16edo: Difference between revisions

Line 6:

| ja = 16平均律

}}

'''16-EDO''' is the [[Equal_division_of_the_octave|equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent|cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7/4|7/4]] which is six cents sharp, and a [[5/4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.

'''16-EDO''' is the [[Equal_division_of_the_octave|equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent|cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7/4|7/4]] which is only six cents sharp, and a [[5/4|5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.

~~=Images=~~

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

=Intervals=

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the melodic meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. (But see below in "Chord Names".)

The second approach is to preserve the harmonic meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".

(Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic Notation|below]])

Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic Notation|below]]

{| class="wikitable"

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*based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

=Chord ~~names~~=

=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~~. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals.~~ (~~See [[Ups_and_Downs_Notation#Other EDOs|xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs]]~~).

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"

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| style="text-align:center;" | DM7

| style="text-align:center;" | D minor seven

|}

See [[Ups and Downs Notation#Chords and Chord Progressions|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"

|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

|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

|}

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{| class="wikitable"

|-

! | ~~Comma~~

! | [[Ratio]]

! | Monzo

! | [[Monzo]]

! | ~~Value (~~Cents)

! | [[Cent|Cents]]

![[Color notation/Temperament Names|Color Name]]

! | Name 1

! | Name 2

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| |<nowiki> | -7 3 1 </nowiki>>

| style="text-align:right;" | 92.18

| style="text-align:center;" |Layobi

| style="text-align:center;" | Major Chroma

| style="text-align:center;" | Major Limma

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| |<nowiki> | 3 4 -4 </nowiki>>

| style="text-align:right;" | 62.57

| style="text-align:center;" |Quadgu

| style="text-align:center;" | Major Diesis

| style="text-align:center;" | Diminished Comma

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| |<nowiki> | -10 -1 5 </nowiki>>

| style="text-align:right;" | 29.61

| style="text-align:center;" |Laquinyo

| style="text-align:center;" | Small Diesis

| style="text-align:center;" | Magic Comma

~~| style="text-align:center;" |~~

|-

~~| style="text-align:center;" |~~

~~| |<nowiki> | 23 6 -14 </nowiki>>~~

~~| style="text-align:right;" | 3.34~~

~~| style="text-align:center;" | Vishnuzma~~

~~| style="text-align:center;" | Semisuper~~

| style="text-align:center;" |

|-

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| |<nowiki> | 2 2 -1 -1 </nowiki>>

| style="text-align:right;" | 48.77

| style="text-align:center;" |Rugu

| style="text-align:center;" | Septimal Quarter Tone

| style="text-align:center;" |

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| |<nowiki> | -9 1 2 1 </nowiki>>

| style="text-align:right;" | 43.41

| style="text-align:center;" |Lazoyoyo

| style="text-align:center;" | Avicennma

| style="text-align:center;" | Avicenna's Enharmonic Diesis

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| |<nowiki> | 1 0 2 -2 </nowiki>>

| style="text-align:right;" | 34.98

| style="text-align:center;" |Biruyo

| style="text-align:center;" | Tritonic Diesis

| style="text-align:center;" | Jubilisma

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| |<nowiki> | -9 3 -3 4 </nowiki>>

| style="text-align:right;" | 22.23

| style="text-align:center;" |Laquadzo-atrigu

| style="text-align:center;" | Squalentine

| style="text-align:center;" |

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| |<nowiki> | 0 -2 5 -3 </nowiki>>

| style="text-align:right;" | 21.18

| style="text-align:center;" |Triru-aquinyo

| style="text-align:center;" | Gariboh

| style="text-align:center;" |

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| |<nowiki> | 1 2 -3 1 </nowiki>>

| style="text-align:right;" | 13.79

| style="text-align:center;" |Zotrigu

| style="text-align:center;" | Septimal Semicomma

| style="text-align:center;" | Starling Comma

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| |<nowiki> | -10 1 0 3 </nowiki>>

| style="text-align:right;" | 8.43

| style="text-align:center;" |Latrizo

| style="text-align:center;" | Gamelisma

| style="text-align:center;" |

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| |<nowiki> | 11 1 -3 -2 </nowiki>>

| style="text-align:right;" | 5.36

| style="text-align:center;" |Sarurutrigu

| style="text-align:center;" | Porwell

| style="text-align:center;" |

|-

| style="text-align:center;" |

| |<nowiki> | 23 6 -14 </nowiki>>

| style="text-align:right;" | 3.34

| style="text-align:center;" |Sasepbiru

| style="text-align:center;" | Vishnuzma

| style="text-align:center;" | Semisuper

| style="text-align:center;" |

|-

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| |<nowiki> | -3 -1 -1 0 2 </nowiki>>

| style="text-align:right;" | 14.37

| style="text-align:center;" |Lologu

| style="text-align:center;" | Biyatisma

| style="text-align:center;" |

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| |<nowiki> | 4 0 -2 -1 1 </nowiki>>

| style="text-align:right;" | 9.86

| style="text-align:center;" |Lorugugu

| style="text-align:center;" | Valinorsma

| style="text-align:center;" |

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| |<nowiki> | -7 -1 1 1 1 </nowiki>>

| style="text-align:right;" | 4.50

| style="text-align:center;" |Lozoyo

| style="text-align:center;" | Keenanisma

| style="text-align:center;" |

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| |<nowiki> | -3 2 -1 2 -1 </nowiki>>

| style="text-align:right;" | 3.93

| style="text-align:center;" |Luzozogu

| style="text-align:center;" | Werckisma

| style="text-align:center;" |

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| |<nowiki> | -4 -3 2 -1 2 </nowiki>>

| style="text-align:right;" | 0.57

| style="text-align:center;" |Loloruyoyo

| style="text-align:center;" | Lehmerisma

| style="text-align:center;" |

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|}

=~~'''~~Armodue Theory (4-line staff)~~'''~~=

=Armodue Theory (4-line staff)=

[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.

Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.

=Images=

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

=Books/Literature=

@@ Line 6: / Line 6: @@
 | ja = 16平均律
 }}
-'''16-EDO''' is the [[Equal_division_of_the_octave|equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent|cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7/4|7/4]] which is six cents sharp, and a [[5/4|5/4]] which is eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.
+'''16-EDO''' is the [[Equal_division_of_the_octave|equal division of the octave]] into sixteen narrow chromatic semitones each of 75 [[cent|cent]]s exactly. It is not especially good at representing most low-integer musical intervals, but it has a [[7/4|7/4]] which is only six cents sharp, and a [[5/4|5/4]] which is only eleven cents flat. Four steps of it gives the 300 cent minor third interval, the same of that 12-EDO, giving it four diminished seventh chords exactly like those of [[12edo|12-EDO]], and a diminished triad on each scale step.
-=Images=
-[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
 =Intervals=
-edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.
+edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the <u>melodic</u> meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5. (But see below in "Chord Names".)
 The second approach is to preserve the <u>harmonic</u> meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 16edo "on the fly".
-(Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic Notation|below]])
+Alternatively, one can use Armodue nine-nominal notation; see [[16edo#Hexadecaphonic Notation|below]]
 {| class="wikitable"
@@ Line 256: / Line 252: @@
 *based on treating 16-EDO as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.
-=Chord names=
+=Chord Names=
-edo 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. First change sharps to flats, then find every interval from the root, then exchange major for minor and aug for dim, then name the chord from the intervals. (See [[Ups_and_Downs_Notation#Other EDOs|xenharmonic.wikispaces.com/Ups+and+Downs+Notation#Other%20EDOs]]).
+edo 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"
@@ Line 338: / Line 334: @@
 | style="text-align:center;" | DM7
 | style="text-align:center;" | D minor seven
+|}
+See [[Ups and Downs Notation#Chords and Chord Progressions|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"
+|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
 |}
@@ Line 751: / Line 839: @@
 {| class="wikitable"
 |-
-! | Comma
+! | [[Ratio]]
-! | Monzo
+! | [[Monzo]]
-! | Value (Cents)
+! | [[Cent|Cents]]
+![[Color notation/Temperament Names|Color Name]]
 ! | Name 1
 ! | Name 2
@@ Line 761: / Line 850: @@
 | |<nowiki> | -7 3 1 </nowiki>&gt;
 | style="text-align:right;" | 92.18
+| style="text-align:center;" |Layobi
 | style="text-align:center;" | Major Chroma
 | style="text-align:center;" | Major Limma
@@ Line 768: / Line 858: @@
 | |<nowiki> | 3 4 -4 </nowiki>&gt;
 | style="text-align:right;" | 62.57
+| style="text-align:center;" |Quadgu
 | style="text-align:center;" | Major Diesis
 | style="text-align:center;" | Diminished Comma
@@ Line 775: / Line 866: @@
 | |<nowiki> | -10 -1 5 </nowiki>&gt;
 | style="text-align:right;" | 29.61
+| style="text-align:center;" |Laquinyo
 | style="text-align:center;" | Small Diesis
 | style="text-align:center;" | Magic Comma
-| style="text-align:center;" |
-|-
-| style="text-align:center;" |
-| |<nowiki> | 23 6 -14 </nowiki>&gt;
-| style="text-align:right;" | 3.34
-| style="text-align:center;" | Vishnuzma
-| style="text-align:center;" | Semisuper
 | style="text-align:center;" |
 |-
@@ Line 789: / Line 874: @@
 | |<nowiki> | 2 2 -1 -1 </nowiki>&gt;
 | style="text-align:right;" | 48.77
+| style="text-align:center;" |Rugu
 | style="text-align:center;" | Septimal Quarter Tone
 | style="text-align:center;" |
@@ Line 796: / Line 882: @@
 | |<nowiki> | -9 1 2 1 </nowiki>&gt;
 | style="text-align:right;" | 43.41
+| style="text-align:center;" |Lazoyoyo
 | style="text-align:center;" | Avicennma
 | style="text-align:center;" | Avicenna's Enharmonic Diesis
@@ Line 803: / Line 890: @@
 | |<nowiki> | 1 0 2 -2 </nowiki>&gt;
 | style="text-align:right;" | 34.98
+| style="text-align:center;" |Biruyo
 | style="text-align:center;" | Tritonic Diesis
 | style="text-align:center;" | Jubilisma
@@ Line 810: / Line 898: @@
 | |<nowiki> | -9 3 -3 4 </nowiki>&gt;
 | style="text-align:right;" | 22.23
+| style="text-align:center;" |Laquadzo-atrigu
 | style="text-align:center;" | Squalentine
 | style="text-align:center;" |
@@ Line 817: / Line 906: @@
 | |<nowiki> | 0 -2 5 -3 </nowiki>&gt;
 | style="text-align:right;" | 21.18
+| style="text-align:center;" |Triru-aquinyo
 | style="text-align:center;" | Gariboh
 | style="text-align:center;" |
@@ Line 824: / Line 914: @@
 | |<nowiki> | 1 2 -3 1 </nowiki>&gt;
 | style="text-align:right;" | 13.79
+| style="text-align:center;" |Zotrigu
 | style="text-align:center;" | Septimal Semicomma
 | style="text-align:center;" | Starling Comma
@@ Line 831: / Line 922: @@
 | |<nowiki> | -10 1 0 3 </nowiki>&gt;
 | style="text-align:right;" | 8.43
+| style="text-align:center;" |Latrizo
 | style="text-align:center;" | Gamelisma
 | style="text-align:center;" |
@@ Line 838: / Line 930: @@
 | |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
 | style="text-align:right;" | 5.36
+| style="text-align:center;" |Sarurutrigu
 | style="text-align:center;" | Porwell
 | style="text-align:center;" |
+| style="text-align:center;" |
+|-
+| style="text-align:center;" |
+| |<nowiki> | 23 6 -14 </nowiki>&gt;
+| style="text-align:right;" | 3.34
+| style="text-align:center;" |Sasepbiru
+| style="text-align:center;" | Vishnuzma
+| style="text-align:center;" | Semisuper
 | style="text-align:center;" |
 |-
@@ Line 845: / Line 946: @@
 | |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
 | style="text-align:right;" | 14.37
+| style="text-align:center;" |Lologu
 | style="text-align:center;" | Biyatisma
 | style="text-align:center;" |
@@ Line 852: / Line 954: @@
 | |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
 | style="text-align:right;" | 9.86
+| style="text-align:center;" |Lorugugu
 | style="text-align:center;" | Valinorsma
 | style="text-align:center;" |
@@ Line 859: / Line 962: @@
 | |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
 | style="text-align:right;" | 4.50
+| style="text-align:center;" |Lozoyo
 | style="text-align:center;" | Keenanisma
 | style="text-align:center;" |
@@ Line 866: / Line 970: @@
 | |<nowiki> | -3 2 -1 2 -1 </nowiki>&gt;
 | style="text-align:right;" | 3.93
+| style="text-align:center;" |Luzozogu
 | style="text-align:center;" | Werckisma
 | style="text-align:center;" |
@@ Line 873: / Line 978: @@
 | |<nowiki> | -4 -3 2 -1 2 </nowiki>&gt;
 | style="text-align:right;" | 0.57
+| style="text-align:center;" |Loloruyoyo
 | style="text-align:center;" | Lehmerisma
 | style="text-align:center;" |
@@ Line 878: / Line 984: @@
 |}
-='''Armodue Theory (4-line staff)'''=
+=Armodue Theory (4-line staff)=
 [http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his "Armodue" theory for 16edo (esadekaphonic), including compositions.
 Translations of parts of the Armodue pages can be found [[Armodue|here]] on this wiki.
+=Images=
+[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]
 =Books/Literature=