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== Theory ==
== Theory ==
49edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth and the {{w|e (mathematical constant)|e-based}} analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit.
49edo is very much on the sharp side of things, with sharp tunings of [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]]. It is the [[optimal patent val]] for [[superpyth]] temperament in the 7- and 11-limit, [[Archytas family #Archytas|archytas]] ([[7-limit]]), and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments, being almost exactly equal to {{frac|3|10}}-comma superpyth and the {{w|e (mathematical constant)|''e''-based}} analog of [[Lucy tuning]]. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]] and [[1375/1372]] in the 11-limit.


=== Harmonics ===
=== Harmonics ===
Line 13: Line 13:
== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
! #
|-
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios*
! [[Ups and downs notation|Ups and Downs Notation]]
! [[Ups and downs notation]]
|-
|-
| 0
| 0
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== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
Using [[Helmholtz–Ellis notation|Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation]]:
Using [[Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation|ups and downs]]:
{{Sharpness-sharp7}}
{{Sharpness-sharp7}}


===Sagittal notation===
=== Sagittal notation ===
====Evo flavor====
==== Evo flavor ====
 
<imagemap>
<imagemap>
File:49-EDO_Evo_Sagittal.svg
File:49-EDO_Evo_Sagittal.svg
Line 289: Line 289:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:49-EDO_Revo_Sagittal.svg
File:49-EDO_Revo_Sagittal.svg
Line 304: Line 303:
== 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]]
[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]]
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|49}}
{{Q-odd-limit intervals|49}}
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|-
|-
| ϕ
| ϕ
| &minus;0.437
| −0.437
| 34
| 34
|-
|-
| ϕ<sup>ϕ<sup>-1</sup></sup>
| ϕ<sup>ϕ<sup>-1</sup></sup>
| &minus;0.592
| −0.592
| 21
| 21
|}
|}
Line 339: Line 339:


=== 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>&minus;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 ==
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| {{monzo| 78 -49 }}
| {{monzo| 78 -49 }}
| {{mapping| 49 78 }}
| {{mapping| 49 78 }}
| &minus;2.60
| −2.60
| 2.60
| 2.60
| 10.62
| 10.62
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| 15625/15552, 20480/19683
| 15625/15552, 20480/19683
| {{mapping| 49 78 114 }}
| {{mapping| 49 78 114 }}
| &minus;2.53
| −2.53
| 2.12
| 2.12
| 8.69
| 8.69
Line 370: Line 370:
| 64/63, 245/243, 3125/3087
| 64/63, 245/243, 3125/3087
| {{mapping| 49 78 114 138 }}
| {{mapping| 49 78 114 138 }}
| &minus;2.85
| −2.85
| 1.92
| 1.92
| 7.87
| 7.87
Line 377: Line 377:
| 64/63, 100/99, 245/243, 1331/1323
| 64/63, 100/99, 245/243, 1331/1323
| {{mapping| 49 78 114 138 170 }}
| {{mapping| 49 78 114 138 170 }}
| &minus;2.97
| −2.97
| 1.74
| 1.74
| 7.11
| 7.11
Line 476: Line 476:


== Zeta properties ==
== Zeta properties ==
===Zeta peak index===
=== Zeta peak index ===
{| class="wikitable"
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
|-
!ZPI
! colspan="3" | Tuning
!Steps per octave
! colspan="3" | Strength
!Step size (cents)
! colspan="2" | Closest EDO
!Height
! colspan="2" | Integer limit
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|-
|[[233zpi]]
! ZPI
|49.1412057629230
! Steps per octave
|24.4194252332612
! Step size (cents)
|5.691547
! Height
|0.920677
! Integral
|15.624024
! Gap
|49edo
! EDO
|1196.55183642980
! Octave (cents)
|7
! Consistent
|7
! Distinct
|-
| [[233zpi]]
| 49.1412057629230
| 24.4194252332612
| 5.691547
| 0.920677
| 15.624024
| 49edo
| 1196.55183642980
| 7
| 7
|}
|}
== Scales ==
== Scales ==
=== MOS scales ===
=== MOS scales ===
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; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune - ''Man'' (cover)] (2023)
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023)


[[Category:Archytas]]
[[Category:Archytas]]

Revision as of 01:52, 19 January 2025

← 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, 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

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)

Subsets and supersets

Since 49 factors into 72, 49edo contains 7edo as its only nontrivial subset. 49edo is the first square edo with a non-enfactored diatonic fifth.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.000 1/1 D
1 24.490 50/49 ^D, vE♭
2 48.980 28/27, 36/35, 49/48, 81/80 ^^D, E♭
3 73.469 22/21, 25/24, 33/32 ^3D, ^E♭
4 97.959 16/15, 21/20 v3D♯, ^^E♭
5 122.449 15/14 vvD♯, ^3E♭
6 146.939 12/11 vD♯, v3E
7 171.429 10/9, 11/10 D♯, vvE
8 195.918 28/25 ^D♯, vE
9 220.408 8/7, 9/8 E
10 244.898 125/108, 144/125 ^E, vF
11 269.388 7/6 F
12 293.878 25/21, 33/28 ^F, vG♭
13 318.367 6/5 ^^F, G♭
14 342.857 11/9 ^3F, ^G♭
15 367.347 27/22 v3F♯, ^^G♭
16 391.837 5/4 vvF♯, ^3G♭
17 416.327 14/11 vF♯, v3G
18 440.816 9/7 F♯, vvG
19 465.306 125/96, 162/125 ^F♯, vG
20 489.796 4/3, 21/16 G
21 514.286 75/56 ^G, vA♭
22 538.776 15/11, 27/20 ^^G, A♭
23 563.265 11/8 ^3G, ^A♭
24 587.755 7/5 v3G♯, ^^A♭
25 612.245 10/7 vvG♯, ^3A♭
26 636.735 16/11 vG♯, v3A
27 661.244 22/15, 40/27 G♯, vvA
28 685.714 112/75 ^G♯, vA
29 710.204 3/2, 32/21 A
30 734.694 125/81, 192/125 ^A, vB♭
31 759.184 14/9 ^^A, B♭
32 783.673 11/7 ^3A, ^B♭
33 808.163 8/5 v3A♯, ^^B♭
34 832.653 44/27 vvA♯, ^3B♭
35 857.143 18/11 vA♯, v3B
36 881.633 5/3 A♯, vvB
37 906.122 42/25, 56/33 ^A♯, vB
38 930.612 12/7 B
39 955.102 125/72, 216/125 ^B, vC
40 979.592 7/4, 16/9 C
41 1004.082 25/14 ^C, vD♭
42 1028.571 9/5, 20/11 ^^C, D♭
43 1053.061 11/6 ^3C, ^D♭
44 1077.551 28/15 v3C♯, ^^D♭
45 1102.041 15/8, 40/21 vvC♯, ^3D♭
46 1126.531 21/11, 48/25, 64/33 vC♯, v3D
47 1151.020 27/14, 35/18, 96/49, 160/81 C♯, vvD
48 1175.510 49/25 ^C♯, vD
49 1200.000 2/1 D

* Based on 49edo's 11-limit patent val ⟨49 78 114 138 170] mapping

Notation

Ups and downs notation

Using Helmholtz–Ellis accidentals, 49edo can be notated using ups and downs:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8033/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation513/51281/8033/32

Approximation to JI

alt : Your browser has no SVG support.
Selected 19-limit intervals approximated in 49edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 49edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 49edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/9, 18/13 0.117 0.5
11/7, 14/11 1.181 4.8
15/11, 22/15 1.825 7.5
7/6, 12/7 2.517 10.3
5/3, 6/5 2.726 11.1
15/13, 26/15 2.843 11.6
15/14, 28/15 3.006 12.3
11/6, 12/11 3.698 15.1
11/9, 18/11 4.551 18.6
13/11, 22/13 4.668 19.1
7/5, 10/7 5.243 21.4
5/4, 8/5 5.523 22.6
9/7, 14/9 5.732 23.4
13/7, 14/13 5.849 23.9
11/10, 20/11 6.424 26.2
13/8, 16/13 7.875 32.2
9/8, 16/9 7.992 32.6
3/2, 4/3 8.249 33.7
13/12, 24/13 8.366 34.2
15/8, 16/15 10.718 43.8
7/4, 8/7 10.766 44.0
9/5, 10/9 10.975 44.8
13/10, 20/13 11.092 45.3
11/8, 16/11 11.947 48.8
15-odd-limit intervals in 49edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/7, 14/11 1.181 4.8
15/11, 22/15 1.825 7.5
7/6, 12/7 2.517 10.3
5/3, 6/5 2.726 11.1
15/14, 28/15 3.006 12.3
11/6, 12/11 3.698 15.1
11/9, 18/11 4.551 18.6
7/5, 10/7 5.243 21.4
5/4, 8/5 5.523 22.6
9/7, 14/9 5.732 23.4
11/10, 20/11 6.424 26.2
13/8, 16/13 7.875 32.2
3/2, 4/3 8.249 33.7
7/4, 8/7 10.766 44.0
9/5, 10/9 10.975 44.8
11/8, 16/11 11.947 48.8
13/10, 20/13 13.398 54.7
15/8, 16/15 13.772 56.2
13/12, 24/13 16.124 65.8
9/8, 16/9 16.498 67.4
13/7, 14/13 18.641 76.1
13/11, 22/13 19.822 80.9
15/13, 26/15 21.647 88.4
13/9, 18/13 24.373 99.5

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.

Direct approximation
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 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

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 1\49 24.5 99/98 Sengagen
1 4\49 98.0 16/15 Passion
1 6\49 146.9 12/11 Bohpier
1 8\49 195.9 28/25 Didacus
1 11\49 269.4 7/6 Infraorwell
1 12\49 293.9 25/21 Kleiboh
1 13\49 318.4 6/5 Catalan
1 16\49 391.8 5/4 Magus
1 17\49 416.3 14/11 Sqrtphi
1 18\49 440.8 9/7 Clyde
1 19\49 465.3 55/36 Semisept
1 20\49 489.8 4/3 Superpyth
7 20\49
(1\49) 		489.8
(24.5) 		4/3
(250/243) 		Sevond (49)
4/3
(25/24) 		Seville (49c)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Zeta properties

Zeta peak index

Tuning Strength Closest EDO Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap EDO Octave (cents) Consistent Distinct
233zpi 49.1412057629230 24.4194252332612 5.691547 0.920677 15.624024 49edo 1196.55183642980 7 7


Scales

MOS scales

Main article: List of MOS scales in 49edo

Instruments

Lumatone

See Lumatone mapping for 49edo

Music

Mercury Amalgam
Bryan Deister
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