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== Theory ==
== Theory ==
16edo's fifth is too flat to be a reasonable approximation of 3/2, but still within the range where it can serve the role of a fifth, similarly to 9edo. If it is treated as a diatonic fifth, it generates a scale that is similar to diatonic, but where minor is larger than major, which is called an [[antidiatonic]] scale (with 5 small steps and 2 large steps). It is thus often useful to switch "minor" and "major" from their standard diatonic meanings in interval names (but see below in [[#Intervals]]). If treated as [[3/2]], this fifth produces the [[mavila]] temperament, as stacking it four times produces a [[6/5]] minor third as opposed to the standard [[5/4]] major third found in meantone temperaments. (This tempers out [[135/128]], as that is the difference between 6/5 and 81/64, the interval generated by a stack of 4 untempered fifths. Note that 5/4 is tempered to 32/27.)
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.  


This minor third is 300 cents (supporting the [[diminished]] temperament), however in this tuning its inaccuracy comes more from the error on the fifth than on the 5/4 major third, which is much closer to (and flat of) just intonation at 375 cents. However, 16edo's best low prime is 7, which is mapped with only 6 cents of error. 13 is also mapped decently accurately (if 12edo's error on 5/4 is acceptable). This contributes to a good tuning of certain [[interseptimal]] intervals, specifically 13/10 and 20/13. As such, 16edo can most reasonably be considered a 2.5.7.13 temperament (2.3.5.7.13, if the mavila fifth is acceptable as a 3/2).
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]]).  


7/5 is mapped to the tritone of 600 cents, and 14/13 to the neutral second of 150 cents. 35/32 is mapped to the same interval, and 28/25 is mapped to 225 cents.
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.  


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: by choosing a tuning with greater error on lower primes as opposed to higher ones, one can create a much more consistent feeling than if the highest errors are on higher primes. In essence, 16edo's 3/1, 5/1, and 7/1 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.  


12edo's symmetrical diminished seventh chord may be constructed in 16edo by stacking minor thirds.  
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.  
16edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.  
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 temperaments, and thus slendric temperaments. 16edo and 15edo also both have 3 types of seconds and 2 types of thirds (excluding arto/tendo thirds).
However, 15edo's fifth is sharp while 16's is flat.


=== 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 ===
Line 38: Line 33:
=== Composition theory ===
=== Composition theory ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue harmony]]


{{Todo|inline=1| expand }}
{{Todo|inline=1| expand }}


== 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}}  


Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]
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 60: Line 174:
| 0
| 0
| 0
| 0
| 1/1
| unison
| unison
| D
| D
Line 70: Line 183:
| 1
| 1
| 75
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| aug 1, dim 2nd
| D♯, E♭
| D♯, E♭
Line 80: 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♯
Line 100: Line 210:
| 4
| 4
| 300
| 300
| 19/16, 32/27
| minor 3rd
| minor 3rd
| F♭
| F♭
Line 110: Line 219:
| 5
| 5
| 375
| 375
| 5/4, 16/13, 26/21
| major 3rd
| major 3rd
| F
| F
Line 120: Line 228:
| 6
| 6
| 450
| 450
| 13/10, 35/27
| aug 3rd,<br>dim 4th
| aug 3rd,<br>dim 4th
| F♯, G♭
| F♯, G♭
Line 130: Line 237:
| 7
| 7
| 525
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| perfect 4th
| G
| G
Line 140: 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 150: Line 255:
| 9
| 9
| 675
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| perfect 5th
| A
| A
Line 160: 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 170: Line 273:
| 11
| 11
| 825
| 825
| 8/5, 13/8, 21/13
| minor 6th
| minor 6th
| B
| B
Line 180: Line 282:
| 12
| 12
| 900
| 900
| 27/16, 32/19
| major 6th
| major 6th
| B♯
| B♯
Line 190: Line 291:
| 13
| 13
| 975
| 975
| 7/4
| minor 7th
| minor 7th
| C♭
| C♭
Line 200: Line 300:
| 14
| 14
| 1050
| 1050
| 64/35
| major 7th
| major 7th
| C
| C
Line 210: 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 220: Line 318:
| 16
| 16
| 1200
| 1200
| 2/1
| 8ve
| 8ve
| D
| D
Line 228: 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 330: 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 446: 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 ===
Line 654: Line 729:
| [[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:
Line 706: Line 786:
[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 721: 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 743: 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 844: 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 850: 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/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]]
Line 871: 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/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 892: 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 904: 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 910: 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 924: 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"