49edo: Difference between revisions

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Adopt template: EDO intro; style; +associated ratios of the temps; -redundant categories
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{{Infobox ET}}
{{Infobox ET}}
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 [[cent]]s each.
{{EDO intro|49}}


== Theory ==
== Theory ==
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=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|49}}
{{Harmonics in equal|49}}


=== Miscellany ===
=== Miscellany ===
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| D
| 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


== Acoustic ϕ and ϕ<sup>ϕ<sup>-1</sup></sup> ==
== Acoustic ϕ and ϕ<sup>ϕ<sup>-1</sup></sup> ==
Line 280: Line 280:
! Interval
! Interval
! Error (abs, [[Cent|¢]])
! Error (abs, [[Cent|¢]])
!#\49
! #\49
|-
|-
| ϕ / ϕ<sup>ϕ<sup>-1</sup></sup> = ϕ<sup>(2-ϕ)</sup>
| ϕ / ϕ<sup>ϕ<sup>-1</sup></sup> = ϕ<sup>(2-ϕ)</sup>
| 0.155
| 0.155
|13
| 13
|-
|-
| ϕ
| ϕ
|  -0.437
|  -0.437
|34
| 34
|-
|-
| ϕ<sup>ϕ<sup>-1</sup></sup>
| ϕ<sup>ϕ<sup>-1</sup></sup>
| -0.592
| -0.592
|21
| 21
|}
|}


Not until [[592edo|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).
Not until [[592edo|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 ===
=== Music ===
[https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕ<sup>ϕ<sup>-1</sup></sup>. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5
* [https://www.youtube.com/watch?v=vZyAm-D3nlk&ab_channel=Sevish Sevish - Star Nursery] uses a scale based on [[acoustic phi]] and ϕ<sup>ϕ<sup>-1</sup></sup>. 49edo provides a suitable approximation for this scale with the mode: 5 3 5 5 3 5 3 5


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Sstretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-4"
|+ Rank-2 temperaments by generators
|+ Rank-2 temperaments by generators
! Periods<br>per octave
! Periods<br>per 8ve
! Generator
! Generator
! Associated Ratio
! Temperaments
! Temperaments
|-
|-
| 1
| 1
| 1\49
| 1\49
| 99/98
| [[Sengagen]]
| [[Sengagen]]
|-
|-
| 1
| 1
| 4\49
| 4\49
| 16/15
| [[Passion]]
| [[Passion]]
|-
|-
| 1
| 1
| 6\49
| 6\49
| 12/11
| [[Bohpier]]
| [[Bohpier]]
|-
|-
| 1
| 1
| 8\49
| 8\49
| 28/25
| [[Didacus]]
| [[Didacus]]
|-
|-
| 1
| 1
| 11\49
| 11\49
| 7/6
| [[Infraorwell]]
| [[Infraorwell]]
|-
|-
| 1
| 1
| 12\49
| 12\49
| 25/21
| [[Kleiboh]]
| [[Kleiboh]]
|-
|-
| 1
| 1
| 13\49
| 13\49
| [[Hanson]] / [[catalan]]
| 6/5
| [[Catalan]]
|-
|-
| 1
| 1
| 16\49
| 16\49
| 5/4
| [[Magus]]
| [[Magus]]
|-
|-
| 1
| 1
| 17\49
| 17\49
| 14/11
| [[Sqrtphi]]
| [[Sqrtphi]]
|-
|-
| 1
| 1
| 18\49
| 18\49
| 9/7
| [[Clyde]]
| [[Clyde]]
|-
|-
| 1
| 1
| 19\49
| 19\49
| 55/36
| [[Semisept]]
| [[Semisept]]
|-
|-
| 1
| 1
| 20\49
| 20\49
| [[Archy]] / [[superpyth]]
| 4/3
| [[Superpyth]]
|-
|-
| 7
| 7
| 20\49
| 20\49
| [[Sevond]]<br>[[Seville]]
| 4/3
| [[Sevond]] (49)<br>[[Seville]] (49c)
|}
|}


[[Category:49edo]]
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Superpyth]]
[[Category:Superpyth]]
[[Category:Archytas]]
[[Category:Archytas]]
[[Category:Ares]]
[[Category:Ares]]

Revision as of 05:46, 4 September 2023

← 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

Template:EDO intro

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 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

Approximation of odd harmonics in 49edo
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

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.

Direct mapping
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

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

Rank-2 temperaments by generators
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)
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