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Kite's ups and downs notation: Difference between revisions

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A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:
A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:
{| class="wikitable" style="text-align:center;"  
{| class="wikitable"
|+
|+
!
!
!interval between
two notes
!note plus
an interval
!sum of two
intervals
|-
!conventional
!conventional
! colspan="2" |with ups and downs
|C to E = M3
!(cancelling)
|C + M3 = E
!(combining)
|M2 + M2 = M3
|-
|-
|interval between two notes
! rowspan="2" |with ups
|C to E = M3
 
and downs
|^C to E = vM3
|^C to E = vM3
|^C + M3 = ^E
|^M2 + M2 = ^M3
|-
|C to ^E = ^M3
|C to ^E = ^M3
|C + ^M3 = ^E
|M2 + vM2 = vM3
|-
!(cancelling)
|^C to ^E = M3
|^C to ^E = M3
|^C + vM3 = E
|^M2 + vM2 = M3
|-
!(combining)
|^C to vE = vvM3
|^C to vE = vvM3
|-
|note plus an interval
|C + M3 = E
|^C + M3 = ^E
|C + ^M3 = ^E
|^C + vM3 = E
|^C + ^M3 = ^^E
|^C + ^M3 = ^^E
|-
|sum of two intervals
|M2 + M2 = M3
|^M2 + M2 = ^M3
|M2 + vM2 = vM3
|^M2 + vM2 = M3
|vM2 + vM2 = vvM3
|vM2 + vM2 = vvM3
|}
|}
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In perfect EDOs (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following list of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".
In perfect EDOs (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following list of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".


An up or down between the chord root and the chord type (e.g. C^m7) raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A mid-something chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord with a 6th below the root is affected: '''<u>6th</u>''' - root - '''<u>3rd</u>''' - 5th - '''<u>7th</u>''' - 9th - '''<u>11th</u>''' - 13th.
An up or down between the chord root and the chord type (e.g. C^m7) raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A mid-something chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord with a 6th below the root is affected: '''<u>6th</u>''' - root - '''<u>3rd</u>''' - 5th - '''<u>7th</u>''' - 9th - '''<u>11th</u>''' - 13th. Note that the 6th is affected, but the 13th is not.


The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.
The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in larger EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.


Every conventional chord can accept such an up or down, with one exception: it's pointless to down a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down-5", it means C(v5) = C E vG.
Every conventional chord can accept such an up or down, with one exception: it's pointless to down a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid. But C(v5) is valid, and if someone says "C down-5", it means C(v5) = C E vG.
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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. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is <u>down</u>major, 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. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is <u>down</u>major, it doesn't work in that progression.


Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under).
Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over, thus 7/4). In 19-edo, it sounds gu (5-under, thus 9/5).


==Ups and downs solfege==
==Ups and downs solfege==
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down in two ways, single-vowel solfege and double-vowel solfege. The former is more singable than the latter, especially for quick runs. But it doesn't work well for fixed-do, since it lacks sharps.
=== Single-vowel solfege ===
The initial consonant remains as before: D, R, M, F, S, L and T
All consonants other than D and F have an alternate form that is flattened: R becomes Fr (mnemonic: f for flat), M -> N, S -> Sh, L -> Fl and T -> Th (unvoiced as in think).
The vowels are pronounced as in Spanish or Italian. The vowel indicates up/down/plain. The vowel sequence can vary for different EDOs.
==== 41-EDO example ====
Vowel sequence: u = ^, a = plain, o = v, i = vv. Mnemonic: u = '''<u>u</u>'''p, a = pl'''<u>a</u>'''in, o = d'''<u>o</u>'''wn, i = m'''<u>i</u>'''d.
* Fro = '''f'''lat-'''r'''e-d'''o'''wn = vm2
* Fra = '''f'''lat-'''r'''e-pl'''a'''in = m2
* Fu = '''f'''lat-'''r'''e-'''u'''p = ^m2
* Ri = '''r'''e-m'''i'''d = ~2
* Ro = '''r'''e-d'''o'''wn = vM2
* Ra = '''r'''e-pl'''a'''in = M2
* Ru = '''r'''e-'''u'''p = ^M2
A new consonant P indicates a sharpened 4th. (Mnemonic: Sha and Tha sharpen to Sa and Ta, so if Fa were spelled Pha, it would sharpen to Pa.) This lets us name 11/8, 45/32 and 10/7 as 4ths (Fi, Po and Pa).
* unisons: Da Du
* 2nds: Fro Fra Fru Ri Ro Ra Ru
* 3rds: No Na Nu Mi Mo Ma Mu
* 4ths: Fo Fa Fu
* tritones: Fi/Sho Po/Sha Pa/Shu Pu/Si
* 5ths: So Sa Su
* 6ths: Flo Fla Flu Li Lo La Lu
* 7ths: Tho Tha Thu Ti To Ta Tu
* 8ves: Do Da (Du)
To find the octave inverse of any interval:
* change the '''degree''' as usual: 2nd <--> 7th, 3rd <--> 6th, and 4th <--> 5th
* change the '''quality''' as usual: major <--> minor, aug <--> dim, but perfect and mid are unchanged
* change the '''vowel''' as expected: -o <--> -u,  but -a and -i are unchanged
For example, Fro = minor-Re-down becomes major-Ti-up = Tu. The rule for changing the quality means the ~4 and the ~5 must be either Fi & Si, or else Pi & Shi. The former is chosen because then the 6 mid intervals Ri Mi Fi Si Li Ti all use the default consonants.


Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
* Plain major: Da Ra Ma Fa Sa La Ta Da
* Plain phrygian: Da Fra Na Fa Sa Fla Tha Da
* Upmajor: Da Ra Mu Fa Sa Lu Tu Da
* Upminor: Da Ra Nu Fa Sa Flu Thu Da
* Downmajor: Da Ra Mo Fa Sa Lo To Da
* Downminor: Da Ra No Fa Sa Flo Tho Da
* 3rd mode of [[243/242#Neutral|Lulu/Rastmic[7]]]: Da Ra Mi Fa Sa Li Ti Da
* 2nd mode: Da Ra Mi Fi Sa Li Ti Da
* Harmonics 8-16: Da Ra Mo Fi Sa Li Tho To Da


=== Double-vowel solfege ===
The initial consonant remains as before: D, R, M, F, S, L and T
The initial consonant remains as before: D, R, M, F, S, L and T


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