49edo: Difference between revisions

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! Cents
! Cents
! Approximate Ratios (*)
! Approximate Ratios (*)
! Notation
|-
|-
| 0
| 0
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
| D
|-
|-
| 1
| 1
| 24.490
| 24.490
| [[50/49]]
| [[50/49]]
| ^D
|-
|-
| 2
| 2
| 48.980
| 48.980
| [[81/80]], [[28/27]], [[36/35]], [[49/48]]
| [[81/80]], [[28/27]], [[36/35]], [[49/48]]
| Eb/^^D
|-
|-
| 3
| 3
| 73.469
| 73.469
| [[25/24]], [[22/21]], [[33/32]]
| [[25/24]], [[22/21]], [[33/32]]
| ^Eb/^^^D
|-
|-
| 4
| 4
| 97.959
| 97.959
| [[16/15]], [[21/20]]
| [[16/15]], [[21/20]]
| ^^Eb/Fb/vvvD#
|-
|-
| 5
| 5
| 122.449
| 122.449
| [[15/14]]
| [[15/14]]
| ^^^Eb/vvD#
|-
|-
| 6
| 6
| 146.939
| 146.939
| [[12/11]]
| [[12/11]]
| vvvE/vD#
|-
|-
| 7
| 7
| 171.429
| 171.429
| [[10/9]], [[11/10]]
| [[10/9]], [[11/10]]
| vvE/D#
|-
|-
| 8
| 8
| 195.918
| 195.918
| [[28/25]]
| [[28/25]]
| vE
|-
|-
| 9
| 9
| 220.408
| 220.408
| [[9/8]], [[8/7]]
| [[9/8]], [[8/7]]
| E
|-
|-
| 10
| 10
| 244.898
| 244.898
|
|
| ^E/vF
|-
|-
| 11
| 11
| 269.388
| 269.388
| [[7/6]]
| [[7/6]]
| F
|-
|-
| 12
| 12
| 293.878
| 293.878
| [[33/28]]
| [[33/28]]
| ^F
|-
|-
| 13
| 13
| 318.367
| 318.367
| [[6/5]]
| [[6/5]]
| ^^F/Gb
|-
|-
| 14
| 14
| 342.857
| 342.857
| [[11/9]]
| [[11/9]]
| ^^^F/^Gb
|-
|-
| 15
| 15
| 367.347
| 367.347
| [[27/22]]
| [[27/22]]
| vvvF#/^^Gb
|-
|-
| 16
| 16
| 391.837
| 391.837
| [[5/4]]
| [[5/4]]
| vvF#/E#
|-
|-
| 17
| 17
| 416.327
| 416.327
| [[14/11]]
| [[14/11]]
| vF#
|-
|-
| 18
| 18
| 440.816
| 440.816
| [[9/7]]
| [[9/7]]
| F#
|-
|-
| 19
| 19
| 465.306
| 465.306
|  
|  
| ^F#
|-
|-
| 20
| 20
| 489.796
| 489.796
| [[4/3]], [[21/16]]
| [[4/3]], [[21/16]]
| G
|-
|-
| 21
| 21
| 514.286
| 514.286
| [[75/56]]
| [[75/56]]
| ^G/vAb
|-
|-
| 22
| 22
| 538.776
| 538.776
| [[27/20]], [[15/11]]
| [[27/20]], [[15/11]]
| Ab/^^G
|-
|-
| 23
| 23
| 563.265
| 563.265
| [[11/8]]
| [[11/8]]
| ^Ab/^^^G
|-
|-
| 24
| 24
| 587.755
| 587.755
| [[7/5]]
| [[7/5]]
| ^^Ab/vvvG#
|-
|-
| 25
| 25
| 612.245
| 612.245
| [[10/7]]
| [[10/7]]
| vvG#/^^^Ab
|-
|-
| 26
| 26
| 636.735
| 636.735
| [[16/11]]
| [[16/11]]
| vG#/vvvA
|-
|-
| 27
| 27
| 661.244
| 661.244
| [[40/27]], [[22/15]]
| [[40/27]], [[22/15]]
| G#/vvA
|-
|-
| 28
| 28
| 685.714
| 685.714
| [[112/75]]
| [[112/75]]
| vA/^G#
|-
|-
| 29
| 29
| 710.204
| 710.204
| [[3/2]], [[32/21]]
| [[3/2]], [[32/21]]
| A
|-
|-
| 30
| 30
| 734.694
| 734.694
|  
|
| ^A/vBb
|-
|-
| 31
| 31
| 759.184
| 759.184
| [[14/9]]
| [[14/9]]
| Bb/^^A
|-
|-
| 32
| 32
| 783.673
| 783.673
| [[11/7]]
| [[11/7]]
| ^Bb/vCb/^^^A
|-
|-
| 33
| 33
| 808.163
| 808.163
| [[8/5]]
| [[8/5]]
| Cb/^^Bb/vvvA#
|-
|-
| 34
| 34
| 832.653
| 832.653
| [[44/27]]
| [[44/27]]
| ^^^Bb/^Cb/vvA#
|-
|-
| 35
| 35
| 857.143
| 857.143
| [[18/11]]
| [[18/11]]
| vvvB/^^Cb/vA#
|-
|-
| 36
| 36
| 881.633
| 881.633
| [[5/3]]
| [[5/3]]
| vvB/^^^Cb/A#
|-
|-
| 37
| 37
| 906.122
| 906.122
| [[56/33]]
| [[56/33]]
| vB/vvvC
|-
|-
| 38
| 38
| 930.612
| 930.612
| [[12/7]]
| [[12/7]]
| B/vvC
|-
|-
| 39
| 39
| 955.102
| 955.102
|  
|  
| ^B/vC
|-
|-
| 40
| 40
| 979.592
| 979.592
| [[16/9]], [[7/4]]
| [[16/9]], [[7/4]]
| C/^^B
|-
|-
| 41
| 41
| 1004.082
| 1004.082
| [[25/14]]
| [[25/14]]
| ^C/^^^B
|-
|-
| 42
| 42
| 1028.571
| 1028.571
| [[9/5]], [[20/11]]
| [[9/5]], [[20/11]]
| ^^C/vvvB#/Db
|-
|-
| 43
| 43
| 1053.061
| 1053.061
| [[11/6]]
| [[11/6]]
| ^^^C/vvB#/^Db
|-
|-
| 44
| 44
| 1077.551
| 1077.551
| [[28/15]]
| [[28/15]]
| vvvC#/vB#/^^Db
|-
|-
| 45
| 45
| 1102.041
| 1102.041
| [[15/8]], [[40/21]]
| [[15/8]], [[40/21]]
| vvC#/B#/^^^Db
|-
|-
| 46
| 46
| 1126.531
| 1126.531
| [[48/25]], [[21/11]], [[64/33]]
| [[48/25]], [[21/11]], [[64/33]]
| 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
|-
|-
| 48
| 48
| 1175.510
| 1175.510
| [[49/25]]
| [[49/25]]
| vD
|-
|-
| 49
| 49
| 1200.000
| 1200.000
| [[2/1]]
| [[2/1]]
| D
|}
|}
(*) Based on 49edo's 11-limit patent val {{val|49 78 114 138 170}} mapping
(*) Based on 49edo's 11-limit patent val {{val|49 78 114 138 170}} mapping

Revision as of 18:08, 14 April 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

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 (*) 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 ^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 ^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 ^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 ^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

Rank-2 temperaments by generators
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
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