Kite's ups and downs notation

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="Ups and Downs" Notation= 

Ups and Downs is a notation system developed by Kite that works very well 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 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 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 what looks ascending on the page sounds descending and vice versa. 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 something like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing too 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. The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So in B, we get B-C-C^-C#v-C#-D-D^-D#v-D#-E etc.

The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation.

You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.

Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-EDO, Db can be spelled C^ or B#v or even B^^ ("B double-up"). However avoid using both C# and Db, as the ascending Db-C# looks descending.

The chain of fifths runs:
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc.
This chain can be expressed in relative notation:
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc.
Superimposing one chain onto the other allows one to name any interval. C-Eb = m3 because m3 means "raised by 3 fourths". Likewise Ab-A = A1, etc.

22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^
D-F# = M3, D-F#v = vM3, Dv-F#v = M3
M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3

Staff notation is a breeze, just put an up or down to the left of the note and any standard accidental it might have. All 22 possible keys can be written out. The tonic is usually a mid note, i.e. not up or down. Some of the key signatures have double-sharps or double-flats in them.

So that's 22-EDO. This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)

EDOs come in 5 categories, based on the size of the fifth:
supersharp EDOs, with fifths wider than 720¢
pentatonic EDOs, with a fifth = 720¢
"sweet" EDOs, so-called because the fifth hits the "sweet spot" between 720¢ and 686¢
heptatonic EDOs, with a fifth = four sevenths of an octave = 686¢
superflat EDOs or Mavila EDOs, with a fifth less than 686¢

This is in addition to the trivial EDOs, 1, 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. This section will cover sweet EDOs and the other categories will be covered in other sections.

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 sweet EDOs that don't need ups and downs are 12, 19, 26, 33, 40 and 47. In fact, every 7th sweet EDO. There are also 3 such superflat EDOs, 9, 16 and 23. All other EDOs need ups and downs, except for 5-EDO and 7-EDO.

17-EDO:
Black and white keys: C _ _ D _ _ E F _ _ G _ _ A _ _ B C
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5
Many other variations are possible, much freedom of spelling.
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B



An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. For example, 36-EDO has a fifth of 21 steps, and 21 isn't coprime with 36, because they have a common divisor of 3. 36-EDO is notated like so:
C

36-EDO is said to close at 12 (1/3 of 36), because the circle of fifths has only 12 notes. There are actually 3 circles in 36-EDO, which are notated as the mid one, the up one and the down one:

, the Closed EDOs make it impossible to have every possible tonic be a mid note.

Original HTML content:

<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>
 <br />
Ups and Downs is a notation system developed by Kite that works very well 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 mid symbol &quot;~&quot; which undoes ups and downs.<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 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 />
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 what looks ascending on the page sounds descending and vice versa. 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 something like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing too 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 />
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. The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So in B, we get B-C-C^-C#v-C#-D-D^-D#v-D#-E etc.<br />
<br />
The basic pattern for 22-EDO is P1-m2-^m2-vM2-M2-m3-^m3-vM3-M3-P4-d5-^d5-vP5-P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation.<br />
<br />
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br />
<br />
Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-EDO, Db can be spelled C^ or B#v or even B^^ (&quot;B double-up&quot;). However avoid using both C# and Db, as the ascending Db-C# looks descending.<br />
<br />
The chain of fifths runs:<br />
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc.<br />
This chain can be expressed in relative notation:<br />
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc.<br />
Superimposing one chain onto the other allows one to name any interval. C-Eb = m3 because m3 means &quot;raised by 3 fourths&quot;. Likewise Ab-A = A1, etc.<br />
<br />
22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:<br />
C + M3 = E, C + vM3 = Ev, C^ + M3 = E^<br />
D-F#  <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="M3, D-F#v"></a><!-- ws:end:WikiTextHeadingRule:2 --> M3, D-F#v </h1>
 vM3, Dv-F#v = M3<br />
M2 + m2 = m3, M2 + ^m2 = ^m3, vM2 + m2 = vm3<br />
<br />
Staff notation is a breeze, just put an up or down to the left of the note and any standard accidental it might have. All 22 possible keys can be written out. The tonic is usually a mid note, i.e. not up or down. Some of the key signatures have double-sharps or double-flats in them.<br />
<br />
So that's 22-EDO. This notation works for almost every EDO. 9, 11, 16, and 23 have weird interval arithmetic because of the narrow fifth, but they can be notated. 13 and 18 are best notated using the narrower of the 2 possible fifths, which makes them like 9, 11, 16 and 23. 8-EDO is hard. It works with pentatonic notation, if you don't mind learning pentatonic interval arithmetic. (Big if!)<br />
<br />
EDOs come in 5 categories, based on the size of the fifth:<br />
supersharp EDOs, with fifths wider than 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 />
heptatonic EDOs, with a fifth = four sevenths of an octave = 686¢<br />
superflat EDOs or Mavila EDOs, with a fifth less than 686¢<br />
<br />
This is in addition to the trivial EDOs, 1, 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. This section will cover sweet EDOs and the other categories will be covered in other sections.<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 sweet EDOs that don't need ups and downs are 12, 19, 26, 33, 40 and 47. In fact, every 7th sweet EDO. There are also 3 such superflat EDOs, 9, 16 and 23. All other EDOs need ups and downs, except for 5-EDO and 7-EDO.<br />
<br />
17-EDO:<br />
Black and white keys: C _ _ D _ _ E F _ _ G _ _ A _ _ B C<br />
Relative notation: P1 m2 vM2 M2 m3 vM3 M3 P4 d5 vP5 P5 m6 vM6 M6 m7 vM7 M7 P8<br />
or with upminors instead of downmajors: P1 m2 ^m2 M2 m3 ^m3 M3 P4 d5 ^d5 P5 m6 ^m6 M6 m7 ^m7 M7 P8<br />
The d5 could instead be an A4: P4 ^P4 A4 P5 or P4 vA4 A4 P5<br />
Many other variations are possible, much freedom of spelling.<br />
In C, with downmajors: C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br />
In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B<br />
<br />
<br />
<br />
An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. For example, 36-EDO has a fifth of 21 steps, and 21 isn't coprime with 36, because they have a common divisor of 3. 36-EDO is notated like so:<br />
C<br />
<br />
36-EDO is said to close at 12 (1/3 of 36), because the circle of fifths has only 12 notes. There are actually 3 circles in 36-EDO, which are notated as the mid one, the up one and the down one:<br />
<br />
, the Closed EDOs make it impossible to have every possible tonic be a mid note.</body></html>