Kite's ups and downs notation: Difference between revisions

== Definition ==

Ups and downs can also notate any [[Tour of Regular Temperaments|rank-2 temperament]], although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a [[comma]] that has been tempered, but not tempered out. For example, in [[Porcupine|Triyo aka Porcupine]], an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyo is supported by both 15edo and 22edo, and both edos map 81/80 to one EDOstep. Thus if Triyo is tuned to 15edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22edo, the up means both 81/80 and 1\22. If not tuned to an edo at all, then the up only means 81/80. Thus a piece written in Triyo can be converted to a piece written in 22edo by simply writing "22edo" on the top of the page.  

Ups and downs can also be used to notate rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11. See [[Ups and Downs Notation for Rank-3 JI]].  

For more on edo notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf '''Notation guide for edos 5-72'''], which also covers chord names, slash chords, staff notation, key signatures, and scale trees.  

To understand the ups and downs notation, let's start with an edo that doesn't need it. 19edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDOstep. C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19edo as long as you remember that C# and Db are different notes.

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.

Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22edo intervals: P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^4/d5 - vA4/^d5 - A4/v5 - P5 etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22edo key. The '''plain''' notes (those without ups or downs) always form a chain of fifths.

A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:

|+

!

|-

!conventional

|C to E = M3

|C + M3 = E

|-

|^C to E = vM3

|^M2 + M2 = ^M3

|-

|C to ^E = ^M3

|C + ^M3 = ^E

|-

!(cancelling)

|^C + vM3 = E

|^M2 + vM2 = M3

|-

|^C to vE = vvM3

|^C + ^M3 = ^^E

The same logic holds for a note minus an interval (C - vm3 = ^A) or one interval minus another interval (M3 - vM2 = ^M2).

Up and down are short for up-arrow and down-arrow, and arrow refers to both. Sometimes the name of a notation symbol comes to mean that which the symbol indicates. Just as "bar" (the vertical line that separates measures) has come to mean "measure", "[[arrow]]" has also come to mean "EDOstep".

In larger edos, triple-arrows, quadruple-arrows, etc. can occur. Up, dup, trup and quup all rhyme, as do dud, trud and quud.  

|+symbols and words for multiple arrows

!written

!2 arrows

|vv

| rowspan="4" |"-d" for down  

"-p" for up

|trud

|-

|'''<u>qu</u>'''adruple-'''<u>up</u>'''

!4 arrows

|-

|>

|quid

|}

|+

|-

|up

|quip

>

^>

|dupquip

|trupquip

^^^>

v>>

|quipquip

|dupquipquip

^^>>

|-

|down

v

vv

vvv

|quud

^<

<

v<

|dudquid

|dudquidquid

vv<<

|}

Lifts and drops (/ and \) can be used for microinflections of less than an edostep, since they look like part of an arrow.

For staff notation, put an arrow to the left of the note and any sharp or flat it might have. Like sharps and flats, an arrow applies to any similar note that follows in the measure. If C is upped, any other C in the same octave inherits the up. If an up-C is followed by a down-C, the down-arrow cancels the up-arrow.   

There are several possible ways to handle this issue. The default is the simplest way, to explicitly specify both arrows and accidentals every time. Thus any accidental or arrow cancels any previous ones. An arrow by itself implies a natural sign.

|+

! rowspan="2" |start with

|-

!C

|^

|^^

!^C

|<big>♮</big>

|^^

|-

!^^C

|<big>♮</big>

|^

|#

|^#

|^^#

|-

|<big>♮</big>

|^

|^^

|^#

|^^#

|^^

|#

|^^#

|-

!^^C#

|<big>♮</big>

|#

|^#

|}

See [[Kite Guitar originals#Cancelling rules]] for another way.

For more on staff notation, see the [http://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf Notation Guide for EDOs 5-72].  

Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some edos. For example, 19edo has the key of Bbb with a key signature of Bbb Ebb Ab Db Gb Cb Fb. Some edos have upped/downed tonics, e.g. 24edo has the key of vD with a key signature of F# C# (v). The (v) is a "global down" that downs all 7 notes of the vD scale. See also [[Kite Guitar originals#Scales and key signatures]] for the use of '''arrow stacks'''.

It might seem more natural to place the arrow after the note, for example B^ or Bb^. But the arrow must come first, to make chord names unambiguous. Otherwise B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)

The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.

Edo intervals are often written as 7\22. This can also be written as vM3\22. This is useful when comparing edos, e.g. vM3\22 vs. vM3\15.

|D#/Eb

|'''E'''

|'''G'''

|G#/Ab

|'''D'''

|-

|A1/m2

|m3

|M3

|P5

|m6

|m7

|P8

|}

|vEb/^F#

|Eb/vF

|'''D'''

|-

|P8

|-

major wider than minor  

|'''D'''

|-

|P8

|}

| rowspan="2" |'''[[14-edo|14edo]]'''

sharp-0

|-

|8

|}

| rowspan="2" |'''[[15-edo|15edo]]'''

sharp-3   

|-

|}

16edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either AA2 or dd2.  

| rowspan="2" |sharp lowers the pitch,

major narrower than minor

|-

|-

| rowspan="2" | sharp raises the pitch,

major wider than minor

|-

| P8

|}

| rowspan="2" |'''[[17edo]]'''

sharp-2   

|-

|}

18b-edo contains 2 rings of 9edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic intervals: either ^^A1 and vvM2, or vvA1 and vvm2.

| rowspan="4" |'''[[18-edo|18b-edo]]'''

flat-2

major is narrower

|-

|-

major is wider

|-

|}

19edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: dd2.

| rowspan="2" |'''[[19-edo|19edo]]'''

sharp-1

|-

|}

20edo contains 4 rings of 5edo: an up-ring, a down-ring, a dup/dud-ring, and a plain-ring. Enharmonic intervals: v⁴A1 and m2.

| rowspan="2" |'''[[20-edo|20edo]]'''

sharp-4

|-

|}

Because every 21edo interval is perfect, the quality can be omitted. 21edo contains 3 rings of 7edo: an up-ring, a down-ring and a plain-ring. Enharmonic intervals: A1 and v³m2.

| rowspan="2" |'''[[21-edo|21edo]]'''

sharp-0

|-

|}

22edo is sharp-3. Enharmonic intervals: v³A1 and vm2.

| rowspan="2" |'''[[22-edo|22edo]]'''

sharp-3

|-

| vA1/^m2

|}

23edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A³2 or d³2.

| rowspan="2" |sharp lowers,

major is narrower

|-

|-

major is wider

|-

|}

24edo contains 2 rings of 12edo: an up/down-ring and a plain-ring. Enharmonic intervals: vvA1 and d2.

{| class="wikitable" style="text-align:center;"  

sharp-2

|-

|}

Chord names are based on jazz chord names. See Jim Aiken's book ''A Player's Guide to Chords & Harmony''. Alterations are enclosed in parentheses, additions never are. Alterations always come last in the chord name. Examples:

In [[Sharpness|sharp-0]] edos aka perfect edos (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following lists 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".

Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, <u>'''never use lower case roman numerals'''</u> for minor chords, because both vIIm and VIIm would be written vii.  

<span style="display: block; text-align: left;">The major chord and various alterations of it:</span>

* C vvE G = Cvv = "C dud" or "C dudmajor" (in sharp-4 edos, C~ = "C mid", in sharp-6 edos, C^~ = "C upmid")

This table shows how altering the 3rd or the 5th affects the name of the triad. The conventional abbreviations for aug and dim are + and ^o. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.

!C D G

! colspan="2" |C E G#

|-

!nothing

|Ca

|C+

|-

|Cv4

|Cv2

|Cm(v5)

|C4(v5)

|Cd(v5)

|C^o(v5)

|-

!3rd, 5th

|Cv2(v5)

|Cva(v5)

|Cv^o(v5)

|}

Many edos have notes between the major 3rd and the perfect 4th, creating triads impossible in 12edo, such as:

The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.  

A global arrow occurs between the chord root and the conventional chord type (e.g. C^m7). It 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 global-mid chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord 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.  

Every conventional chord can accept a global arrow, with one exception: it's pointless for 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 five", it means C(v5) = C E vG.

If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th. An added 7th is usually preceded by a comma (the actual punctuation mark, not an interval), which is spoken as "add":

All 7th chords follow this same pattern. Likewise, if the 6th is not a perfect 4th or aug 4th above the 3rd, it's an add-6 chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can both be replaced with "dim add-7", written Cd,7.   

!dom7

!min7

! colspan="2" |dim7

!maj6

|CM7

|C7

|Cm7(b5)

|C^ø

!3rd

|Cv,M7

|Cvd,7

|Cvm,7(b5)

|Cvm,6

|-

|CM7(v5)

|C7(v5)

|Cm6(v5)

|-

|C,v7

|Cmv7

!3rd, 5th

|Cv,M7(v5)

|Cvm,7(v5)

|Cvd,7(v5)

|Cvm7

|Cvdv7

|Cv^ø

|Cvd7

|}

Various unusual tetrads:

As in conventional chord naming, a sharp-9 or flat-9 chord is always named as a 7th chord with an added 9th. Thus B D# F# A C is named B7b9 (not Bb9 which would be Bb D F A C). Likewise C#7b9 not C#b9, even thought the latter is clearly the same flat-9 chord as the former. Likewise Cm7b9 not Cmb9, etc.

!maj9