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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|49}}
{{ED intro}}


== 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. It [[tempering out|tempers out]] [[64/63]], [[245/243]], and [[3125/3087]] in the 7-limit, and [[100/99]], [[540/539]], and [[1375/1372]] in the 11-limit. In the 13-limit, its [[patent val]] {{val| 49 78 114 138 170 181 }}, has a rather flat (by relative error) harmonic [[13/1|13]], which leads to inconsistent mappings; but using the 49f val {{val| 49 78 114 138 170 182 }} improves 13-limit consistency, and in this val it tempers out [[364/363]] and [[847/845]].


=== Harmonics ===
=== Harmonics ===
Line 9: Line 9:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 49 factors into {{factorization|49}}, 49edo contains [[7edo]] as its only nontrivial subset. 49edo is the first square edo with a [[enfactoring|non-enfactored]] diatonic fifth.  
Since 49 factors into primes as 7<sup>2</sup>, 49edo contains [[7edo]] as its only nontrivial subset. 49edo is the first square edo with a [[enfactoring|non-enfactored]] diatonic fifth. Doubling it produces [[98edo]], a respectable (if overly complex) [[meantone]] tuning.  


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
! #
|-
! &#35;
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios*
! [[Ups and downs notation|Ups and Downs Notation]]
! [[Ups and downs notation]]
|-
|-
| 0
| 0
Line 65: Line 66:
| 9
| 9
| 220.408
| 220.408
| [[8/7]], ''[[9/8]]''
| [[8/7]], ''[[9/8]]'', [[25/22]]
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|-
|-
Line 220: Line 221:
| 40
| 40
| 979.592
| 979.592
| [[7/4]], ''[[16/9]]''
| [[7/4]], ''[[16/9]]'', [[44/25]]
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|-
|-
Line 272: Line 273:
== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
Using [[Helmholtz–Ellis notation|Helmholtz&ndash;Ellis]] accidentals, 49edo can be notated using [[ups and downs notation]]:
49edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Ups and downs sharpness}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{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 292:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:49-EDO_Revo_Sagittal.svg
File:49-EDO_Revo_Sagittal.svg
Line 304: Line 306:
== 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}}
{{Q-odd-limit intervals|49.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 49f val mapping}}


=== Zeta peaks ===
=== Zeta peaks ===
Line 311: Line 315:


== Approximation to irrational intervals ==
== Approximation to irrational intervals ==
=== Acoustic ϕ and ϕ<sup>ϕ<sup>-1</sup></sup> ===
=== Acoustic ϕ and ϕ<sup>ϕ<sup>−1</sup></sup> ===
49edo has a very close approximation of both [[acoustic phi]] and ϕ<sup>ϕ<sup>-1</sup></sup>, a kind of logarithmic phi that divides [[acoustic phi]] logarithmically by phi ([[Logarithmic phi|instead of dividing 2/1]]).
49edo has a very close approximation of both [[acoustic phi]] and [[phith root of phi|ϕ<sup>ϕ<sup>-1</sup></sup>]], a kind of logarithmic phi that divides [[acoustic phi]] logarithmically by phi ([[Logarithmic phi|instead of dividing 2/1]]).


ϕ<sup>ϕ<sup>-1</sup></sup> has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales within [[acoustic phi]].
The  [[phith root of phi|phith root of phi (ϕ<sup>ϕ<sup>-1</sup></sup>)]] has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales within [[acoustic phi]].


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 323: Line 327:
! #\49
! #\49
|-
|-
| ϕ / ϕ<sup>ϕ<sup>-1</sup></sup> = ϕ<sup>(2-ϕ)</sup>
| {{nowrap|ϕ / ϕ<sup>ϕ<sup>−1</sup></sup> {{=}} ϕ<sup>(2 − ϕ)</sup>}}
| 0.155
| 0.155
| 13
| 13
|-
|-
| 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 343:


=== 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 ==
Line 356: Line 360:
| {{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
Line 363: Line 367:
| 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 374:
| 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 381:
| 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 473: Line 477:
| style="text-align: left;" | [[Seville]] (49c)
| style="text-align: left;" | [[Seville]] (49c)
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
== Octave stretch or compression ==
49edo's [[prime]]s 3, 5, 7 and 11 are all tuned sharp, so 49edo can benefit from [[octave shrinking]]. Some compressed-octave tunings of 49edo include (least to most compression): [[ed12|176ed12]], [[ed5|114ed5]], [[zpi|233zpi]], [[ed6|127ed6]], [[ed7|138ed7]] and [[78edt]].
 
=== Nonoctave temperament ===
The TE-optimized [[Triple BP|triple Bohlen–Pierce scale]] is obtained by taking every second degree of 49edo with the octave compressed by 3.861 cents to 1196.139 cents. It realizes the Tenney–Euclidean regular temperament on the 3.5.7.11.13 subgroup mapped as [⟨78 114 138 170 182]]. Under this compression, the primes map to the 49fgh val in the 23-limit.


== Zeta properties ==
===Zeta peak index===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |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 ==
== Scales ==
=== MOS scales ===
=== MOS scales ===
{{main|List of MOS scales in 49edo}}
{{main|List of MOS scales in 49edo}}
* Bohpier[8]: 6 6 6 6 7 6 6 6
* Catalan[7]: 3 10 3 10 3 10 10 (vaugely diminished-like)
* Catalan[11]: 3 7 3 3 7 3 3 7 3 3 7
* Catalan[19]: 3 3 1 3 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3
* Clyde[5]: 5 13 5 13 13 (mysterious, adventurous)
* Didacus[6]: 8 8 8 8 8 9 (like the whole tone scale)
* Didacus[13]: 1 7 1 7 1 7 1 7 1 7 1 7 1
* Infraorwell[5]: 11 11 5 11 11
* Infraorwell[22]: 1 4 1 4 1 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Kleiboh[5]: 12 12 12 12 1
* Kleiboh[13]: 1 10 1 1 10 1 1 1 10 1 1 10 1
* Magus[7]: 1 15 1 15 1 15 1 (vaguely augmented-like)
* Passion[12]: 4 4 4 4 4 4 5 4 4 4 4 4 (like [[12edo]])
* Passion[23]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
* Sevond[21]/Seville[21]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
* Superpyth[5]: 11 9 9 11 9 (in between minor pentatonic and [[equipentatonic]])
* Superpyth[7]: 9 2 9 9 9 2 9 (Dorian mode; rotate for other modes)
* Superpyth[12]: 2 7 2 7 2 2 7 2 7 2 7 2 (same melodic shape as [[12edo]] but much more [[xenharmonic]] harmonies)
* Superpyth[27]: 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2
=== Other scales ===
* [[6ed7/3#6ed7/3+7edo scale|The 6ed7/3+7edo scale]] ''(non-octave-repeating)''


== Instruments ==
== Instruments ==
; Lumatone
=== Lumatone ===
* [[Lumatone mapping for 49edo]]
 
=== Skip fretting ===
'''Skip fretting system 49 3 7''' is a [[skip fretting]] system for [[49edo]]. All examples are for 5-string bass.
 
; Harmonics
1/1: string 2 open
 
2/1: not easily accessible


See [[Lumatone mapping for 49edo]]
3/2: string 4 fret 5 and string 1 fret 12
 
5/4: string 3 fret 3
 
7/4: string 3 fret 11
 
11/8: string 3 fret 5


== Music ==
== Music ==
=== Modern renderings ===
; {{W|The Cure}}
* [https://www.youtube.com/watch?v=GHslu-ZWspk ''Boys Don't Cry''] (1979) – Lumatone cover by [[YoVariable]] (2025)
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023)
* [https://www.youtube.com/shorts/V8t7MyP2Nuo ''microtonal improv in 49edo''] (2024)
* [https://www.youtube.com/shorts/zb1Z6o-Uvuw ''weathergirl - FLAVOR FOLEY (microtonal cover in 49edo)''] (2025)
* [https://www.youtube.com/shorts/73PfAAWubVs ''I'm Your Captain Now (The Ancients) - The Recovery System (microtonal cover in 49edo)''] (2026) {{todo|research|comment=Identify the original composers.}}
* [https://www.youtube.com/shorts/34w7euOF-Ss ''49edo improv''] (2026)
* [https://www.youtube.com/shorts/_yNrDI6nS1I ''49edo riff''] (2026)
* [https://www.youtube.com/shorts/BcBtD3nuEQs ''49edo groove''] (2026)
* [https://www.youtube.com/shorts/VmUIxWb8NCY ''49edo prelude''] (2026)
; [[Mercury Amalgam]]
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation (Demo, January 2022)''] (2023)
* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation''] (2022 demo version)


; [[Bryan Deister]]
; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune - ''Man'' (cover)] (2023)
* [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023)


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

Latest revision as of 10:43, 25 April 2026

← 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 equal divisions of the octave (abbreviated 49edo or 49ed2), also called 49-tone equal temperament (49tet) or 49 equal temperament (49et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 49 equal parts of about 24.5 ¢ each. Each step represents a frequency ratio of 21/49, or the 49th root of 2.

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. It tempers out 64/63, 245/243, and 3125/3087 in the 7-limit, and 100/99, 540/539, and 1375/1372 in the 11-limit. In the 13-limit, its patent val ⟨49 78 114 138 170 181], has a rather flat (by relative error) harmonic 13, which leads to inconsistent mappings; but using the 49f val ⟨49 78 114 138 170 182] improves 13-limit consistency, and in this val it tempers out 364/363 and 847/845.

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 primes as 72, 49edo contains 7edo as its only nontrivial subset. 49edo is the first square edo with a non-enfactored diatonic fifth. Doubling it produces 98edo, a respectable (if overly complex) meantone tuning.

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, 25/22 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, 44/25 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

49edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

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

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

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
15-odd-limit intervals by 49f val mapping
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
3/2, 4/3 8.249 33.7
13/12, 24/13 8.366 34.2
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
15/8, 16/15 13.772 56.2
9/8, 16/9 16.498 67.4
13/8, 16/13 16.615 67.8

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).

The phith root of phi (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

Octave stretch or compression

49edo's primes 3, 5, 7 and 11 are all tuned sharp, so 49edo can benefit from octave shrinking. Some compressed-octave tunings of 49edo include (least to most compression): 176ed12, 114ed5, 233zpi, 127ed6, 138ed7 and 78edt.

Nonoctave temperament

The TE-optimized triple Bohlen–Pierce scale is obtained by taking every second degree of 49edo with the octave compressed by 3.861 cents to 1196.139 cents. It realizes the Tenney–Euclidean regular temperament on the 3.5.7.11.13 subgroup mapped as [⟨78 114 138 170 182]]. Under this compression, the primes map to the 49fgh val in the 23-limit.

Scales

MOS scales

Main article: List of MOS scales in 49edo
  • Bohpier[8]: 6 6 6 6 7 6 6 6
  • Catalan[7]: 3 10 3 10 3 10 10 (vaugely diminished-like)
  • Catalan[11]: 3 7 3 3 7 3 3 7 3 3 7
  • Catalan[19]: 3 3 1 3 3 3 3 1 3 3 3 1 3 3 3 3 1 3 3
  • Clyde[5]: 5 13 5 13 13 (mysterious, adventurous)
  • Didacus[6]: 8 8 8 8 8 9 (like the whole tone scale)
  • Didacus[13]: 1 7 1 7 1 7 1 7 1 7 1 7 1
  • Infraorwell[5]: 11 11 5 11 11
  • Infraorwell[22]: 1 4 1 4 1 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
  • Kleiboh[5]: 12 12 12 12 1
  • Kleiboh[13]: 1 10 1 1 10 1 1 1 10 1 1 10 1
  • Magus[7]: 1 15 1 15 1 15 1 (vaguely augmented-like)
  • Passion[12]: 4 4 4 4 4 4 5 4 4 4 4 4 (like 12edo)
  • Passion[23]: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
  • Sevond[21]/Seville[21]: 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1 1 5 1
  • Superpyth[5]: 11 9 9 11 9 (in between minor pentatonic and equipentatonic)
  • Superpyth[7]: 9 2 9 9 9 2 9 (Dorian mode; rotate for other modes)
  • Superpyth[12]: 2 7 2 7 2 2 7 2 7 2 7 2 (same melodic shape as 12edo but much more xenharmonic harmonies)
  • Superpyth[27]: 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2

Other scales

Instruments

Lumatone

Skip fretting

Skip fretting system 49 3 7 is a skip fretting system for 49edo. All examples are for 5-string bass.

Harmonics

1/1: string 2 open

2/1: not easily accessible

3/2: string 4 fret 5 and string 1 fret 12

5/4: string 3 fret 3

7/4: string 3 fret 11

11/8: string 3 fret 5

Music

Modern renderings

The Cure

21st century

Bryan Deister
Mercury Amalgam
Cam Taylor
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