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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-21 20:21:41 UTC</tt>.<br>
 
: The original revision id was <tt>212617994</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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]].
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 49 equal temperament divides the octave into 49 equal parts of 24.490 cents each. it is very much on the sharp side of things, with sharp tunings of 3, 5, 7, and 11. It is the [[optimal patent val]] for [[Archytas clan|superpyth temperament]] in the 7 and 11 limits, archytas (64/63) 7-limit planar temperament, and [[Archytas clan|ares]] 11-limit planar temperament. It tempers out 64/63, 245/243 and 3125/3087 in the 7-limit, and 100/99 and 1375/1372 in the 11-limit.</pre></div>
{{Harmonics in equal|49}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;49edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 49 equal temperament divides the octave into 49 equal parts of 24.490 cents each. it is very much on the sharp side of things, with sharp tunings of 3, 5, 7, and 11. It is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Archytas%20clan"&gt;superpyth temperament&lt;/a&gt; in the 7 and 11 limits, archytas (64/63) 7-limit planar temperament, and &lt;a class="wiki_link" href="/Archytas%20clan"&gt;ares&lt;/a&gt; 11-limit planar temperament. It tempers out 64/63, 245/243 and 3125/3087 in the 7-limit, and 100/99 and 1375/1372 in the 11-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== 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 ==
{| class="wikitable center-all right-2 left-3"
|-
! &#35;
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.000
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 24.490
| [[50/49]]
| {{UDnote|step=1}}
|-
| 2
| 48.980
| ''[[28/27]]'', [[36/35]], ''[[49/48]]'', ''[[81/80]]''
| {{UDnote|step=2}}
|-
| 3
| 73.469
| [[22/21]], [[25/24]], ''[[33/32]]''
| {{UDnote|step=3}}
|-
| 4
| 97.959
| ''[[16/15]]'', [[21/20]]
| {{UDnote|step=4}}
|-
| 5
| 122.449
| [[15/14]]
| {{UDnote|step=5}}
|-
| 6
| 146.939
| [[12/11]]
| {{UDnote|step=6}}
|-
| 7
| 171.429
| [[10/9]], [[11/10]]
| {{UDnote|step=7}}
|-
| 8
| 195.918
| [[28/25]]
| {{UDnote|step=8}}
|-
| 9
| 220.408
| [[8/7]], ''[[9/8]]'', [[25/22]]
| {{UDnote|step=9}}
|-
| 10
| 244.898
| 125/108, 144/125
| {{UDnote|step=10}}
|-
| 11
| 269.388
| [[7/6]]
| {{UDnote|step=11}}
|-
| 12
| 293.878
| [[25/21]], [[33/28]]
| {{UDnote|step=12}}
|-
| 13
| 318.367
| [[6/5]]
| {{UDnote|step=13}}
|-
| 14
| 342.857
| [[11/9]]
| {{UDnote|step=14}}
|-
| 15
| 367.347
| [[27/22]]
| {{UDnote|step=15}}
|-
| 16
| 391.837
| [[5/4]]
| {{UDnote|step=16}}
|-
| 17
| 416.327
| [[14/11]]
| {{UDnote|step=17}}
|-
| 18
| 440.816
| [[9/7]]
| {{UDnote|step=18}}
|-
| 19
| 465.306
| 125/96, ''162/125''
| {{UDnote|step=19}}
|-
| 20
| 489.796
| [[4/3]], ''[[21/16]]''
| {{UDnote|step=20}}
|-
| 21
| 514.286
| [[75/56]]
| {{UDnote|step=21}}
|-
| 22
| 538.776
| [[15/11]], ''[[27/20]]''
| {{UDnote|step=22}}
|-
| 23
| 563.265
| [[11/8]]
| {{UDnote|step=23}}
|-
| 24
| 587.755
| [[7/5]]
| {{UDnote|step=24}}
|-
| 25
| 612.245
| [[10/7]]
| {{UDnote|step=25}}
|-
| 26
| 636.735
| [[16/11]]
| {{UDnote|step=26}}
|-
| 27
| 661.244
| [[22/15]], ''[[40/27]]''
| {{UDnote|step=27}}
|-
| 28
| 685.714
| [[112/75]]
| {{UDnote|step=28}}
|-
| 29
| 710.204
| [[3/2]], ''[[32/21]]''
| {{UDnote|step=29}}
|-
| 30
| 734.694
| ''125/81'', 192/125
| {{UDnote|step=30}}
|-
| 31
| 759.184
| [[14/9]]
| {{UDnote|step=31}}
|-
| 32
| 783.673
| [[11/7]]
| {{UDnote|step=32}}
|-
| 33
| 808.163
| [[8/5]]
| {{UDnote|step=33}}
|-
| 34
| 832.653
| [[44/27]]
| {{UDnote|step=34}}
|-
| 35
| 857.143
| [[18/11]]
| {{UDnote|step=35}}
|-
| 36
| 881.633
| [[5/3]]
| {{UDnote|step=36}}
|-
| 37
| 906.122
| [[42/25]], [[56/33]]
| {{UDnote|step=37}}
|-
| 38
| 930.612
| [[12/7]]
| {{UDnote|step=38}}
|-
| 39
| 955.102
| 125/72, 216/125
| {{UDnote|step=39}}
|-
| 40
| 979.592
| [[7/4]], ''[[16/9]]'', [[44/25]]
| {{UDnote|step=40}}
|-
| 41
| 1004.082
| [[25/14]]
| {{UDnote|step=41}}
|-
| 42
| 1028.571
| [[9/5]], [[20/11]]
| {{UDnote|step=42}}
|-
| 43
| 1053.061
| [[11/6]]
| {{UDnote|step=43}}
|-
| 44
| 1077.551
| [[28/15]]
| {{UDnote|step=44}}
|-
| 45
| 1102.041
| ''[[15/8]]'', [[40/21]]
| {{UDnote|step=45}}
|-
| 46
| 1126.531
| [[21/11]], [[48/25]], ''[[64/33]]''
| {{UDnote|step=46}}
|-
| 47
| 1151.020
| ''[[27/14]]'', [[35/18]], ''[[96/49]]'', ''[[160/81]]''
| {{UDnote|step=47}}
|-
| 48
| 1175.510
| [[49/25]]
| {{UDnote|step=48}}
|-
| 49
| 1200.000
| [[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]].
 
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Direct approximation
|-
! Interval
! Error (abs, [[Cent|¢]])
! #\49
|-
| {{nowrap|ϕ / ϕ<sup>ϕ<sup>−1</sup></sup> {{=}} ϕ<sup>(2 − ϕ)</sup>}}
| 0.155
| 13
|-
| ϕ
| −0.437
| 34
|-
| ϕ<sup>ϕ<sup>−1</sup></sup>
| −0.592
| 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).
 
=== 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
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 78 -49 }}
| {{mapping| 49 78 }}
| −2.60
| 2.60
| 10.62
|-
| 2.3.5
| 15625/15552, 20480/19683
| {{mapping| 49 78 114 }}
| −2.53
| 2.12
| 8.69
|-
| 2.3.5.7
| 64/63, 245/243, 3125/3087
| {{mapping| 49 78 114 138 }}
| −2.85
| 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
| 7.11
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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]]
|-
| rowspan="2" | 7
| rowspan="2" | 20\49<br />(1\49)
| 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:Archytas]]
[[Category:Ares]]
[[Category:Listen]]
[[Category:Superpyth]]
Retrieved from "https://en.xen.wiki/w/49edo"