Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 558118067 - Original comment: ** |
Wikispaces>TallKite **Imported revision 558195589 - 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>2015-09-03 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2015-09-03 15:33:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>558195589</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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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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 adds up to one EDO-step. So C# is right next to C, and your 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 adds up to one EDO-step. So C# is right next to C, and your 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. | ||
In contrast, 22-EDO is hard to notate because 7 fifths are __three__ EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing | In contrast, 22-EDO is hard to notate because 7 fifths are __three__ EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key! | ||
The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. | The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. | ||
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C: no sharps | C: no sharps | ||
G: 1 sharp | G: 1 sharp on F | ||
D: 2 sharps | D: 2 sharps on F and C | ||
... | ... | ||
C#: 7 sharps | C#: 7 sharps | ||
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pentatonic EDOs, with a fifth = 720¢ | pentatonic EDOs, with a fifth = 720¢ | ||
"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢ | "sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢ | ||
"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | |||
superflat EDOs or Mavila EDOs, with a fifth less than 686¢ | superflat EDOs or Mavila EDOs, with a fifth less than 686¢ | ||
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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. | ||
**__53-EDO__:** (5 keys per sharp/flat) | |||
Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C | |||
=__Naming Chords__= | =__Naming Chords__= | ||
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The second 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. That's because 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. | The second 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. That's because 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. | ||
The chain of fifths in | The chain of fifths in "perfect" EDOs (3/2 maps to 4\7): | ||
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc. | P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 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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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. | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. 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 they are two different concepts that must be kept distinct. In this redefined notation, B - F# isn't a perfect fifth because it's really B - F^. | ||
The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. 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. Such EDOs are dealt with below. | The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. 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. Such EDOs are dealt with below. | ||
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keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D | keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D | ||
(the chain of fifths is always centered on D) | (the chain of fifths is always centered on D) | ||
chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8 | chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8 | ||
chord roots: I ^bII vII II/bIII ^bIII vIII IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII | chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII | ||
0-3-9 = m or sus2 | 0-3-9 = m or sus2 | ||
0-4-9 = ^m | 0-4-9 = ^m | ||
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"Fifth-less" EDOs (8, 11, 13 and 18) | "Fifth-less" EDOs (8, 11, 13 and 18) | ||
Fourthward EDOs (9, 16 and 23) | Fourthward EDOs (9, 16 and 23) | ||
"Perfect" EDOs (7, 14, 21, 28 and 35) | |||
Pentatonic EDOs (5, 10, 15, 20, 25 and 30) | Pentatonic EDOs (5, 10, 15, 20, 25 and 30) | ||
"Sweet" EDOs (all others) | |||
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). | |||
===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__=== | ===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__=== | ||
**__8edo__:** (generator = 1\8 2nd) | **__8edo__:** (generator = 1\8 = perfect 2nd = 150¢) | ||
D E F G * A B C D | D E F G * A B C D | ||
D - E - F - G - G#/Ab - A -B - C - D | |||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8 | ||
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc. | seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc. | ||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc. | ||
__**11edo**__: (generator = 3\11 3rd) | __**11edo**__: (generator = 3\11 = perfect 3rd) | ||
D * E F * G A * B C * D | D * E F * G A * B C * D | ||
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D | |||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8 | P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8 | ||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | ||
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 | ||
__**13edo**__**:** (generator = 2\13 2nd) | __**13edo**__**:** (generator = 2\13 = perfect 2nd) | ||
D * E * F * G A * B * C * D | D * E * F * G A * B * C * D | ||
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D | |||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | ||
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | ||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | ||
**__18edo__:** (generator = 5\18 = 3rd) | **__18edo__:** (generator = 5\18 = perfect 3rd) | ||
D * * E * F * * G * A * * B * C * * D | D * * E * F * * G * A * * B * C * * D | ||
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8 | P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8 | ||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc. | ||
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 | ||
===__Fourthward EDOs (9, 16 and 23)__=== </pre></div> | ===__Fourthward EDOs (9, 16 and 23)__=== | ||
All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc. | |||
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc. | |||
**__9edo__:** D E * F G A B * C D | |||
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D | |||
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8 | |||
**__16edo__:** D * E * * F * G * A * B * * C * D | |||
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D | |||
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8 | |||
**__23edo__:** D * * E * * * F * * G * * A * * B * * * C * * D | |||
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D | |||
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8 | |||
===__"Perfect" EDOs (7, 14, 21, 28 and 35)__=== | |||
All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc. | |||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc. | |||
**__7edo__:** D E F G A B C D | |||
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 | |||
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 | |||
**__14edo__:** D * E * F * G * A * B * C * D | |||
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D | |||
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8 | |||
**__21edo__:** D * * E * * F * * G * * A * * B * * C * * D | |||
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D | |||
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | |||
**__28edo__:** D * * * E * * * F * * * G * * * A * * * B * * * C * * * D | |||
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc. | |||
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc. | |||
**__35edo__:** D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D | |||
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc. | |||
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.</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:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | <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:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | ||
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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 adds up to one EDO-step. So C# is right next to C, and your 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 adds up to one EDO-step. So C# is right next to C, and your 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 /> | ||
<br /> | <br /> | ||
In contrast, 22-EDO is hard to notate because 7 fifths are <u>three</u> EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing | In contrast, 22-EDO is hard to notate because 7 fifths are <u>three</u> EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!<br /> | ||
<br /> | <br /> | ||
The solution is to use the sharp symbol to mean &quot;raised by 7 fifths&quot;, and to use the up symbol &quot;^&quot; to mean &quot;sharpened by one EDO-step&quot;. 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced &quot;D-flat-up, D-down&quot;, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.<br /> | The solution is to use the sharp symbol to mean &quot;raised by 7 fifths&quot;, and to use the up symbol &quot;^&quot; to mean &quot;sharpened by one EDO-step&quot;. 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced &quot;D-flat-up, D-down&quot;, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.<br /> | ||
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<br /> | <br /> | ||
C: no sharps<br /> | C: no sharps<br /> | ||
G: 1 sharp<br /> | G: 1 sharp on F<br /> | ||
D: 2 sharps<br /> | D: 2 sharps on F and C<br /> | ||
...<br /> | ...<br /> | ||
C#: 7 sharps<br /> | C#: 7 sharps<br /> | ||
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pentatonic EDOs, with a fifth = 720¢<br /> | pentatonic EDOs, with a fifth = 720¢<br /> | ||
&quot;sweet&quot; EDOs, so-called because the fifth hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | &quot;sweet&quot; EDOs, so-called because the fifth 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 /> | |||
superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br /> | superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
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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.<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 /> | ||
<br /> | <br /> | ||
<strong><u>53-EDO</u>:</strong> (5 keys per sharp/flat)<br /> | |||
Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C<br /> | |||
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The second 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. That's because 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.<br /> | The second 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. That's because 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.<br /> | ||
<br /> | <br /> | ||
The chain of fifths in | The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br /> | ||
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 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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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.<br /> | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. 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 they are two different concepts that must be kept distinct. In this redefined notation, B - F# isn't a perfect fifth because it's really B - F^.<br /> | ||
<br /> | <br /> | ||
The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. 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. Such EDOs are dealt with below.<br /> | The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. 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. Such EDOs are dealt with below.<br /> | ||
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keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D<br /> | keyboard/fretboard: D * * E/F * * G * * A * * B/C * * D<br /> | ||
(the chain of fifths is always centered on D)<br /> | (the chain of fifths is always centered on D)<br /> | ||
chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8<br /> | chord components: P1 ^m2 vM2 M2/m3 ^m3 vM3 M3/P4 ^P4 vP5 P5 ^m6 vM6 M6/m7 ^m7 vM7 P8<br /> | ||
chord roots: I ^bII vII II/bIII ^bIII vIII IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII<br /> | chord roots: I ^bII vII II/bIII ^bIII vIII III/IV ^IV vV V ^bVI vVI VI/bVII ^bVII vVII<br /> | ||
0-3-9 = m or sus2<br /> | 0-3-9 = m or sus2<br /> | ||
0-4-9 = ^m<br /> | 0-4-9 = ^m<br /> | ||
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Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region 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 this diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region 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 /> | ||
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&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)<br /> | &quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)<br /> | ||
Fourthward EDOs (9, 16 and 23)<br /> | Fourthward EDOs (9, 16 and 23)<br /> | ||
&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)<br /> | |||
Pentatonic EDOs (5, 10, 15, 20, 25 and 30)<br /> | Pentatonic EDOs (5, 10, 15, 20, 25 and 30)<br /> | ||
&quot;Sweet&quot; EDOs (all others)<br /> | |||
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo).<br /> | |||
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<strong><u>8edo</u>:</strong> (generator = 1\8 2nd) <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 /> | ||
D - E - F - G - G#/Ab - A -B - C - D<br /> | |||
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8<br /> | ||
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.<br /> | seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.<br /> | ||
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb<br /> | A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.<br /> | ||
<br /> | <br /> | ||
<u><strong>11edo</strong></u>: (generator = 3\11 3rd) <br /> | <u><strong>11edo</strong></u>: (generator = 3\11 = perfect 3rd)<br /> | ||
D * E F * G A * B C * D <br /> | D * E F * G A * B C * D<br /> | ||
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D<br /> | |||
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br /> | P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8<br /> | ||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | ||
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 /> | ||
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<u><strong>13edo</strong></u><strong>:</strong> (generator = 2\13 2nd)<br /> | <u><strong>13edo</strong></u><strong>:</strong> (generator = 2\13 = perfect 2nd)<br /> | ||
D * E * F * G A * B * C * D<br /> | D * E * F * G A * B * C * D<br /> | ||
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br /> | |||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | ||
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | ||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | ||
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<strong><u>18edo</u>:</strong> (generator = 5\18 = 3rd)<br /> | <strong><u>18edo</u>:</strong> (generator = 5\18 = perfect 3rd)<br /> | ||
D * * E * F * * G * A * * B * C * * D<br /> | D * * E * F * * G * A * * B * C * * D<br /> | ||
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D<br /> | |||
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br /> | P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8<br /> | ||
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.<br /> | ||
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All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.<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 /> | |||
<br /> | |||
<strong><u>9edo</u>:</strong> D E * F G A B * C D<br /> | |||
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D<br /> | |||
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8<br /> | |||
<br /> | |||
<strong><u>16edo</u>:</strong> D * E * * F * G * A * B * * C * D<br /> | |||
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D<br /> | |||
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8<br /> | |||
<br /> | |||
<strong><u>23edo</u>:</strong> D * * E * * * F * * G * * A * * B * * * C * * D<br /> | |||
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D<br /> | |||
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8<br /> | |||
<br /> | |||
<br /> | |||
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All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | |||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br /> | |||
<br /> | |||
<strong><u>7edo</u>:</strong> D E F G A B C D<br /> | |||
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8<br /> | |||
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8<br /> | |||
<br /> | |||
<strong><u>14edo</u>:</strong> D * E * F * G * A * B * C * D<br /> | |||
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D<br /> | |||
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8<br /> | |||
<br /> | |||
<strong><u>21edo</u>:</strong> D * * E * * F * * G * * A * * B * * C * * D<br /> | |||
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D<br /> | |||
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | |||
<br /> | |||
<strong><u>28edo</u>:</strong> D * * * E * * * F * * * G * * * A * * * B * * * C * * * D<br /> | |||
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A 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 /> | |||
<strong><u>35edo</u>:</strong> D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D<br /> | |||
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.<br /> | |||
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.</body></html></pre></div> | |||