Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 593579316 - Original comment: ** |
Wikispaces>TallKite **Imported revision 593581404 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-29 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-09-29 04:54:28 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>593581404</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=__"Ups and Downs" Notation for 22-EDO__= | =__"Ups and Downs" Notation for 22-EDO__= | ||
Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs | Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs. | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes. | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes. | ||
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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | ||
"fifth-less" EDOs, with fifths wider than 720¢ | "fifth-less" EDOs, with fifths wider than 720¢ | ||
pentatonic EDOs, with a fifth = 720¢ | "pentatonic" EDOs, with a fifth = 720¢ | ||
"regular" EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢ | "regular" EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢ | ||
"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | "perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | ||
fourthwards | "fourthwards" EDOs, with a fifth less than 686¢ | ||
This is in addition to the trivial EDOs | This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | ||
[[image:The Scale Tree.png width="800" height="1002"]] | [[image:The Scale Tree.png width="800" height="1002"]] | ||
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Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later. | Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a "natural" (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later. | ||
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | ||
The mid symbol "~" means "exactly midway between major and minor", hence neutral. In other words, mid is a quality like major or perfect. This only applies to certain "neutral EDOs" in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and "mid" replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces upmajor and downminor. | |||
**__17-EDO__:** (2 keys per sharp/flat) | **__17-EDO__:** (2 keys per sharp/flat) | ||
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JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber. | JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber. | ||
24-EDO is an example of a | 24-EDO is an example of a multi-ring EDO. An EDO is multi-ring if the keyspan of the fifth (generator) isn't coprime with the keyspan of the octave, and 1-ring if it is. 24-EDO has a fifth of 14 steps, and is 2-ring because its circle of fifths has only 12 notes. There are 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one: | ||
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# | Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#/Ab - Eb | ||
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^ | Eb^ - Bb^ - F^ - C^ - G^ - D^ - A^ - E^ - B^ - F#^ - C#^ - G#^/Ab^ - Eb^ | ||
Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes. | Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes. | ||
In | In 1-ring EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we could use B#. But multi-ring EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs: | ||
C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc. | C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc. | ||
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JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber. | JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber. | ||
=__Naming Chords__= | =__Naming Chords__= | ||
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Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. | Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc. | ||
Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22! | |||
There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals. | |||
The chain of fifths in "perfect" EDOs (3/2 maps to 4\7): | The chain of fifths in "perfect" EDOs (3/2 maps to 4\7): | ||
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21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because B - F# | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# looks like a perfect fifth. | ||
The | The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth. | ||
The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning. | |||
In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G. | |||
In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards: | |||
M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc. | |||
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc. | |||
16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8 | |||
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C | |||
Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. | |||
M2 + M2 --> m2 + m2 = dim3 --> aug3 | |||
D to F# --> D to Fb = dim3 --> aug3 | |||
Eb + m3 --> E# + M3 = G## --> Gbb | |||
Both approaches have their merit, but the first one will be used from here on. | |||
=__22edo chord names__= | =__22edo chord names__= | ||
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=__**Summary of EDO notation**__= | =__**Summary of EDO notation**__= | ||
===__"Regular" EDOs__=== | |||
===__"Regular" | (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher) | ||
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | ||
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P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc. | P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc. | ||
===__"Perfect" EDOs__=== | |||
(7, 14, 21, 28 and 35) | |||
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc. | All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc. | ||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc. | F - C - G - D - A - E - B - F - C - G - D - A - E - B etc. | ||
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1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc. | 1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc. | ||
===__Fourthward EDOs__=== | |||
(9, 11, 13b, 16, 18b and 23) | |||
All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | ||
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D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D | D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D | ||
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8 | P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8 | ||
problematic because M3 = 2\11 is narrower than m2 = 3\11 | |||
**__13b-edo__:** C D * Cb Db, # = vvv | **__13b-edo__:** C D * Cb Db, # = vvv | ||
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P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8 | P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8 | ||
**__18b-edo__:** C/D# * Cb/D | **__18b-edo__:** # = vv, C/D# * Cb/D | ||
D * E * * * F * G * A * B * * * C * D | D * E * * * F * G * A * B * * * C * D | ||
D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D | D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D | ||
P1 - ^P1/vM2 - | P1 - ^P1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m2/d8 - P8 | ||
Mid "~" is midway between major and minor, which equates it to upmajor and downminor. | |||
**__23edo__:** C Cb * D# D, # = v | **__23edo__:** C Cb * D# D, # = v | ||
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===__Pentatonic | ===__Pentatonic EDOs__=== | ||
(5, 10, 15, 20, 25 and 30) | |||
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc. | ||
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P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8 | P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8 | ||
===__**"Fifth-less" EDOs (8, 11b, 13 and 18) | ===__**"Fifth-less" EDOs**__=== | ||
(8, 11b, 13 and 18) | |||
**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢) | **__8edo__:** (generator = 1\8 = perfect 2nd = 150¢) | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | ||
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc. | ==__Pentatonic notation__== | ||
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc. | |||
**__Alternative pentatonic notation for pentatonic EDOs__** | |||
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc. | |||
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc. | |||
All intervals are perfect, so quality can be omitted. | |||
__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D | |||
D E G A C D | |||
1 - s3 - 4d - 5d - s7 - 8d | |||
__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D | |||
D * E * G * A * C * D | |||
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D | |||
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d | |||
__**15edo**__**:** zero keys per sharp/flat: C/C# * * Db/D | |||
D * * E * * G * * A * * C * * D | |||
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D | |||
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d | |||
etc. | |||
__**Alternate pentatonic notation for EDOs 8, 13 and 18**__ | |||
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc. | |||
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc. | |||
__**8edo**__**:** (generator = 5\8 = perfect 5thoid) C C#/Db D | __**8edo**__**:** (generator = 5\8 = perfect 5thoid) C C#/Db D | ||
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P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9 | P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9 | ||
requires learning octatonic interval arithmetic and staff notation | requires learning octatonic interval arithmetic and staff notation | ||
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th): | 11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th): | ||
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P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9 | P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9 | ||
requires learning octatonic interval arithmetic and notation | requires learning octatonic interval arithmetic and notation | ||
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th) | 18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th) | ||
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__**Alternate notation for other edos:**__ | __**Alternate notation for other edos:**__ | ||
23b-edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid: | |||
D * * * * E * * * G * * * * A * * * C * * * * D | D * * * * E * * * G * * * * A * * * C * * * * D | ||
35b-edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth: | |||
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D | D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D | ||
42b-edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth: | |||
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D | D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D | ||
| Line 1,341: | Line 1,338: | ||
||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule:46:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#x&quot;Ups and Downs&quot; Notation for 22-EDO">&quot;Ups and Downs&quot; Notation for 22-EDO</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 1em;"><a href="#Other EDOs">Other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 1em;"><a href="#Naming Chords">Naming Chords</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 1em;"><a href="#x22edo chord names">22edo chord names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 2em;"><a href="#toc4"></a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 2em;"><a href="#toc5"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 2em;"><a href="#toc6"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="#Chord names in other EDOs">Chord names in other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 4em;"><a href="#Chord names in other EDOs---Example EDOs:">Example EDOs:</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 1em;"><a href="#Cross-EDO considerations">Cross-EDO considerations</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 1em;"><a href="#Scale Fragments">Scale Fragments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 1em;"><a href="#Summary of EDO notation">Summary of EDO notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Regular&quot; EDOs">&quot;Regular&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Perfect&quot; EDOs">&quot;Perfect&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 3em;"><a href="#Summary of EDO notation--Fourthward EDOs">Fourthward EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 3em;"><a href="#Summary of EDO notation--Pentatonic EDOs">Pentatonic EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 3em;"><a href="#Summary of EDO notation--&quot;Fifth-less&quot; EDOs">&quot;Fifth-less&quot; EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Pentatonic notation">Pentatonic notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 1em;"><a href="#EDOs with an inaccurate 3/2">EDOs with an inaccurate 3/2</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 1em;"><a href="#Rank-2 Notation">Rank-2 Notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#toc21"> </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 1em;"><a href="#Generators other than a fifth">Generators other than a fifth</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --></div> | ||
<!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation for 22-EDO"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>&quot;Ups and Downs&quot; Notation for 22-EDO</u></h1> | |||
<br /> | <br /> | ||
Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs | Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs.<br /> | ||
<br /> | <br /> | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.<br /> | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.<br /> | ||
| Line 1,450: | Line 1,446: | ||
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | ||
&quot;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | &quot;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | ||
pentatonic EDOs, with a fifth = 720¢<br /> | &quot;pentatonic&quot; EDOs, with a fifth = 720¢<br /> | ||
&quot;regular&quot; EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | &quot;regular&quot; EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | ||
&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | &quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | ||
fourthwards | &quot;fourthwards&quot; EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
This is in addition to the trivial EDOs | This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4119:&lt;img src=&quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1002px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4119 --><br /> | ||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call <strong>kites</strong>, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
<br /> | <br /> | ||
| Line 1,465: | Line 1,461: | ||
<br /> | <br /> | ||
Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.<br /> | Every EDO larger than 7edo will appear on only one of these three mirror-pairs of kites. The only exception is perfect EDOs, which appear on the spine of every heptatonic kite. This means that every non-perfect EDO above 7edo has a &quot;natural&quot; (not requiring ups and downs) notation, generated by either the 2nd, the 3rd, or the 5th. For now we'll assume that the fifth is the notation's generator. More on alternate generators later.<br /> | ||
<br /> | <br /> | ||
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | ||
<br /> | <br /> | ||
The mid symbol &quot;~&quot; means &quot;exactly midway between major and minor&quot;, hence neutral. In other words, mid is a quality like major or perfect. This only applies to certain &quot;neutral EDOs&quot; in which the sharp equals an even number of EDOsteps. For example, in every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.), a sharp is two EDOsteps, upminor equals downmajor, and &quot;mid&quot; replaces both terms. In 20edo, 27edo, 34edo, 41edo, etc., a sharp is four EDOsteps, and mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces upmajor and downminor.<br /> | |||
<br /> | <br /> | ||
<strong><u>17-EDO</u>:</strong> (2 keys per sharp/flat)<br /> | <strong><u>17-EDO</u>:</strong> (2 keys per sharp/flat)<br /> | ||
| Line 1,488: | Line 1,482: | ||
JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br /> | JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.<br /> | ||
<br /> | <br /> | ||
24-EDO is an example of a | 24-EDO is an example of a multi-ring EDO. An EDO is multi-ring if the keyspan of the fifth (generator) isn't coprime with the keyspan of the octave, and 1-ring if it is. 24-EDO has a fifth of 14 steps, and is 2-ring because its circle of fifths has only 12 notes. There are 2 unconnected circles of fifths in 24-EDO, which are notated as the mid one and the up one:<br /> | ||
Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#<br /> | Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#/Ab - Eb<br /> | ||
Eb^-Bb^-F^-C^-G^-D^-A^-E^-B^-F#^-C#^-G#^<br /> | Eb^ - Bb^ - F^ - C^ - G^ - D^ - A^ - E^ - B^ - F#^ - C#^ - G#^/Ab^ - Eb^<br /> | ||
Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes.<br /> | Just as G# could alternatively be written as Ab, all the up notes could alternatively be written as down notes.<br /> | ||
<br /> | <br /> | ||
In | In 1-ring EDOs, we can require that the tonic be a mid note. For example in 22-EDO, rather than using C#v as a tonic, we could use B#. But multi-ring EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:<br /> | ||
C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.<br /> | C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.<br /> | ||
<br /> | <br /> | ||
| Line 1,510: | Line 1,504: | ||
JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber.<br /> | JI associations: Perfect = white, major = fifthward white, minor = fourthward white, downmajor = yellow, upminor = green, downminor = blue, upmajor = red, double-downmajor = double-upminor = jade or amber.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Naming Chords</u></h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Naming Chords</u></h1> | ||
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Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | ||
<br /> | <br /> | ||
Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!<br /> | |||
<br /> | <br /> | ||
The | There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.<br /> | ||
<br /> | <br /> | ||
The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br /> | The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br /> | ||
| Line 1,539: | Line 1,520: | ||
21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because B - F# | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# looks like a perfect fifth.<br /> | ||
<br /> | |||
The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.<br /> | |||
<br /> | <br /> | ||
The | The third special case is when the edo's 5th is narrower than 4\7, as in 16edo. There are two approaches. One preserves the harmonic (chain-of-fifths) meaning of sharp/flat, major/minor and aug/dim, and the other preserves the melodic meaning.<br /> | ||
<br /> | <br /> | ||
In the first approach, major is still fifthwards, which makes it narrower than minor. Aug is narrower than dim. This makes interval arithmetic and chord names unaffected. M2 + M2 is still M3, and a C minor chord is still C Eb G.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x22edo chord names"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>22edo chord names</u></h1> | In the 2nd approach, major is still wider than minor, so major is not fifthwards but fourthwards. The chain of fifths runs backwards:<br /> | ||
<br /> | M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 - A1 etc.<br /> | ||
Let's review the 22edo interval names:<br /> | F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.<br /> | ||
0\22 = P1<br /> | 16edo: P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br /> | ||
16edo: C - C#/Db - D - D#/Eb - E - E# - Fb - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C<br /> | |||
<br /> | |||
Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again.<br /> | |||
M2 + M2 --&gt; m2 + m2 = dim3 --&gt; aug3<br /> | |||
D to F# --&gt; D to Fb = dim3 --&gt; aug3<br /> | |||
Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br /> | |||
<br /> | |||
Both approaches have their merit, but the first one will be used from here on.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x22edo chord names"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>22edo chord names</u></h1> | |||
<br /> | |||
Let's review the 22edo interval names:<br /> | |||
0\22 = P1<br /> | |||
1\22 = m2<br /> | 1\22 = m2<br /> | ||
2\22 = ^m2<br /> | 2\22 = ^m2<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:22 --><u><strong>Summary of EDO notation</strong></u></h1> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:22 --><u><strong>Summary of EDO notation</strong></u></h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Summary of EDO notation--&quot;Regular&quot; EDOs"></a><!-- ws:end:WikiTextHeadingRule:24 --><u>&quot;Regular&quot; EDOs</u></h3> | |||
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Summary of EDO notation--&quot;Regular&quot; EDOs | (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)<br /> | ||
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | ||
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P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.<br /> | P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs | <br /> | ||
<br /> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs"></a><!-- ws:end:WikiTextHeadingRule:26 --><u>&quot;Perfect&quot; EDOs</u></h3> | ||
(7, 14, 21, 28 and 35)<br /> | |||
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.<br /> | All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.<br /> | ||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br /> | F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br /> | ||
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1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.<br /> | 1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Summary of EDO notation--Fourthward EDOs | <br /> | ||
<br /> | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Summary of EDO notation--Fourthward EDOs"></a><!-- ws:end:WikiTextHeadingRule:28 --><u>Fourthward EDOs</u></h3> | ||
(9, 11, 13b, 16, 18b and 23)<br /> | |||
All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | ||
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D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D<br /> | D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D<br /> | ||
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8<br /> | P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8<br /> | ||
problematic because M3 = 2\11 is narrower than m2 = 3\11<br /> | |||
<br /> | <br /> | ||
<strong><u>13b-edo</u>:</strong> C D * Cb Db, # = vvv<br /> | <strong><u>13b-edo</u>:</strong> C D * Cb Db, # = vvv<br /> | ||
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P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8<br /> | P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8<br /> | ||
<br /> | <br /> | ||
<strong><u>18b-edo</u>:</strong> C/D# * Cb/D | <strong><u>18b-edo</u>:</strong> # = vv, C/D# * Cb/D<br /> | ||
D * E * * * F * G * A * B * * * C * D<br /> | D * E * * * F * G * A * B * * * C * D<br /> | ||
D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D<br /> | D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D<br /> | ||
P1 - ^P1/vM2 - | P1 - ^P1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m2/d8 - P8<br /> | ||
Mid &quot;~&quot; is midway between major and minor, which equates it to upmajor and downminor.<br /> | |||
<br /> | <br /> | ||
<strong><u>23edo</u>:</strong> C Cb * D# D, # = v<br /> | <strong><u>23edo</u>:</strong> C Cb * D# D, # = v<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Summary of EDO notation--Pentatonic EDOs | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Summary of EDO notation--Pentatonic EDOs"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Pentatonic EDOs</u></h3> | ||
<br /> | (5, 10, 15, 20, 25 and 30)<br /> | ||
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.<br /> | ||
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P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br /> | P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
< | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs"></a><!-- ws:end:WikiTextHeadingRule:32 --><u><strong>&quot;Fifth-less&quot; EDOs</strong></u></h3> | ||
(8, 11b, 13 and 18)<br /> | |||
<br /> | <br /> | ||
<strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br /> | <strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br /> | ||
D E F G * A B C D<br /> | D E F G * A B C D<br /> | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | ||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Summary of EDO notation-Pentatonic notation"></a><!-- ws:end:WikiTextHeadingRule:34 --><u>Pentatonic notation</u></h2> | |||
<br /> | <br /> | ||
<strong><u>Alternative pentatonic notation for pentatonic EDOs</u></strong><br /> | |||
<br /> | |||
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.<br /> | |||
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.<br /> | |||
All intervals are perfect, so quality can be omitted.<br /> | |||
<br /> | |||
<u><strong>5edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# Db/D<br /> | |||
D E G A C D<br /> | |||
1 - s3 - 4d - 5d - s7 - 8d<br /> | |||
<br /> | |||
<u><strong>10edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# * Db/D<br /> | |||
D * E * G * A * C * D<br /> | |||
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D<br /> | |||
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d<br /> | |||
<br /> | |||
<u><strong>15edo</strong></u><strong>:</strong> zero keys per sharp/flat: C/C# * * Db/D<br /> | |||
D * * E * * G * * A * * C * * D<br /> | |||
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D<br /> | |||
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d<br /> | |||
etc.<br /> | |||
<br /> | |||
<u><strong>Alternate pentatonic notation for EDOs 8, 13 and 18</strong></u><br /> | |||
<br /> | |||
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.<br /> | All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.<br /> | ||
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.<br /> | Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h1&gt; --><h1 id="toc18"><a name="EDOs with an inaccurate 3/2"></a><!-- ws:end:WikiTextHeadingRule:36 --><u>EDOs with an inaccurate 3/2</u></h1> | ||
<br /> | <br /> | ||
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | ||
| Line 3,617: | Line 3,621: | ||
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9<br /> | P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9<br /> | ||
requires learning octatonic interval arithmetic and staff notation<br /> | requires learning octatonic interval arithmetic and staff notation<br /> | ||
<br /> | <br /> | ||
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):<br /> | 11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):<br /> | ||
| Line 3,656: | Line 3,654: | ||
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9<br /> | P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9<br /> | ||
requires learning octatonic interval arithmetic and notation<br /> | requires learning octatonic interval arithmetic and notation<br /> | ||
<br /> | <br /> | ||
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)<br /> | 18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)<br /> | ||
| Line 3,668: | Line 3,661: | ||
<br /> | <br /> | ||
<u><strong>Alternate notation for other edos:</strong></u><br /> | <u><strong>Alternate notation for other edos:</strong></u><br /> | ||
23b-edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:<br /> | |||
D * * * * E * * * G * * * * A * * * C * * * * D<br /> | D * * * * E * * * G * * * * A * * * C * * * * D<br /> | ||
35b-edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:<br /> | |||
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D<br /> | D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D<br /> | ||
42b-edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:<br /> | |||
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D<br /> | D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | ||
| Line 3,706: | Line 3,699: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc20"><a name="Rank-2 Notation"></a><!-- ws:end:WikiTextHeadingRule:40 --><u>Rank-2 Notation</u></h1> | ||
<br /> | <br /> | ||
Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | ||
| Line 5,896: | Line 5,889: | ||
Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &quot;height&quot;; one is up, another is down, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods">xenharmonic.wikispaces.com/Naming+Rank-2+Scales</a><br /> | Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &quot;height&quot;; one is up, another is down, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods">xenharmonic.wikispaces.com/Naming+Rank-2+Scales</a><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><!-- ws:end:WikiTextHeadingRule:42 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Generators other than a fifth"></a><!-- ws:end:WikiTextHeadingRule:44 --><u>Generators other than a fifth</u></h1> | ||
<br /> | <br /> | ||
The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | ||