sevond
Seville
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