16edo: Difference between revisions

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== 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]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.

~~=== Tuning theory ===~~

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~~ 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~~.

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]].

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 ===

=== 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 ===

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=== Composition theory ===

* [[User:VectorGraphics/16edo theory|Vector's approach]]

* [[Armodue harmony]]

== Intervals ==

~~16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.~~

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".)

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).

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.

Alternatively, one can use Armodue nine-nominal notation.

~~{| class="wikitable"~~

|+

!

~~!P1-M3-P5 ~ 4:5:6~~

~~!P1-M3-P5 = C-E-G on C~~

|-

~~!Diatonic notation~~

~~|NO~~

~~|YES~~

|-

~~!Antidiatonic notation~~

~~|YES~~

~~|NO~~

|}

Alternatively, one can use Armodue nine-nominal notation~~; see [[Armodue theory]]~~

{| class="wikitable center-all"

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[[: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 ==

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[[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]]

~~'''~~Lumatone mapping~~'''~~

=== Lumatone mapping ===

See: [[Lumatone mapping for 16edo]]

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; [[Bryan Deister]]

* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)

* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)

; [[E8 Heterotic]]

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* [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/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)

; [[Herman Miller]]

@@ Line 11: / Line 11: @@
 == 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]]. A more accurate restriction is [[mabilic]], which discards the inaccurate mapping of 3 while keeping the fifth as a generator.
-=== Tuning theory ===
+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 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.
-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]].
+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]].
+edo 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.
+edo works as a tuning for [[extraclassical tonality]], due to its ultramajor third of 450 cents.
 === Odd harmonics ===
 {{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 ===
@@ Line 25: / Line 36: @@
 === Composition theory ===
 * [[User:VectorGraphics/16edo theory|Vector's approach]]
+* [[Armodue harmony]]
 {{Todo|inline=1| expand }}
 == Intervals ==
-edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.
+{{Mavila}}
-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".)
-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).
-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.
+Alternatively, one can use Armodue nine-nominal notation.
-{| class="wikitable"
-|+
-!
-!P1-M3-P5 ~ 4:5:6
-!P1-M3-P5 = C-E-G on C
-|-
-!Diatonic notation
-|NO
-|YES
-|-
-!Antidiatonic notation
-|YES
-|NO
-|}
-Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]
 {| class="wikitable center-all"
@@ Line 452: / Line 445: @@
 [[: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 ==
@@ Line 749: / Line 728: @@
 [[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]]
-'''Lumatone mapping'''
+=== Lumatone mapping ===
 See: [[Lumatone mapping for 16edo]]
@@ Line 857: / Line 836: @@
 ; [[Bryan Deister]]
 * [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)
+* [https://www.youtube.com/watch?v=cUgbkkIvy0g ''Waltz in 16edo''] (2025)
 ; [[E8 Heterotic]]
@@ Line 881: / Line 861: @@
 * [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/shorts/I4-URAGgQMQ ''Baroque Micropiece in 16edo''] (2024)
 ; [[Herman Miller]]