Kite's ups and downs notation: Difference between revisions

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.  

'''<u>This page only discusses notation of edos.</u>''' However, the notation of chords and chord progressions applies to all situations. For notation of rank-2 and rank-3 temperaments, see the [[pergen|pergens]] article.   

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.  

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.

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

|-

| M2 + M2 = M3

|-

| ^C + M3 = ^E

|-

| M2 + vM2 = vM3

|-

| ^C to ^E = M3

|-

! (combining)

|}

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".

From the [[Pergen|pergens]] article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic unisons are like vanishing commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of EUs needed always equals the difference between the notation's rank and the tuning's rank."  

Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two EUs are needed to define the notation: v³A1 and vm2. Either EU can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v³A1 = vE. Any combination of these two EUs is also an EU, for example their sum v⁴M2. Thus ^^F = ^^F + v⁴M2 =  vvG (double-down G, or '''dud''' G for short, rhymes with "cud").  

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

|-

|-

| dud

|-

|-

|-

|}

(In addition to dup, trup, etc. there is dub, trip, quad and quin, used for multiple sharps/flats and multiple lifts/drops, e.g. dubsharp or triplift.)

Very large edos can go well beyond 5 arrows. The sequence of names resembles tally counting I, II, III, IIII, <s>||||</s>. But the sequence of ''symbols'' resembles roman numerals I, II, III, IV, V. Thus 4 ups is spoken quup but written v>.

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

|-

|-

! 4

! 5

! 15

|-

|down

|}

For very large edos in which commas like 81/80 and 64/63 are many edosteps, the color notation accidental pairs yo/gu and zo/ru can be "edoized" to stand for a certain number of edosteps. For example, in [[311edo]], 81/80 is 6 edosteps. Thus g means ^> and y means v<. The colors can be combined with arrows as in upyo or dudgu (^y or vvg). Likewise, 64/63 is 7 edosteps, thus r means ^^> and z means vv<.

===Staff Notation===

But what happens when accidentals are mixed with arrows? What if the key signature makes that upped C be sharp? Or what if there is a C with a sharp just before the upped C? Does the up-arrow override or "cancel" the sharp? And what if an upped C is followed by a sharpened C?

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.

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

|-

|-

|-

| #

| ^#

| ^^#

|-

|-

! C#

|-

|-

| ^

| ^^

|}

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].  

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.

Sharp-1, flat-2, etc. refer to the [[sharpness]], the number of arrows made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.  

There are four flat-N edos on this list. 16edo and 23edo are flat-1, 18b is flat-2 and 13b is flat-3. There are two ways to notate such edos: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. Both notations are shown. In the 2nd notation, note that a fifth above B is Fb, not F#.   

| F#/Gb

|-

| P1

| M2

| M6

| M7

|}

There are two ways to notate 13b-edo. The enharmonic unisons for the 1st notation are ^³A1 and vM2. For the 2nd they are v³A1 and vm2.

| Bb/vC

|-

| m3

|-

| ^E/Fb

| vB#/^Cb

|-

| P4

| vM6/^m7

| M6/vM7

|}

Because every 14edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14edo contains 2 rings of 7edo: an up/down-ring and a plain-ring. Enharmonic unisons: A1 and vvm2.

| ^F/vG

|-

| 1

| 3

| ^3/v4

| ^6/v7

|}

15edo contains 3 rings of 5edo: an up-ring, a down-ring, and a plain-ring. Enharmonic unisons: v³A1 and m2.

| ^A

|-

| vM3

| P5

| ^m7

| P8

|}

| Db/E#

| F#

| Gb/A#

| Ab/B#

|-

| M3

| m3

| M6

| m6/A7

| M7

| m7

|-

| C#/Db

|-

| m3

| P4

| P5

| m6

| M6/d7

| P8

|}

17edo is sharp-2 and thus has mid intervals. Enharmonic unisons: vvA1 and vm2.

| ^G/Ab

| G#/vA

| ^A/Bb

|-

| P1

| ^1/m2

| M2

| ^4/~4/d5

| A4/v5/~5

|}

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

| ^D/vE

| ^E

|-

| ~2

| ~3

| ^4/v5

|-

| E#/Fb

|-

| ~3

| M3

| P4

| ^5/vm6

| ~7

| P8

|}

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

| Eb

| C#

|-

| A3

| A4

| P5

| M6

| m7

|}

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

| vG

|-

| ~2

| ~3

| vM3

| ^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 unisons: A1 and v³m2.

| ^F

| ^G

| ^A

| vC

|-

|}

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

| vD#/^Eb

| D#/vE

| ^G/Ab

| vG#/^Ab

|-

| ^1/m2

| vA1/^m2

| m3

| vM3

| P4

|}

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

| Db

| F#

| Bbb/Cx

| Cb

|-

| A5

| m6

| M7

|-

| Gb

|-

| d2

| A2/d3

| m3

| P4

| d5

| P5

| m6

| A6/d7

| m7

| M7

| A7

|}

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

| ^G/vAb

|-

| P1

| ^1/vm2

| A1/m2

| M2

| ^M2/vm3

| M3

| ^M3/v4

| ^4/~4

| A4/d5

| v5/~5

|}

== Chords and chord progressions==

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 E G = C = "C" or "C major" (in perfect edos, "C" or "C perfect")

* 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.

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

|-