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<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
<nowiki>*</nowiki> based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
== Chord names ==
16edo chords can be named using ups and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated (see below).
{| class="wikitable center-all"
|-
! | chord
! | JI ratios
! colspan="3" | harmonic name
! colspan="3" | melodic name
|-
| 0-5-9
| 4:5:6
| D F A
| Dm
| D minor
| D F A
| D
| D major
|-
| 0-4-9
| 10:12:15
| D F# A
| D
| D major
| D Fb A
| Dm
| D minor
|-
| 0-4-8
| 5:6:7
| D F# A#
| Daug
| D augmented
| D Fb Ab
| Ddim
| D diminished
|-
| 0-5-10
|
| D F Ab
| Ddim
| D diminished
| D F A#
| Daug
| D augmented
|-
| 0-5-9-13
| 4:5:6:7
| D F A C#
| Dm(M7)
| D minor-major
| D F A Cb
| D7
| D seven
|-
| 0-5-9-12
|
| D F A Bb
| Dm(b6)
| D minor flat-six
| D F A B#
| D6
| D six
|-
| 0-5-9-14
|
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
| 0-4-9-13
|
| D F# A C#
| DM7
| D major seven
| D Fb A Cb
| DM7
| D minor seven
|}
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). See [[Ups and Downs Notation #Chords and Chord Progressions]] for more examples.
Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:
{| class="wikitable" style="text-align:center;"
!initial question
!reverse everything
!do the math
!reverse again
|-
|M2 + M2
|m2 + m2
|dim3
|aug3
|-
|D to F#
|D to Fb
|dim3
|aug3
|-
|D to F
|D to F
|m3
|M3
|-
|Eb + m3
|E# + M3
|G##
|Gbb
|-
|Eb + P5
|E# + P5
|B#
|Bb
|-
|A minor chord
|A major chord
|A C# E
|A Cb E
|-
|Eb major chord
|E# minor chord
|E# G# B#
|Eb Gb Db
|-
|Gm7 = G + m3 + P5 + m7
|G + M3 + P5 + M7
|G B D F#
|G B D Fb
|-
|Ab7aug = Ab + M3 + A5 + m7
|A# + m3 + d5 + M7
|A# C# E G##
|Ab Cb E Gbb
|-
|what chord is D F A#?
|D F Ab
|D + m3 + d5
|D + M3 + A5 = Daug
|-
|what chord is C E Gb Bb?
|C E G# B#
|C + M3 + A5 + A7
|C + m3 + d5 + d7 = Cdim7
|-
|C major scale = C + M2 + M3
+ P4 + P5 + M6 + M7 + P8
|C + m2 + m3 + P4
+ P5 + m6 + m7 + P8
|C Db Eb F
G Ab Bb C
|C D# E# F
G A# B# C
|-
|C minor scale = C + M2 + m3
+ P4 + P5 + m6 + m7 + P8
|C + m2 + M3 + P4
+ P5 + M6 + M7 + P8
|C Db E F
G A B C
|C D# E F
G A B C
|-
|what scale is A B# Cb D
E F Gb A?
|A Bb C# D
E F G# A
|A + m2 + M3 + P4
+ P5 + m6 + M7
|A + M2 + m3 + P4
+ P5 + M6 + m7 = A dorian
|}


== JI approximation ==
== JI approximation ==
Line 467: Line 292:


[[:File:16ed2-001.svg|16ed2-001.svg]]
[[:File:16ed2-001.svg|16ed2-001.svg]]
== Octave Theory ==
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.
16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).
16edo is also a tuning for the no-threes 7-limit temperament tempering out [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288], which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "'''Magic family of scales'''".
[[Easley Blackwood Jr]] writes of 16edo:
"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"
From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174 cents, and take the 300-cent minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5 cents), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19.


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


Line 587: Line 395:
For translations of parts of the Armodue pages see the [[Armodue]] on this wiki
For translations of parts of the Armodue pages see the [[Armodue]] on this wiki


== Rank-2 temperaments ==
=== Chord names ===
*[[List of 16et rank two temperaments by badness]]
16edo chords can be named using ups and downs. Using harmonic interval names, the names are easy to find, but they bear little relationship to the sound. 4:5:6 is a minor chord and 10:12:15 is a major chord! Using melodic names, the chord names will match the sound, but finding the name is much more complicated (see below).


Important mosses include:
{| class="wikitable center-all"
*[[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
*[[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* Pathological [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)
*[[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)
*[[mavila]] superdiatonic 7L2s 222212221 (9\16, 1\1)
*[[gorgo]] 5L1s 333331 (3\16, 1\1)
*[[lemba]] 4L2s 332332 (3\16, 1\2)
* Pathological [[1L 12s]] 4 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[1L 13s]] 3 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[2L 12s]] 2 1 1 1 1 1 1 2 1 1 1 1 1 1 (1\16, 1\2)
* Pathological [[1L 14s]] 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
 
Temperaments listed by generator size:
 
{| class="wikitable center-1 center-2"
|-
|-
! Periods<br>per octave
! | chord
! Generator
! | JI ratios
! Temperaments
! colspan="3" | harmonic name
! colspan="3" | melodic name
|-
|-
| 1
| 0-5-9
| 1\16
| 4:5:6
|[[Valentine]], [[slurpee]]
| D F A
| Dm
| D minor
| D F A
| D
| D major
|-
|-
| 1
| 0-4-9
| 3\16
| 10:12:15
|[[Gorgo]]
| D F# A
| D
| D major
| D Fb A
| Dm
| D minor
|-
|-
| 1
| 0-4-8
| 5\16
| 5:6:7
|[[magic]]/muggles
| D F# A#
| Daug
| D augmented
| D Fb Ab
| Ddim
| D diminished
|-
|-
| 1
| 0-5-10
| 7\16
|  
|[[Mavila]]/armodue
| D F Ab
| Ddim
| D diminished
| D F A#
| Daug
| D augmented
|-
|-
| 2
| 0-5-9-13
| 1\16
| 4:5:6:7
|[[Bipelog]]
| D F A C#
| Dm(M7)
| D minor-major
| D F A Cb
| D7
| D seven
|-
|-
| 2
| 0-5-9-12
| 3\16
|  
|[[Lemba]], [[astrology]]
| D F A Bb
| Dm(b6)
| D minor flat-six
| D F A B#
| D6
| D six
|-
|-
| 4
| 0-5-9-14
| 1\16
|  
|[[Diminished]]/demolished
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
|-
| 8
| 0-4-9-13
| 1\16
|  
|  
| D F# A C#
| DM7
| D major seven
| D Fb A Cb
| DM7
| D minor seven
|}
|}
Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). See [[Ups and Downs Notation #Chords and Chord Progressions]] for more examples.


'''Mavila'''
Using melodic names, interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:


{| class="wikitable"
{| class="wikitable" style="text-align:center;"
!initial question
!reverse everything
!do the math
!reverse again
|-
|M2 + M2
|m2 + m2
|dim3
|aug3
|-
|D to F#
|D to Fb
|dim3
|aug3
|-
|D to F
|D to F
|m3
|M3
|-
|Eb + m3
|E# + M3
|G##
|Gbb
|-
|Eb + P5
|E# + P5
|B#
|Bb
|-
|A minor chord
|A major chord
|A C# E
|A Cb E
|-
|Eb major chord
|E# minor chord
|E# G# B#
|Eb Gb Db
|-
|Gm7 = G + m3 + P5 + m7
|G + M3 + P5 + M7
|G B D F#
|G B D Fb
|-
|Ab7aug = Ab + M3 + A5 + m7
|A# + m3 + d5 + M7
|A# C# E G##
|Ab Cb E Gbb
|-
|-
| [5]:
|what chord is D F A#?
| 5 2 5 2 2
|D F Ab
|  
|D + m3 + d5
|D + M3 + A5 = Daug
|-
|-
| [7]:
|what chord is C E Gb Bb?
| 3 2 2 3 2 2 2
|C E G# B#
|[[File:MavilaAntidiatonic16edo.mp3]]
|C + M3 + A5 + A7
|C + m3 + d5 + d7 = Cdim7
|-
|-
| [9]:
|C major scale = C + M2 + M3
| 1 2 2 2 1 2 2 2 2
+ P4 + P5 + M6 + M7 + P8
|[[File:MavilaSuperdiatonic16edo.mp3]]
|C + m2 + m3 + P4
|}
+ P5 + m6 + m7 + P8
See also [[Mavila Temperament Modal Harmony]].
|C Db Eb F
 
G Ab Bb C
'''Diminished'''
|C D# E# F
 
G A# B# C
{| class="wikitable"
|-
|-
| [8]:
|C minor scale = C + M2 + m3
| 1 3 1 3 1 3 1 3
+ P4 + P5 + m6 + m7 + P8
|[[File:htgt16edo.mp3]]
|C + m2 + M3 + P4
+ P5 + M6 + M7 + P8
|C Db E F
G A B C
|C D# E F
G A B C
|-
|-
| [12]:
|what scale is A B# Cb D
| 1 1 2 1 1 2 1 1 2 1 1 2
E F Gb A?
|  
|A Bb C# D
E F G# A
|A + m2 + M3 + P4
+ P5 + m6 + M7
|A + M2 + m3 + P4
+ P5 + M6 + m7 = A dorian
|}
|}


'''Magic'''
== Octave Theory ==


[7]: 1 4 1 4 1 4 1
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75 cents, is smaller than ideal. Its very flat 3/2 of 675 cents [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150-cent "3/4-tone" equal division of the traditional 300-cent minor third.


[10]: 1 3 1 1 3 1 1 1 3 1
16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600¢), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).


[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
16edo is also a tuning for the no-threes 7-limit temperament tempering out [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288], which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "'''Magic family of scales'''".


'''Cynder/Gorgo'''
[[Easley Blackwood Jr]] writes of 16edo:


[5]: 3 3 4 3 3
"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"


[6]: 3 3 1 3 3 3
From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174 cents, and take the 300-cent minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5 cents), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .
 
[11]: 1 2 1 2 1 2 1 2 1 2 1
 
'''Lemba/Astrology'''
 
[4]: 3 5 3 5
 
[6]: 3 2 3 3 2 3
 
[10]: 2 1 2 1 2 2 1 2 1 2
 
== Metallic Harmony ==
 
Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not a great option for typical harmonic timbres.
 
However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use


it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant.
The interval between the 28th &amp; 19th harmonics, 28:19, measures approximately 671.3 cents, which is 3.7 cents away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7 cents just, 525.0 cents in 16edo). A perhaps more consonant open voicing is 7:16:19
 
Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".
 
=== MOS scales supporting Metallic Harmony in 16edo ===
The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.
 
Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.
 
See [[Metallic Harmony]].


== Commas ==
== Commas ==
Line 848: Line 718:
|}
|}
<references />
<references />
== Rank-2 temperaments ==
*[[List of 16et rank two temperaments by badness]]
Important mosses include:
*[[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
*[[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* Pathological [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)
*[[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)
*[[mavila]] superdiatonic 7L2s 222212221 (9\16, 1\1)
*[[gorgo]] 5L1s 333331 (3\16, 1\1)
*[[lemba]] 4L2s 332332 (3\16, 1\2)
* Pathological [[1L 12s]] 4 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[1L 13s]] 3 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
* Pathological [[2L 12s]] 2 1 1 1 1 1 1 2 1 1 1 1 1 1 (1\16, 1\2)
* Pathological [[1L 14s]] 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1\16, 1\1)
Temperaments listed by generator size:
{| class="wikitable center-1 center-2"
|-
! Periods<br>per octave
! Generator
! Temperaments
|-
| 1
| 1\16
|[[Valentine]], [[slurpee]]
|-
| 1
| 3\16
|[[Gorgo]]
|-
| 1
| 5\16
|[[magic]]/muggles
|-
| 1
| 7\16
|[[Mavila]]/armodue
|-
| 2
| 1\16
|[[Bipelog]]
|-
| 2
| 3\16
|[[Lemba]], [[astrology]]
|-
| 4
| 1\16
|[[Diminished]]/demolished
|-
| 8
| 1\16
|
|}
'''Mavila'''
{| class="wikitable"
|-
| [5]:
| 5 2 5 2 2
|
|-
| [7]:
| 3 2 2 3 2 2 2
|[[File:MavilaAntidiatonic16edo.mp3]]
|-
| [9]:
| 1 2 2 2 1 2 2 2 2
|[[File:MavilaSuperdiatonic16edo.mp3]]
|}
See also [[Mavila Temperament Modal Harmony]].
'''Diminished'''
{| class="wikitable"
|-
| [8]:
| 1 3 1 3 1 3 1 3
|[[File:htgt16edo.mp3]]
|-
| [12]:
| 1 1 2 1 1 2 1 1 2 1 1 2
|
|}
'''Magic'''
[7]: 1 4 1 4 1 4 1
[10]: 1 3 1 1 3 1 1 1 3 1
[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1
'''Cynder/Gorgo'''
[5]: 3 3 4 3 3
[6]: 3 3 1 3 3 3
[11]: 1 2 1 2 1 2 1 2 1 2 1
'''Lemba/Astrology'''
[4]: 3 5 3 5
[6]: 3 2 3 3 2 3
[10]: 2 1 2 1 2 2 1 2 1 2
== Metallic harmony ==
Because 16edo does not approximate 3/2 well at all, triadic harmony based on heptatonic thirds is not a great option for typical harmonic timbres.
However, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use
it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050 cents). Stacking these two intervals reaches 2025¢, or a minor 6th plus an octave. Thus the out-of-tune 675¢ interval is bypassed, and all the dyads in the triad are consonant.
Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, 0-975-2025¢, and a large one, 0-1050-2025¢. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at 0-975-1950¢, and a wide symmetrical triad at 0-1050-2100¢. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".
=== MOS scales supporting metallic harmony in 16edo ===
The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025¢. In Mavila[9], hard and soft triads cease to share a triad class, as 975¢ is a major 8th, while 1050¢ is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.
Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.
''See: [[Metallic Harmony]].''


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