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Kite's ups and downs notation: Difference between revisions

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=<u>A 22edo example</u>=
=<u>A 22edo example</u>=


Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with highs and lows, it works with almost all rank 2 tunings (see the [[pergen|pergens]] page). 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".
Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works with almost all EDOs. When extended with lifts and drops (/ and \), it works with all rank 2 tunings (see the [[pergen|pergens]] page). 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". (The third symbol is the mid symbol, "~".)


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.
Line 16: Line 16:
Relative notation 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 notes without ups or downs always form a chain of fifths.
Relative notation 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 notes without ups or downs always form a 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|Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.
You can loosely relate the ups and downs to JI: major = ru or fifthward wa, downmajor = yo, upminor = gu, minor = zo or fourthwards wa. Or simply up = gu, down = yo, and mid = wa, zo or ru. (See [[Kite's_color_notation|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# interval appears descending.
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# interval appears descending.
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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
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 yellow and green because of the poor approximation of the 5-limit. However major = red or fifthward white, minor = blue or fourthward white, and downmajor = upminor = jade or amber.
One can't associate ups and downs with yo and gu because of the poor approximation of the 5-limit. However major = ru or fifthward wa, minor = zo or fourthward wa, and downmajor = upminor = jade or amber.


'''<u>24-EDO</u>:''' (2 keys per sharp/flat)
'''<u>24-EDO</u>:''' (2 keys per sharp/flat)
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In C: C C^/Dbv C#/Db Dv D D^/Ebv D#/Eb Ev E E^/Fv F F^ F#/Gb Gv G etc.
In C: C C^/Dbv C#/Db Dv D D^/Ebv D#/Eb Ev E E^/Fv F F^ F#/Gb Gv G etc.


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


24-EDO is an example of a '''multi-ring''' EDO. An EDO is multi-ring if the keyspan of the generator (usually the fifth) isn't coprime with the keyspan of the octave, and '''single-ring''' or 1-ring if it is. 24-EDO has a fifth of 14 steps, and is 2-ring because there are 2 unconnected circles of 12 fifths. They are notated as the mid one and the up one:
24-EDO is an example of a '''multi-ring''' EDO. An EDO is multi-ring if the keyspan of the generator (usually the fifth) isn't coprime with the keyspan of the octave, and '''single-ring''' or 1-ring if it is. 24-EDO has a fifth of 14 steps, and is 2-ring because there are 2 unconnected circles of 12 fifths. They are notated as the mid one and the up one:
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In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.


JI associations: Perfect = white, major = yellow or fifthward white, minor = green or fourthward white, downminor = blue, upmajor = red, downmajor = upminor = jade or amber (same as 24-EDO).
JI associations: Perfect = wa, major = yo or fifthward wa, minor = gu or fourthward wa, downminor = zo, upmajor = ru, downmajor = upminor = jade or amber (same as 24-EDO).


'''<u>41-EDO</u>:''' (4 keys per sharp/flat)
'''<u>41-EDO</u>:''' (4 keys per sharp/flat)
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In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.
In C: C C^ Dbv Db Db^ D D^ Ebv Eb Eb^ E E^ Fv F F^ F# Gb Gb^ G etc.


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 = wa, major = fifthward wa, minor = fourthward wa, downmajor = yo, upminor = gu, downminor = zo, upmajor = ru, double-downmajor = double-upminor = jade or amber.


=<u>22edo Chord Names</u>=
=<u>22edo Chord Names</u>=
Line 1,074: Line 1,074:
='''<u>Cross-EDO considerations</u>'''=
='''<u>Cross-EDO considerations</u>'''=


In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound ru and 19edo major chords sound yo.


A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a red chord.
A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any edo, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22edo, the chord that you need sounds like a ru chord.


In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yellow. If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it doesn't work in that progression.
In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. If 22edo's downmajor chord 0-7-13 = 0¢-382¢-709¢ were called "major", you wouldn't know that it doesn't work in that progression.


Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds blue. In 19edo, it sounds green. If you want a blue 7th in 19edo, you have to use the downminor 7th, which will cause shifts or drifts in the progression.
Another example: I7 - bVII7 - IV7 - I7. To make this work, the 7th in the I7 chord must be a minor 7th. in 22edo, that 7th sounds zo. In 19edo, it sounds gu. If you want a zo 7th in 19edo, you have to use the downminor 7th, which will cause shifts or drifts in the progression.


=<u>'''Scale Fragments'''</u>=
=<u>'''Scale Fragments'''</u>=
Line 1,806: Line 1,806:
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8


perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white
perfect = wa, major = ru, yo and fifthward wa, minor = gu, zo and fourthwards wa


'''<u>17edo</u>:''' sharp = 2 keys: C Db C# D
'''<u>17edo</u>:''' sharp = 2 keys: C Db C# D
Line 1,824: Line 1,824:
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
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 = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.
perfect = wa, major = yo and fifthward wa, minor = gu and fourthward wa, aug/dim = ru/zo.


'''<u>22edo</u>:''' sharp = 3 keys: C Db * C# D
'''<u>22edo</u>:''' sharp = 3 keys: C Db * C# D
Line 2,332: Line 2,332:
etc.
etc.


=<u>Rank-2 Scales: 8ve Periods</u>=
=<u>Ups and downs solfege</u>=
 
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
 
The initial consonant remains as before: D, R, M, F, S, L and T


<span style="font-size: 150%;">This section is somewhat obsolete, see the [[pergen|pergens]] page instead.</span>
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb


Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about '''frameworks''' like 12-tone. The generator chain is called a '''genchain'''. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1):
The vowels are pronounced as in Spanish or Italian
 
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u


12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv


12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B C
The 2nd vowel is separated from the first by an "h", a "w", or a "y"


When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, easily notated:
Thus C#v is Deo, pronounced as Deho or Dewo or Deyo.


7-tone: Eb * F C G D A * B = C D Eb F G A B C
This suffices for many but not all edos, as some require triple sharps or quadruple ups.


5-tone: Bb * C G D * E = C D E G Bb C
Fixed-do solfege:


12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb


For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.
Da = C, Dae = C^, Dai = C^^, Dao = Cv, Dau = Cvv, Daa = C^^^


All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:
De = C#, Dee = C#^, Dei = C#^^, Deo = C#v, Deu = C#vv, Dea = C#^^^


17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C
etc.


To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a '''keyspan''' (always +1) but also a '''genspan''', which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.
Moveable-do solfege:


Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.


17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim


Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 12-tone.
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1


22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:
Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1


{| class="wikitable"
etc.
|-
 
| style="text-align:center;" | genspan from C
Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim
| style="text-align:center;" | keyspan from C
 
Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2
 
Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2
 
etc.
 
=<u>Rank-2 Scales: 8ve Periods (OBSOLETE)</u>=
 
<span style="font-size: 150%;">'''<big>This section is obsolete, see the [[pergen|pergens]] page instead.</big>'''</span>
 
Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about '''frameworks''' like 12-tone. The generator chain is called a '''genchain'''. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1):
 
12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C
 
12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B C
 
When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, easily notated:
 
7-tone: Eb * F C G D A * B = C D Eb F G A B C
 
5-tone: Bb * C G D * E = C D E G Bb C
 
12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C
 
For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.
 
All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:
 
17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C
 
To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a '''keyspan''' (always +1) but also a '''genspan''', which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.
 
Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).
 
17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
 
Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 12-tone.
 
22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:
 
{| class="wikitable"
|-
| style="text-align:center;" | genspan from C
| style="text-align:center;" | keyspan from C
| |  
| |  
| |  
| |  
Line 3,098: Line 3,148:
There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. (In theory, 39-tone's C^ could be an octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.
There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. (In theory, 39-tone's C^ could be an octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.


=<u>Rank-2 Scales: Non-8ve Periods</u>=
=<u>Rank-2 Scales: Non-8ve Periods (OBSOLETE)</u>=


<span style="font-size: 24px;">This section is somewhat obsolete, see the [[pergen|pergens]] page instead.</span>
<span style="font-size: 24px;">'''<big>This section is obsolete, see the [[pergen|pergens]] page instead.</big>'''</span>


Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
Line 3,202: Line 3,252:
The scale: C C#v D Ev F F#v G Av A Bv C
The scale: C C#v D Ev F F#v G Av A Bv C


=<u>Ups and downs solfege</u>=
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
The initial consonant remains as before: D, R, M, F, S, L and T
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
The vowels are pronounced as in Spanish or Italian
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y"
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.
This suffices for many but not all edos, as some require triple sharps or quadruple ups.
Fixed-do solfege:
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^
etc.
Moveable-do solfege:
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1
etc.
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2
etc.
[[Category:notation]]
[[Category:notation]]
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