Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2014-01-31 06:51:33 UTC</tt>.

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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.

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!

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 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 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 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.

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 advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

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 apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a (tempered) pythagorean chain of fifths.

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.

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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:

__**Interval arithmetic**__

In ups and downs notation, as in conventional notation, 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~~.

To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means "raised by 4 fifths" and E is 4 fifths away from C. Likewise, C + M3 = E.

C - G - D - A - E

P1-P5-M2-M6-M3

To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:

m3-m7-P4-P1

P1-P5-M2

Line up the lower P1 with m3 and look for what lies above M2.

22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:

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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.~~

There are some exceptions. Take this scale:

C Db Db^ Dv D Eb Eb^ Ev E F Gb Gb^ Gv G Ab Ab^ Av A Bb Bb^ Bv B C

Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which looks like a min 3rd. Here's this scale's chain of 5ths:

Gb^ Db^ Ab^ Eb^ Bb^ Gb Db Ab Eb Bb F C G D A E B Gv Dv Av Ev Bv

The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. Because 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:

Gx Dx Ax F# C# G# D# A# F C G D A Fb Cb Gb Db Ab Fbb Cbb Gbb Dbb

C C# Dbb Db D D# Dx Fbb Fb F F# Gbb Gb G G# Gx Ab A A# Ax Cbb Cb C

Now that's an awful lot of sharps and flats, but that does make a neat and tidy notation (except for the Gbb-Gx fifth). And it exists as an alternative, embedded within our standard notation, with a key signature with circled X's on the B and E spots.

So the chain of fifths has a few spots to watch out for. You have to remember that B-something to G-something is sometimes a fifth, sometimes a sixth. A little tricky, but manageable. Analogous to 12-ET, where G# to Eb is a fifth that looks like a sixth.

__**Staff Notation**__

For staff notation, just put an up or down to the left of the note and any standard accidental it might have. To write Db^ followed by Db in the same measure, use the mid sign: Db^ Db~. All 22 possible keys can be written out. The tonic is always a mid note, i.e. not up or down. Just as conventionally each black key produces both a sharp key and a flat key (Db major and C# minor), each of the 15 black keys of 22-EDO produces both, and there are 37 possible keys. The 2 most remote are Bbbb and F###, and triple-sharps and triple-flat keys seem rather extreme. Avoiding those, we have 35 possible tonics that run from Fbb to Bx. Some of the key signatures will have double-sharps or double-flats in them, or even triple-sharps.

C: no sharps

C#: 7 sharps

G#: 6 sharps, 1 double-sharp on F

D#: 5 sharps, 2 double-sharps on F and C

B#: 2 sharps, 5 double-sharps on F , C, G, D and A

Bx: 2 double-sharps on E and B, 5 triple-sharps on F, C, G, D and A

__**Other EDOs**__

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!)

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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~~.

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.

17-~~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

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In B, with upminors: B C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B

One can't associate ups and downs with JI as easily because of the poor approximation of the 5-limit. However major = red or fifthward white and minor = blue or fourthward white.

**__24-EDO__:**

black and white keys: C _ _ _ D _ _ _ E _ F _ _ _ G _ _ _ A _ _ _ B _ C

Relative notation: P1 vm2 m2 vM2 M2 vm3 m3 vM3 M3 vP4 P4 ^P4 d5 vP5 P5 etc.

Many alternate spellings available, for example vm3 = ^M2, vM3 = ^m3, ^P4 = vd5, etc.

In C: C Dbv Db Dv D Ebv Eb Ev E Fv F F^ Gb Gv G etc.

24-EDO is an example of a closed EDO. An EDO is closed if the keyspan of the fifth isn't coprime with the keyspan of the octave, and open if it is. 24-EDO has a fifth of 14 steps, and 14 isn't coprime with 24, because they have a common divisor of 2. 24-EDO is said to close at 12 (1/2 of 24), because the circle of fifths has only 12 notes. There are actually 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#^

Just as G# could be written as Ab, all the up notes could be written as down notes.

~~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~~:

In open 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 use B#. But closed EDOs force the use of tonics that are not a mid note. For example, the key of C^ runs:

C

C^ Db Db^ D D^ Eb Eb^ E E^ F F^ F^^ Gb^ G G^ etc.

~~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:~~

JI associations: Major = yellow or fifthward white, minor = green or fourthward white, upmajor = red, downminor = blue, downmajor = upminor = jade or amber.

, ~~the Closed EDOs make it impossible to have every possible tonic be a mid note~~.</pre></div>

**__31-EDO__:**

Black and white keys: C * * * * D * * * * E * * F * * * * G * * * * A * * * * B * * C

relative notation: P1 ^P1 A1 m2 ^m2 M2 ^M2 A2 m3 ^m3 M3 ^M3 vP4 P4 ^P4 A4 d5 ^d5 P5 etc.

alternate spellings: A1=vm2, ^m2=vM2, ^M3=vP4, ^P4=vA4, etc.

In C: C C^ C# Db Db^ D D^ D# Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.</pre></div>

<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><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a>&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.

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!

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 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 &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.

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.

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 &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation.

The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.

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. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a (tempered) pythagorean chain of fifths.

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.

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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.