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16edo: Difference between revisions

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→Music: ''16edo waltz'' (2025): Add the reason for keeping the short, in case anyone was thinking of eliminating it
 
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== Theory ==
== Theory ==
The [[3/2|perfect fifth]] of 16edo is 27 cents flat of 3/2, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale, but sharper than [[9edo]]'s fifth, to which it similarly retains the characteristic of being a fifth while being distinctly flat of 3/2. If the fifth is interpreted as 3/2, this befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]]. This leads to some confusion in regards to interval names, as what would be major in diatonic now sounds minor; there are several ways to handle this (see in [[#Intervals]]).  
The [[3/2|perfect fifth]] of 16edo is 27 cents flat of 3/2, flatter than that of [[7edo]] so that it generates an [[2L 5s|antidiatonic]] instead of [[5L 2s|diatonic]] scale, but sharper than [[9edo]]'s fifth, to which it similarly retains the characteristic of being a fifth while being distinctly flat of 3/2. If the fifth is interpreted as 3/2, this befits a tuning of [[mavila]], the [[5-limit]] [[regular temperament|temperament]] that [[tempering out|tempers out]] [[135/128]], such that a stack of four fifths gives a [[6/5]] minor third instead of the familiar [[5/4]] major third as in [[meantone]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.  


In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. As such, 16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error, as in [[Armodue theory]] (i.e. a 3/2 that's 25 cents flat is still recognizable, but a 5/4 that's 25 cents flat loses much of its identity and a 7/4 that's 25 cents flat is completely unrecognizable). In essence, 16edo's 3, 5, and 7 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.  
This leads to some confusion in regards to interval names, as what would be major in diatonic now sounds minor; there are several ways to handle this (see in [[#Intervals]]).  


In terms of higher primes, both 11 and 13 are approximated very flat, with the [[11/8]] not distinguished from [[4/3]], and [[13/8]] not distinguished from [[8/5]].  
In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though it has a [[5/1|5th harmonic]] which is only 11 cents flat, and a [[7/1|7th harmonic]] which is only 6 cents sharp. As such, 16edo can be seen as an approach to tuning that takes advantage of the idea that simpler ratios can be functionally approximated with greater error (i.e. a 3/2 that's 25 cents flat is still recognizable, but a 5/4 that's 25 cents flat loses much of its identity and a 7/4 that's 25 cents flat is completely unrecognizable). In essence, 16edo's 3, 5, and 7 are backwards from 12edo's, with 7 being nearly perfect, 5 being decent, and 3 being distinctly out-of-tune.
 
In terms of higher primes, both 11 and 13 are approximated very flat, with the [[11/8]] not distinguished from [[4/3]], and [[13/8]] not distinguished from [[8/5]]. 16edo represents the no-9 no-15 [[25-odd-limit]] [[consistent]]ly, however.  


Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), which approximates [[6/5]], and thus tempers out 648/625, the [[diminished comma]]. This means that the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]]. Another possible interpretation for this interval is the 19th harmonic, [[19/16]].  
Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), which approximates [[6/5]], and thus tempers out 648/625, the [[diminished comma]]. This means that the familiar [[diminished seventh chord]] may be built on any scale step with four unique tetrads up to [[octave equivalence]]. The minor third is of course not distinguished from the septimal subminor third, [[7/6]], so [[36/35]] and moreover [[50/49]] are tempered out, making 16edo a possible tuning for [[diminished (temperament)|septimal diminished]]. Another possible interpretation for this interval is the 19th harmonic, [[19/16]].  
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=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|16}}
{{Harmonics in equal|16}}
=== Octave stretch ===
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.


=== Subsets and supersets ===
=== Subsets and supersets ===
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== Intervals ==
== Intervals ==
Inconsistent intervals are in ''italics''.
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | [[Cent]]s
! colspan="3" | Approximate ratios
|-
! 7-limit
! 13-limit
! No-3 approach*
|-
| 0
| 0
| [[1/1]]
| 1/1
| 1/1
|-
| 1
| 75
| ''[[15/14]]'', [[21/20]], [[25/24]]
| [[22/21]], [[26/25]]
| [[28/27]], [[27/26]]
|-
| 2
| 150
| ''[[9/8]]'', ''[[16/15]]''
| [[11/10]], [[12/11]], [[13/12]], [[14/13]]
| 12/11, [[35/32]]
|-
| 3
| 225
| [[8/7]], ''[[10/9]]'', [[28/25]]
| [[15/13]], [[25/22]]
| 8/7
|-
| 4
| 300
| [[6/5]], [[7/6]], [[25/21]]
| [[13/11]]
| [[19/16]], [[32/27]]
|-
| 5
| 375
| [[5/4]], ''[[9/7]]''
| [[11/9]], [[16/13]], [[26/21]]
| 5/4, 16/13, 26/21
|-
| 6
| 450
| [[21/16]], [[32/25]]
| [[14/11]], [[13/10]]
| 13/10, [[35/27]]
|-
| 7
| 525
| [[4/3]]
| [[11/8]], ''[[18/13]]'', [[15/11]]
| [[19/14]], [[27/20]], [[35/26]]
|-
| 8
| 600
| [[7/5]], [[10/7]], [[25/18]], [[36/25]]
|
| 7/5, 10/7
|-
| 9
| 675
| [[3/2]]
| [[16/11]], ''[[13/9]]'', [[22/15]]
| [[28/19]], [[40/27]], [[52/35]]
|-
| 10
| 750
| [[25/16]], [[32/21]]
| [[11/7]], [[20/13]]
| 20/13, [[54/35]]
|-
| 11
| 825
| [[8/5]], ''[[14/9]]''
| [[18/11]], [[13/8]], [[21/13]]
| 8/5, 13/8, 21/13
|-
| 12
| 900
| [[5/3]], [[12/7]], [[42/25]]
| [[22/13]]
| [[27/16]], [[32/19]]
|-
| 13
| 975
| [[7/4]], ''[[9/5]]'', [[25/14]]
| [[26/15]], [[44/25]]
| 7/4
|-
| 14
| 1050
| ''[[16/9]]'', ''[[15/8]]''
| [[20/11]], [[11/6]], [[24/13]], [[13/7]]
| [[11/6]], 64/35
|-
| 15
| 1125
| ''[[28/15]]'', [[40/21]], [[48/25]]
| [[21/11]], [[25/13]]
| [[27/14]], [[52/27]]
|-
| 16
| 1200
| [[2/1]]
| 2/1
| 2/1
|}
<nowiki />* Based on treating 16edo as a 2.27.5.7.13.19 subgroup temperament; other approaches are possible. Odd 27 is approximated via [[direct approximation]].
== Notation ==
{{Mavila}}  
{{Mavila}}  


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{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%" | Notation systems for 16edo
|-
|-
! rowspan="2" | Degree
! rowspan="2" | Degree
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Cent]]s
! rowspan="2" | Approximate<br>ratios*
! colspan="6" | Names
! colspan="6" | Names
|-
|-
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| 0
| 0
| 0
| 0
| 1/1
| unison
| unison
| D
| D
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| 1
| 1
| 75
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| aug 1, dim 2nd
| D♯, E♭
| D♯, E♭
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| 2
| 2
| 150
| 150
| 35/32
| minor 2nd
| minor 2nd
| E
| E
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| 3
| 3
| 225
| 225
| 8/7
| major 2nd
| major 2nd
| E♯
| E♯
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| 4
| 4
| 300
| 300
| 19/16, 32/27
| minor 3rd
| minor 3rd
| F♭
| F♭
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| 5
| 5
| 375
| 375
| 5/4, 16/13, 26/21
| major 3rd
| major 3rd
| F
| F
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| 6
| 6
| 450
| 450
| 13/10, 35/27
| aug 3rd,<br>dim 4th
| aug 3rd,<br>dim 4th
| F♯, G♭
| F♯, G♭
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| 7
| 7
| 525
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| perfect 4th
| G
| G
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| 8
| 8
| 600
| 600
| 7/5, 10/7
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
| G♯, A♭
| G♯, A♭
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| 9
| 9
| 675
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| perfect 5th
| A
| A
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| 10
| 10
| 750
| 750
| 20/13, 54/35
| aug 5th,<br>dim 6th
| aug 5th,<br>dim 6th
| A♯, B♭
| A♯, B♭
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| 11
| 11
| 825
| 825
| 8/5, 13/8, 21/13
| minor 6th
| minor 6th
| B
| B
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| 12
| 12
| 900
| 900
| 27/16, 32/19
| major 6th
| major 6th
| B♯
| B♯
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| 13
| 13
| 975
| 975
| 7/4
| minor 7th
| minor 7th
| C♭
| C♭
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| 14
| 14
| 1050
| 1050
| 64/35
| major 7th
| major 7th
| C
| C
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| 15
| 15
| 1125
| 1125
| 27/14, 52/27
| aug 7th,<br>dim 8ve
| aug 7th,<br>dim 8ve
| C♯, D♭
| C♯, D♭
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| 16
| 16
| 1200
| 1200
| 2/1
| 8ve
| 8ve
| D
| D
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| octave
| octave
|}
|}
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.


== Notation ==
16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.
16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.


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This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].


<imagemap>
{{Sagittal chart|}}
File:16-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 471 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:16-EDO_Sagittal.svg]]
</imagemap>


=== Armodue notation (4-line staff) ===
=== Armodue notation (4-line staff) ===
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[[:File:16ed2-001.svg|16ed2-001.svg]]
[[:File:16ed2-001.svg|16ed2-001.svg]]
=== Zeta peak index ===
{{ZPI
| zpi = 51
| steps = 15.9443732426877
| step size = 75.2616601314409
| tempered height = 4.191572
| pure height = 3.476281
| integral = 0.812082
| gap = 13.070433
| octave = 1204.18656210305
| consistent = 6
| distinct = 6
}}


== Octave theory ==
== Octave theory ==
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| Lehmerisma
| Lehmerisma
|}
|}
<references group=note/>


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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| [[Semidim]]
| [[Semidim]]
|}
|}
== Octave stretch or compression ==
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]]. Suitable stretched 16edo tunings include [[zpi|15zpi]] and [[57ed12]].


== Scales ==
== Scales ==
=== MOS scales ===
* {{Main|List of MOS scales in {{PAGENAME}}}}
* {{Main|List of MOS scales in {{PAGENAME}}}}
Important mosses include:
Important mosses include:
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[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1


'''Cynder/Gorgo'''
'''Gorgo'''


[5]: 3 3 4 3 3
[5]: 3 3 4 3 3
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[10]: 2 1 2 1 2 2 1 2 1 2
[10]: 2 1 2 1 2 2 1 2 1 2
=== Other scales ===
* [[User:BudjarnLambeth/Quasipelog theory#Scales]]


== Metallic harmony ==
== Metallic harmony ==
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[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]
[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]


'''Un-annotated diagram'''
'''Interleaved edos'''


Please explain this image. {{todo|annotate}}
A visualization of 16edo being two interleaved copies of [[8edo]] and four interleaved copies of [[4edo]].


[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]


'''Lumatone mapping'''
=== Lumatone mapping ===


See: [[Lumatone mapping for 16edo]]
See: [[Lumatone mapping for 16edo]]
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; [[Beheld]]
; [[Beheld]]
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']
; [[Stevie Boyes]]
* [https://youtu.be/jX190TYgQxc ''Tropical Carnival''] (2018)
; [[David Clifton]] (Cameron Watt Music)
* [https://www.youtube.com/watch?v=wSUlNCw9Pds ''Daydream | Meditative Ambient | 16-EDO''] (2025)
* [https://www.youtube.com/watch?v=ftW6VTdNcbg ''An Unsettling Theme | Horror | 16-EDO''] (2025)
* [https://www.youtube.com/watch?v=he47ywCgEFs ''Foreboding | Horror | 16-EDO''] (2025)


; [[City of the Asleep]]
; [[City of the Asleep]]
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; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/H-abpioYj5k ''improv in 16edo''] (2022)
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)
* [https://www.youtube.com/shorts/PXiWkZ9wDdU ''16edo waltz''] (2025) &mdash; includes Lumatone view
* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025) &mdash; this is the full version of the ''16edo waltz'' above
* [https://www.youtube.com/shorts/UA97cjUN5eE ''Doubt by Bryan Deister (microtonal 16edo snippet)''] (2025)


; [[E8 Heterotic]]
; [[E8 Heterotic]]
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* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=y7D0ZgCZlEg ''Waking with a fever before sunrise''] (2026)


; [[Claudi Meneghin]]
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vIWxP_C0aUM ''Mavila Fugue'']
* ''Mavila Fugue'' ([https://www.youtube.com/watch?v=vIWxP_C0aUM 2020], [https://www.youtube.com/shorts/TLEte0ox3Tw 2026])
* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])
* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])
* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])
* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])
* [https://www.youtube.com/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)
* [https://www.youtube.com/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)
* [https://www.youtube.com/shorts/aQqcbIT7tsg ''MICROPIECE IN 16-EDO FOR BAROQUE CONSORT (Mikrokosmos #21)''] (2026)
* [https://www.youtube.com/shorts/diyc6e-X1hw ''CANON in 16 edo - 3-in-1 on a GROUND, for BAROQUE CONSORT''] (2026)
* [https://www.youtube.com/shorts/cmJoeypG5e0 ''ALLING FIFTHS in 16-edo, with SHEPARD EFFECT''] (2026)


; [[Herman Miller]]
; [[Herman Miller]]
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; [[Jean-Pierre Poulin]]
; [[Jean-Pierre Poulin]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']
; [[Hans Straub]]
* [https://hansstraub.bandcamp.com/album/nights-in-sixteen ''Nights in Sixteen''] (full album) (2026)


; [[Ron Sword]]
; [[Ron Sword]]
Line 901: Line 1,007:
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']
; Stephen Weigel (on 16edo keyboard) with [[Clarissa]] (vocal)
* [https://www.youtube.com/shorts/ZUBE817kwk8 Microtonal cover of ''All I Want for Christmas is You"'' by Mariah Carey] (2024)


; [[Randy Winchester]]
; [[Randy Winchester]]
Line 907: Line 1,016:
; [[Woyten]]
; [[Woyten]]
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)
** [https://www.youtube.com/watch?v=Ta1HzwSJmqk (Transcription of this in 48edo subset notation)], by [[Stephen Weigel]] (2026)


; [[Xotla]]
; [[Xotla]]
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; [[Zewen Senpai]]
; [[Zewen Senpai]]
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']
== Notes ==
<references group=note/>


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/16edo"