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== Theory ==
== Theory ==
In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though there are exceptions: it has a [[7/4|7/1]] which is only six cents sharp, and a [[5/4|5/1]] which is only eleven cents flat. Most low harmonics are tuned very flat, but some such as [[21/16|21]]:[[11/8|22]]:[[23/16|23]]:[[3/2|24]]:[[25/16|25]]:[[13/8|26]] are well in tune with each other. 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.
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.  


Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), and thus the familiar [[diminished seventh chord]] may be built on any scale step with 4 unique tetrads up to [[octave equivalence]].
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 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]].
 
16edo shares several similarities with 15edo. They both share mappings of [[8/7]], [[5/4]], and [[3/2]] in terms of edosteps – in fact, they are both [[valentine]] tunings, and thus [[slendric]] tunings. 16edo and 15edo also both have three types of seconds and two types of thirds (not including arto/tendo thirds). However, 15edo's fifth is sharp while 16's is flat.
 
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.  


=== Odd harmonics ===
=== Odd harmonics ===
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Since 16 factors into primes as 2<sup>4</sup>, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.
Since 16 factors into primes as 2<sup>4</sup>, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.


== Intervals ==
=== Composition theory ===
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue harmony]]


The first and most common defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules (see [[16edo#Interval_arithmetic_examples]]). Note that the notes that form chords are different from in diatonic: for example, a major chord, {{dash|P1, M3, P5|med}}, is approximately 4:5:6 as would be expected, but is notated C-E#-G on C. (But see below in "Chord Names".)
{{Todo|inline=1| expand }}


Alternatively, one can essentially pretend 16edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. The primary purpose of doing this is to allow music notated in 12edo or another diatonic system to be directly translated to 16edo "on the fly" (or to allow support for 16edo in tools that only allow chain-of-fifths notation), and it carries over the way interval arithmetic works from diatonic notation, at the cost of notating the sizes of intervals and the shapes of chords incorrectly: that is, a major chord, P1-M3-P5, is notated C-E-G on C, but is no longer ~4:5:6 (since the third is closer to a minor third).
== Intervals ==
Inconsistent intervals are in ''italics''.


For the sake of clarity, the first notation is commonly called "melodic notation", and the second is called "harmonic notation", but this is a bit of a misnomer as both preserve different features of the notation of harmony.
{| class="wikitable center-1 right-2"
{| class="wikitable"
|-
|+
! rowspan="2" | #
!
! rowspan="2" | [[Cent]]s
!P1-M3-P5 ~ 4:5:6
! colspan="3" | Approximate ratios
!P1-M3-P5 = C-E-G on C
|-
! 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
|-
|-
!Diatonic notation
| 15
|NO
| 1125
|YES
| ''[[28/15]]'', [[40/21]], [[48/25]]
| [[21/11]], [[25/13]]
| [[27/14]], [[52/27]]
|-
|-
!Antidiatonic notation
| 16
|YES
| 1200
|NO
| [[2/1]]
| 2/1
| 2/1
|}
|}
Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]
 
<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}}
 
Alternatively, one can use Armodue nine-nominal notation.


{| 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
|-
|-
Line 59: Line 174:
| 0
| 0
| 0
| 0
| 1/1
| unison
| unison
| D
| D
Line 69: Line 183:
| 1
| 1
| 75
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| aug 1, dim 2nd
| D♯, E♭
| D♯, E♭
Line 79: Line 192:
| 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♭
Line 109: Line 219:
| 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♭
Line 129: Line 237:
| 7
| 7
| 525
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| perfect 4th
| G
| G
Line 139: Line 246:
| 8
| 8
| 600
| 600
| 7/5, 10/7
| aug 4th,<br>dim 5th
| aug 4th,<br>dim 5th
| G♯, A♭
| G♯, A♭
Line 149: Line 255:
| 9
| 9
| 675
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| perfect 5th
| A
| A
Line 159: Line 264:
| 10
| 10
| 750
| 750
| 20/13, 54/35
| aug 5th,<br>dim 6th
| aug 5th,<br>dim 6th
| A♯, B♭
| A♯, B♭
Line 169: Line 273:
| 11
| 11
| 825
| 825
| 8/5, 13/8, 21/13
| minor 6th
| minor 6th
| B
| B
Line 179: Line 282:
| 12
| 12
| 900
| 900
| 27/16, 32/19
| major 6th
| major 6th
| B♯
| B♯
Line 189: Line 291:
| 13
| 13
| 975
| 975
| 7/4
| minor 7th
| minor 7th
| C♭
| C♭
Line 199: Line 300:
| 14
| 14
| 1050
| 1050
| 64/35
| major 7th
| major 7th
| C
| C
Line 209: Line 309:
| 15
| 15
| 1125
| 1125
| 27/14, 52/27
| aug 7th,<br>dim 8ve
| aug 7th,<br>dim 8ve
| C♯, D♭
| C♯, D♭
Line 219: Line 318:
| 16
| 16
| 1200
| 1200
| 2/1
| 8ve
| 8ve
| D
| D
Line 227: Line 325:
| 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.


Line 329: Line 425:
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) ===
Line 445: Line 534:


[[: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
Line 720: Line 801:


[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 ==
Line 742: Line 826:
[[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]]
Line 843: Line 927:
; [[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]]
Line 849: Line 941:


; [[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/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]]
Line 869: Line 965:
* [[: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/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]]
Line 889: Line 992:
; [[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]]
Line 921: Line 1,031:
; [[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"