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'''49-EDO''', or '''49 equal temperament''' divides the octave into 49 equal parts of 24.490 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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 family #Archytas|archytas]] ([[7-limit]]) and [[Archytas family #Ares|ares]] ([[11-limit]]) planar temperaments and almost identical to the 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 in equal|49}}
 
=== Subsets and supersets ===
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]]
|-
|-
| 0
| 0
| 0.000
| 0.000
| [[1/1]]
| [[1/1]]
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 24.490
| 24.490
| [[50/49]]
| [[50/49]]
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 48.980
| 48.980
| [[81/80]], [[28/27]], [[36/35]], [[49/48]]
| ''[[28/27]]'', [[36/35]], ''[[49/48]]'', ''[[81/80]]''
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 73.469
| 73.469
| [[25/24]], [[22/21]], [[33/32]]
| [[22/21]], [[25/24]], ''[[33/32]]''
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 97.959
| 97.959
| [[16/15]], [[21/20]]
| ''[[16/15]]'', [[21/20]]
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 122.449
| 122.449
| [[15/14]]
| [[15/14]]
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 146.939
| 146.939
| [[12/11]]
| [[12/11]]
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 171.429
| 171.429
| [[10/9]], [[11/10]]
| [[10/9]], [[11/10]]
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 195.918
| 195.918
|
| [[28/25]]
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 220.408
| 220.408
| [[9/8]], [[8/7]]
| [[8/7]], ''[[9/8]]'', [[25/22]]
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 244.898
| 244.898
|
| 125/108, 144/125
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 269.388
| 269.388
| [[7/6]]
| [[7/6]]
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 293.878
| 293.878
|
| [[25/21]], [[33/28]]
| {{UDnote|step=12}}
|-
|-
| 13
| 13
| 318.367
| 318.367
| [[6/5]]
| [[6/5]]
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 342.857
| 342.857
| [[11/9]]
| [[11/9]]
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 367.347
| 367.347
| [[27/22]]
| [[27/22]]
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 391.837
| 391.837
| [[5/4]]
| [[5/4]]
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 416.327
| 416.327
| [[14/11]]
| [[14/11]]
| {{UDnote|step=17}}
|-
|-
| 18
| 18
| 440.816
| 440.816
| [[9/7]]
| [[9/7]]
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 465.306
| 465.306
|  
| 125/96, ''162/125''
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 489.796
| 489.796
| [[4/3]], [[21/16]]
| [[4/3]], ''[[21/16]]''
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 514.286
| 514.286
|
| [[75/56]]
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 538.776
| 538.776
| [[27/20]], [[15/11]]
| [[15/11]], ''[[27/20]]''
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 563.265
| 563.265
| [[11/8]]
| [[11/8]]
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 587.755
| 587.755
| [[7/5]]
| [[7/5]]
| {{UDnote|step=24}}
|-
|-
| 25
| 25
| 612.245
| 612.245
| [[10/7]]
| [[10/7]]
| {{UDnote|step=25}}
|-
|-
| 26
| 26
| 636.735
| 636.735
| [[16/11]]
| [[16/11]]
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 661.244
| 661.244
| [[40/27]], [[22/15]]
| [[22/15]], ''[[40/27]]''
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 685.714
| 685.714
|  
| [[112/75]]
| {{UDnote|step=28}}
|-
|-
| 29
| 29
| 710.204
| 710.204
| [[3/2]], [[32/21]]
| [[3/2]], ''[[32/21]]''
| {{UDnote|step=29}}
|-
|-
| 30
| 30
| 734.694
| 734.694
|  
| ''125/81'', 192/125
| {{UDnote|step=30}}
|-
|-
| 31
| 31
| 759.184
| 759.184
| [[14/9]]
| [[14/9]]
| {{UDnote|step=31}}
|-
|-
| 32
| 32
| 783.673
| 783.673
| [[11/7]]
| [[11/7]]
| {{UDnote|step=32}}
|-
|-
| 33
| 33
| 808.163
| 808.163
| [[8/5]]
| [[8/5]]
| {{UDnote|step=33}}
|-
|-
| 34
| 34
| 832.653
| 832.653
| [[44/27]]
| [[44/27]]
| {{UDnote|step=34}}
|-
|-
| 35
| 35
| 857.143
| 857.143
| [[18/11]]
| [[18/11]]
| {{UDnote|step=35}}
|-
|-
| 36
| 36
| 881.633
| 881.633
| [[5/3]]
| [[5/3]]
| {{UDnote|step=36}}
|-
|-
| 37
| 37
| 906.122
| 906.122
|  
| [[42/25]], [[56/33]]
| {{UDnote|step=37}}
|-
|-
| 38
| 38
| 930.612
| 930.612
| [[12/7]]
| [[12/7]]
| {{UDnote|step=38}}
|-
|-
| 39
| 39
| 955.102
| 955.102
|  
| 125/72, 216/125
| {{UDnote|step=39}}
|-
|-
| 40
| 40
| 979.592
| 979.592
| [[16/9]], [[7/4]]
| [[7/4]], ''[[16/9]]'', [[44/25]]
| {{UDnote|step=40}}
|-
|-
| 41
| 41
| 1004.082
| 1004.082
|  
| [[25/14]]
| {{UDnote|step=41}}
|-
|-
| 42
| 42
| 1028.571
| 1028.571
| [[9/5]], [[20/11]]
| [[9/5]], [[20/11]]
| {{UDnote|step=42}}
|-
|-
| 43
| 43
| 1053.061
| 1053.061
| [[11/6]]
| [[11/6]]
| {{UDnote|step=43}}
|-
|-
| 44
| 44
| 1077.551
| 1077.551
| [[28/15]]
| [[28/15]]
| {{UDnote|step=44}}
|-
|-
| 45
| 45
| 1102.041
| 1102.041
| [[15/8]], [[40/21]]
| ''[[15/8]]'', [[40/21]]
| {{UDnote|step=45}}
|-
|-
| 46
| 46
| 1126.531
| 1126.531
| [[48/25]], [[21/11]], [[64/33]]
| [[21/11]], [[48/25]], ''[[64/33]]''
| {{UDnote|step=46}}
|-
|-
| 47
| 47
| 1151.020
| 1151.020
| [[160/81]], [[27/14]], [[35/18]], [[96/49]]
| ''[[27/14]]'', [[35/18]], ''[[96/49]]'', ''[[160/81]]''
| {{UDnote|step=47}}
|-
|-
| 48
| 48
| 1175.510
| 1175.510
| [[49/25]]
| [[49/25]]
| {{UDnote|step=48}}
|-
|-
| 49
| 49
| 1200.000
| 1200.000
| [[2/1]]
| [[2/1]]
| {{UDnote|step=49}}
|}
|}
<nowiki />* Based on 49edo's 11-limit patent val {{val| 49 78 114 138 170 }} mapping
== Notation ==
=== 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}}
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:49-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 589 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
default [[File:49-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:49-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 534 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 140 106 [[513/512]]
rect 140 80 240 106 [[81/80]]
rect 240 80 360 106 [[33/32]]
default [[File:49-EDO_Revo_Sagittal.svg]]
</imagemap>
== Approximation to JI ==
[[File:49ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 49edo]]
=== Interval mappings ===
{{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 ===
The strongest [[The Riemann zeta function and tuning|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 ϕ<sup>ϕ<sup>−1</sup></sup> ===
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]]).
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]].


== Just approximation ==
=== Selected just intervals ===
{| class="wikitable center-all"
{| class="wikitable center-all"
! colspan="2" |
|+ style="font-size: 105%;" | Direct approximation
! prime 2
|-
! prime 3
! Interval
! prime 5
! Error (abs, [[Cent|¢]])
! prime 7
! #\49
! prime 11
|-
! prime 13
| {{nowrap|ϕ / ϕ<sup>ϕ<sup>−1</sup></sup> {{=}} ϕ<sup>(2 − ϕ)</sup>}}
| 0.155
| 13
|-
|-
! rowspan="2" |Error
| ϕ
! absolute (¢)
| −0.437
| 0.0
| 34
| +8.2
| +5.5
| +10.8
| +11.9
| -7.9
|-
|-
! relative (%)
| ϕ<sup>ϕ<sup>−1</sup></sup>
| 0.0
| −0.592
| +33.7
| 21
| +22.6
| +44.0
| +48.8
| -32.2
|}
|}


=== Temperament measures ===
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).
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 49et.
 
{| class="wikitable center-all"
=== Music ===
! colspan="2" |
* [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
! 3-limit
 
! 5-limit
== Regular temperament properties ==
! 7-limit
{| class="wikitable center-4 center-5 center-6"
! 11-limit
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
! colspan="2" |Octave stretch (¢)
! [[TE error|Absolute]] (¢)
| -2.60
! [[TE simple badness|Relative]] (%)
| -2.53
| -2.85
| -2.97
|-
|-
! rowspan="2" |Error
| 2.3
! [[TE error|absolute]] (¢)
| {{monzo| 78 -49 }}
| {{mapping| 49 78 }}
| −2.60
| 2.60
| 2.60
| 10.62
|-
| 2.3.5
| 15625/15552, 20480/19683
| {{mapping| 49 78 114 }}
| −2.53
| 2.12
| 2.12
| 8.69
|-
| 2.3.5.7
| 64/63, 245/243, 3125/3087
| {{mapping| 49 78 114 138 }}
| −2.85
| 1.92
| 1.92
| 7.87
|-
| 2.3.5.7.11
| 64/63, 100/99, 245/243, 1331/1323
| {{mapping| 49 78 114 138 170 }}
| −2.97
| 1.74
| 1.74
|-
! [[TE simple badness|relative]] (%)
| 10.62
| 8.69
| 7.87
| 7.11
| 7.11
|}
|}


== Rank-2 temperaments ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-3"
{| class="wikitable center-all left-5"
! Periods<br>per octave
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Generator
|-
! Temperaments
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
| 1\49
| 1\49
| 24.5
| 99/98
| [[Sengagen]]
| [[Sengagen]]
|-
|-
| 1
| 1
| 4\49
| 4\49
| 98.0
| 16/15
| [[Passion]]
| [[Passion]]
|-
|-
| 1
| 1
| 6\49
| 6\49
| 146.9
| 12/11
| [[Bohpier]]
| [[Bohpier]]
|-
| 1
| 8\49
| 195.9
| 28/25
| [[Didacus]]
|-
|-
| 1
| 1
| 11\49
| 11\49
| 269.4
| 7/6
| [[Infraorwell]]
| [[Infraorwell]]
|-
| 1
| 12\49
| 293.9
| 25/21
| [[Kleiboh]]
|-
|-
| 1
| 1
| 13\49
| 13\49
| 318.4
| 6/5
| [[Catalan]]
| [[Catalan]]
|-
|-
| 1
| 1
| 16\49
| 16\49
| 391.8
| 5/4
| [[Magus]]
| [[Magus]]
|-
| 1
| 17\49
| 416.3
| 14/11
| [[Sqrtphi]]
|-
|-
| 1
| 1
| 18\49
| 18\49
| 440.8
| 9/7
| [[Clyde]]
| [[Clyde]]
|-
|-
| 1
| 1
| 19\49
| 19\49
| 465.3
| 55/36
| [[Semisept]]
| [[Semisept]]
|-
|-
| 1
| 1
| 20\49
| 20\49
| 489.8
| 4/3
| [[Superpyth]]
| [[Superpyth]]
|-
|-
| 7
| rowspan="2" | 7
| 20\49
| rowspan="2" | 20\49<br />(1\49)
| [[Sevond]]/[[seville]]
| rowspan="2" | 489.8<br />(24.5)
| 4/3<br />(250/243)
| [[Sevond]] (49)
|-
| 4/3<br />(25/24)
| style="text-align: left;" | [[Seville]] (49c)
|}
|}
<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.
== Scales ==
=== MOS scales ===
{{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 ==
=== 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
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 ===
; {{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]]
* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation''] (2022 demo version)
; [[Cam Taylor]]
* [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023)


[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:49edo]]
[[Category:Superpyth]]
[[Category:Archytas]]
[[Category:Archytas]]
[[Category:Ares]]
[[Category:Ares]]
[[Category:Listen]]
[[Category:Superpyth]]
Retrieved from "https://en.xen.wiki/w/49edo"