49edo: Difference between revisions

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== 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 ===

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=== 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 72, 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 ==

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| 9

| 220.408

| [[8/7]], ''[[9/8]]''

| [[8/7]], ''[[9/8]]'', [[25/22]]

| {{UDnote|step=9}}

|-

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| 40

| 979.592

| [[7/4]], ''[[16/9]]''

| [[7/4]], ''[[16/9]]'', [[44/25]]

| {{UDnote|step=40}}

|-

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== Notation ==

=== Ups and downs notation ===

~~Using [[Helmholtz–Ellis]] accidentals,~~ 49edo can be notated using [[ups and downs notation|ups and downs]]:

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

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

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=== Interval mappings ===

{{Q-odd-limit intervals|49.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 49f val mapping}}

=== Zeta ~~peak index~~ ===

=== 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]].

~~{{ZPI~~

~~| zpi = 233~~

~~| steps = 49.1412057629230~~

~~| step size = 24.4194252332612~~

~~| tempered height = 5.691547~~

~~| pure height = 0.862596~~

~~| integral = 0.920677~~

~~| gap = 15.624024~~

~~| octave = 1196.55183642980~~

~~| consistent = 7~~

~~| distinct = 7~~

}}

== 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 ([[Logarithmic phi|instead of dividing 2/1]]).

49edo has a very close approximation of both [[acoustic phi]] and [[phith root of phi|ϕϕ-1]], a kind of logarithmic phi that divides [[acoustic phi]] logarithmically by phi ([[Logarithmic 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]].

The [[phith root of phi|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]].

{| class="wikitable center-all"

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| 1.74

| 7.11

~~|- style="border-top: double;"~~

~~| 2.3.7~~

~~| 64/63, 96889010407/94143178827~~

~~| {{mapping| 49 78 138 }}~~

~~| −3.01~~

~~| 2.15~~

~~| 8.97~~

|}

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| 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.

== Scales ==

=== MOS scales ===

* 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 ===

* [[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

~~See [[Lumatone mapping for 49edo]]~~

5/4: string 3 fret 3

~~'''Fretted~~ string ~~instruments'''~~

7/4: string 3 fret 11

~~See [[Skip fretting system 49~~ 3 ~~7]]~~

11/8: string 3 fret 5

== Music ==

; ~~[[Mercury Amalgam]]~~

=== Modern renderings ===

* [https://www.youtube.com/watch?v=~~c_kzhcMMHWM&pp=ygUFNDllZG8%3D~~ ''~~Wrong Generation~~ (~~Demo, January 2022~~)''] (~~2023~~)

; {{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]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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 ===
@@ Line 9: / Line 9: @@
 === 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 ==
@@ Line 66: / Line 66: @@
 | 9
 | 220.408
-| [[8/7]], ''[[9/8]]''
+| [[8/7]], ''[[9/8]]'', [[25/22]]
 | {{UDnote|step=9}}
 |-
@@ Line 221: / Line 221: @@
 | 40
 | 979.592
-| [[7/4]], ''[[16/9]]''
+| [[7/4]], ''[[16/9]]'', [[44/25]]
 | {{UDnote|step=40}}
 |-
@@ Line 273: / Line 273: @@
 == Notation ==
 === Ups and downs notation ===
-Using [[Helmholtz–Ellis]] accidentals, 49edo can be notated using [[ups and downs notation|ups and downs]]:
+edo 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}}
@@ Line 306: / Line 309: @@
 === 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 peak index ===
+=== 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]].
-{{ZPI
-| zpi = 233
-| steps = 49.1412057629230
-| step size = 24.4194252332612
-| tempered height = 5.691547
-| pure height = 0.862596
-| integral = 0.920677
-| gap = 15.624024
-| octave = 1196.55183642980
-| consistent = 7
-| distinct = 7
-}}
 == Approximation to irrational intervals ==
 === Acoustic ϕ and ϕ<sup>ϕ<sup>−1</sup></sup> ===
-edo 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]]).
+edo 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"
@@ Line 393: / Line 384: @@
 | 1.74
 | 7.11
-|- style="border-top: double;"
-| 2.3.7
-| 64/63, 96889010407/94143178827
-| {{mapping| 49 78 138 }}
-| −3.01
-| 2.15
-| 8.97
 |}
@@ Line 493: / Line 477: @@
 | 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 ==
+edo'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 ===
+* [[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: string 2 open
+/1: not easily accessible
+/2: string 4 fret 5 and string 1 fret 12
-See [[Lumatone mapping for 49edo]]
+/4: string 3 fret 3
-'''Fretted string instruments'''
+/4: string 3 fret 11
-See [[Skip fretting system 49 3 7]]
+/8: string 3 fret 5
 == Music ==
-; [[Mercury Amalgam]]
+=== Modern renderings ===
-* [https://www.youtube.com/watch?v=c_kzhcMMHWM&pp=ygUFNDllZG8%3D ''Wrong Generation (Demo, January 2022)''] (2023)
+; {{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]]