Kite's ups and downs notation: Difference between revisions
AthiTrydhen (talk | contribs) |
removed the tables of one-note notation, with Praveen's permission (not appropriate for a page that's an entry-point for newbies). Other minor rewrites as well. |
||
| Line 12: | Line 12: | ||
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 EDOstep. 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 EDOstep. 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. | ||
In contrast, 22-edo is hard to notate because 7 fifths are <u>three</u> EDOsteps, 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 5/4, usually a 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 because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really | In contrast, 22-edo is hard to notate because 7 fifths are <u>three</u> EDOsteps, 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 5/4, usually a 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 because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really confusing. 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 EDOstep". 22-edo can be written C - Db - ^Db - vD - D - Eb - ^Eb - vE - E - F etc. The notes are pronounced up-D-flat, down-D, 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 up or down comes <u>before</u> the note name to make naming chords easy. | The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up-arrow symbol to mean "sharpened by one EDOstep". 22-edo can be written C - Db - ^Db - vD - D - Eb - ^Eb - vE - E - F etc. The notes are pronounced up-D-flat, down-D, 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 up or down comes <u>before</u> the note name to make naming chords easy. | ||
The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - ^C - vC# - C# - D - ^D - vD# - 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 the B scale is B - C - ^C - vC# - C# - D - ^D - vD# - D# - E etc. | ||
| Line 21: | Line 21: | ||
=== Relative notation and interval arithmetic === | === Relative notation and interval arithmetic === | ||
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 22-edo 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 22-edo key. The notes without ups or downs always form a chain of fifths. | 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 22-edo 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 22-edo 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: | A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in: | ||
| Line 71: | Line 71: | ||
Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two enharmonic intervals are needed to define the notation: v<sup>3</sup>A1 and vm2. Either interval can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v<sup>3</sup>A1 = vE. Any combination of these two enharmonic intervals is also an enharmonic interval, for example their sum v<sup>4</sup>M2. Thus ^^F = vvG (double-down G, or '''dud''' G for short, rhymes with "cud"). | Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two enharmonic intervals are needed to define the notation: v<sup>3</sup>A1 and vm2. Either interval can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v<sup>3</sup>A1 = vE. Any combination of these two enharmonic intervals is also an enharmonic interval, for example their sum v<sup>4</sup>M2. Thus ^^F = vvG (double-down G, or '''dud''' G for short, rhymes with "cud"). | ||
In larger edos, triple-ups and triple-downs can occur. These are called '''trup''' and '''trud'''. Quadruple = '''quup''' and '''quud''' ("kwup" and "kwud") and quintuple = quip and quid. | In larger edos, triple-ups and triple-downs can occur. These are called '''trup''' and '''trud'''. Quadruple = '''quup''' and '''quud''' ("kwup" and "kwud") and quintuple = '''quip''' and '''quid'''. | ||
=== Staff Notation === | === Staff Notation === | ||
| Line 81: | Line 81: | ||
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. | 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. | ||
=== "Arrow" as a term for | === "Arrow" as a term for EDOstep === | ||
Up and down are short for up-arrow and down-arrow. 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 come to mean " | Up and down are short for up-arrow and down-arrow. 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 come to mean "EDOstep". | ||
=== Further notes === | === Further notes === | ||
| Line 89: | Line 89: | ||
==Examples: EDOs 12-24== | ==Examples: EDOs 12-24== | ||
Sharp-1, flat-2, etc. refer to the number of | 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. | ||
A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs. | A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs. | ||
13-edo and 18-edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, | 13-edo and 18-edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, as 13b and 18b. | ||
There are four flat-N edos on this list. 16-edo and 23-edo 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#. | |||
12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2. | 12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2. | ||
| Line 127: | Line 129: | ||
|P8 | |P8 | ||
|} | |} | ||
There are two ways to notate 13b-edo | There are two ways to notate 13b-edo. The enharmonic intervals for the 1st notation are ^<sup>3</sup>A1 and vM2. For the 2nd they are v<sup>3</sup>A1 and vm2. | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
| rowspan="4" |'''[[13-edo|13b-edo]]''' | | rowspan="4" |'''[[13-edo|13b-edo]]''' | ||
| Line 270: | Line 272: | ||
| P8 | | P8 | ||
|} | |} | ||
16-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either AA2 or dd2. | |||
16-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either AA2 or dd2 | |||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
| rowspan="4" |'''[[16-edo]]''' | | rowspan="4" |'''[[16-edo]]''' | ||
| Line 432: | Line 394: | ||
| P8 | | P8 | ||
|} | |} | ||
18b-edo contains 2 rings of 9-edo: 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 | 18b-edo contains 2 rings of 9-edo: 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. | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
| rowspan="4" |'''[[18-edo|18b-edo]]''' | | rowspan="4" |'''[[18-edo|18b-edo]]''' | ||
| Line 613: | Line 575: | ||
| vM7 | | vM7 | ||
| P8 | | P8 | ||
|} | |} | ||
Because every 21-edo interval is perfect, the quality can be omitted. 21-edo contains 3 rings of 7-edo: an up-ring, a down-ring and a plain-ring. Enharmonic intervals: A1 and v<sup>3</sup>m2. | Because every 21-edo interval is perfect, the quality can be omitted. 21-edo contains 3 rings of 7-edo: an up-ring, a down-ring and a plain-ring. Enharmonic intervals: A1 and v<sup>3</sup>m2. | ||
| Line 750: | Line 662: | ||
| etc. | | etc. | ||
|} | |} | ||
23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A<sup>3</sup>2 or d<sup>3</sup>2 | 23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A<sup>3</sup>2 or d<sup>3</sup>2. | ||
{| class="wikitable" style="text-align:center;" | {| class="wikitable" style="text-align:center;" | ||
| rowspan="4" |'''[[23-edo]]''' | | rowspan="4" |'''[[23-edo]]''' | ||
| Line 898: | Line 810: | ||
== Extremely Large EDOs == | == Extremely Large EDOs == | ||
In theory, every edo can be notated with ups and downs only. For example, in 159-edo, 11/8 above C would be ^<sup>7</sup>F. But large exponents can be avoided if the edo is multi-ring (if the circle of 5ths doesn't include every note). Ups and downs are used within a ring, and lifts and drops (/ and \, see the [[pergen]] | In theory, every edo can be notated with ups and downs only. For example, in 159-edo, 11/8 above C would be ^<sup>7</sup>F. But large exponents can be avoided if the edo is multi-ring (if the circle of 5ths doesn't include every note). Ups and downs are used within a ring, and lifts and drops (/ and \, see the [[pergen]] article) are used to label each ring. 159-edo has 3 rings of 53-edo. Many people who work with 159-edo are familiar with 53-edo, and can read the lifts and drops as small inflections of the familiar 53-edo notation. 11/8 becomes ^^/F. See the [[159edo notation#Ups-and-Downs-based notation|159edo notation]] page for the complete notation. In this situation, "arrow" refers to 1\53 and "slant" refers to 1\159. | ||
Even if the edo isn't multi-ring, lifts and drops can still be used for single EDOsteps and ups and downs can be used for a group of EDOsteps. See [[311edo#Ups and downs notation]]. | |||
==Chords and Chord Progressions== | ==Chords and Chord Progressions== | ||