Kite's ups and downs notation: Difference between revisions

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== Definition ==

Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that can notate almost every [[EDO|EDO]]. The up symbol "^" and the down symbol "v" indicate raising or lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. ~~half~~-augmented, and the mid 5th (~5) is a ~~half~~-diminished 5th.

Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that can notate almost every [[EDO|EDO]]. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.

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 15-edo and 22-edo, and both EDOs map 81/80 to one EDOstep. Thus if Triyo is tuned to 15-edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22-edo, 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 22-edo by simply writing "22-edo" on the top of the page.

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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.

Relative notation ~~for 22-edo is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - 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.~~

'''Relative notation and interval arithmetic'''

~~'''Enharmonic equivalents'''~~

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.

~~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 vB# or even~~ ^^~~B (double~~-~~up B)~~.

~~'''Staff Notation''~~'

For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't. Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo 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. 24-edo 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. 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].

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

~~'''Interval arithmetic'''~~

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

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

|+

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|sum of two intervals

|M2 + M2 = M3

|M2 + ^M2 = ^M3

|^M2 + M2 = ^M3

|vM2 ~~+ M2~~ = vM3

|M2 + vM2 = vM3

|~~vM2 +~~ ^M2 = M3

|^M2 + vM2 = M3

|vM2 + vM2 = vvM3

|}

'''Enharmonic equivalents'''

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 vB# or even ^^B (double-up B).

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 intervals are like 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 enharmonics 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 enharmonic intervals are needed to define the notation: v3A1 and vm2. Either interval 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 enharmonic intervals is also an enharmonic interval, for example their sum v4M2.

'''Staff Notation'''

For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't. Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo 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. 24-edo 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. 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].

'''Placement of the up or down'''

It might seem more natural to place the up after the note, for example B^ or Bb^. But the up must come first, to make chord names unambiguous. 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.

==Examples: EDOs 12-24==

Sharp-1, flat-2, etc. refer to the number of EDOsteps made by seven 5ths minus four 8ves.

Sharp-1, flat-2, etc. refer to the number of EDOsteps 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 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 (no ups and downs), if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.

12-edo is sharp-1, thus doesn't need ups and downs.

12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.

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

| rowspan="2" |[[12-edo|'''12-edo''']]

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|P8

|}

There are two ways to notate 13b-edo: 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.

There are two ways to notate 13b-edo: 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. The enharmonic intervals for the former notation are ^3A1 and vM2. For the latter they are v3A1 and vm2.

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

| rowspan="4" |'''[[13-edo|13b-edo]]'''

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|}

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

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

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

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

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|8

|}

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

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

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

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

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| P8

|}

16-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it.

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;"

| rowspan="4" |'''[[16-edo]]'''

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| P8

|}

17-edo is sharp-2 and has mid intervals.

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

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

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

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| P8

|}

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

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

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

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

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| P8

|}

20-edo contains 4 rings of 5-edo: an up-ring, a down-ring, a double-up/down-ring, and a plain-ring.

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

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

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

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| 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.

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 v3m2.

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

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

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| 8

|}

22-edo is sharp-3.

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

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

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

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| P8

|}

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

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

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

| rowspan="2" | '''[[24-edo]]'''

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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.

~~There's~~ one exception: it's pointless to apply a global down to a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.

Every conventional chord can accept a global up or down, with one exception: it's pointless to apply a global down to a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.

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, never use lower case roman numerals for minor chords. Write either vIIm or VIIm, never vii.

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'''Ninth chords:'''

In '''bolded''' chords, the comma is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk '''*''', only the first comma is spoken as "add".

In '''bolded''' chords, the comma is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk '''*''', only the first comma is spoken as "add".

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

|+

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The 2nd vowel is optionally separated from the first by an "h", a "w", or a "y"

Thus vC# is Deo, pronounced as Deho or Dewo or Deyo.

Thus vC# is Deo, pronounced as Deo or Deho or Dewo or Deyo.

This suffices for many but not all EDOs, as some require triple sharps or quadruple ups.

@@ Line 2: / Line 2: @@
 == Definition ==
-Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that can notate almost every [[EDO|EDO]]. The up symbol "^" and the down symbol "v" indicate raising or lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. half-augmented, and the mid 5th (~5) is a half-diminished 5th.
+Ups and Downs (or ^v) is a notation system developed by [[KiteGiedraitis|Kite]] that can notate almost every [[EDO|EDO]]. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.
 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 15-edo and 22-edo, and both EDOs map 81/80 to one EDOstep. Thus if Triyo is tuned to 15-edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22-edo, 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 22-edo by simply writing "22-edo" on the top of the page.
@@ Line 20: / Line 20: @@
 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.
-Relative notation for 22-edo is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - 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.
+'''<u>Relative notation and interval arithmetic</u>'''
-'''<u>Enharmonic equivalents</u>'''
+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.
-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 vB# or even ^^B (double-up B).
-'''<u>Staff Notation</u>'''
-For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't. Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo 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. 24-edo 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. 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].
+A core principle of ups and downs notation is that '''interval arithmetic is always preserved'''. Ups and downs are simply added in:
-<u>'''Interval arithmetic'''</u>
-A core principle of ups and downs notation is that '''interval arithmetic is <u>always</u> preserved'''. Ups and downs are just added in:
 {| class="wikitable" style="text-align:center;"
 |+
@@ Line 57: / Line 49: @@
 |sum of two intervals
 |M2 + M2 = M3
-|M2 + ^M2 = ^M3
+|^M2 + M2 = ^M3
-|vM2 + M2 = vM3
+|M2 + vM2 = vM3
-|vM2 + ^M2 = M3
+|^M2 + vM2 = M3
 |vM2 + vM2 = vvM3
 |}
+'''<u>Enharmonic equivalents</u>'''
+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 vB# or even ^^B (double-up B).
+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 intervals are like 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 enharmonics 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 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.
+'''<u>Staff Notation</u>'''
+For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't. Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo 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. 24-edo 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. 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].
+'''<u>Placement of the up or down</u>'''
+It might seem more natural to place the up after the note, for example B^ or Bb^. But the up must come first, to make chord names unambiguous. 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.
 ==Examples: EDOs 12-24==
-Sharp-1, flat-2, etc. refer to the number of EDOsteps made by seven 5ths minus four 8ves.
+Sharp-1, flat-2, etc. refer to the number of EDOsteps 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 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 (no ups and downs), if notes are permitted to be out of order (e.g. 22edo could have C Db B# C#). But multi-ring edos absolutely require ups and downs.
--edo is sharp-1, thus doesn't need ups and downs.
+-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |[[12-edo|'''12-edo''']]
@@ Line 99: / Line 111: @@
 |P8
 |}
-There are two ways to notate 13b-edo: 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.
+There are two ways to notate 13b-edo: 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. The enharmonic intervals for the former notation are ^<sup>3</sup>A1 and vM2. For the latter they are v<sup>3</sup>A1 and vm2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="4" |'''[[13-edo|13b-edo]]'''
@@ Line 168: / Line 180: @@
 |}
-Because every 14-edo interval is perfect, the quality can be omitted. 14-edo contains 2 rings of 7-edo: an up/down-ring and a plain-ring.
+Because every 14-edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14-edo contains 2 rings of 7-edo: an up/down-ring and a plain-ring. Enharmonic intervals: A1 and vvm2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[14-edo]]'''
@@ Line 204: / Line 216: @@
 |8
 |}
--edo contains 3 rings of 5-edo: an up-ring, a down-ring, and a plain-ring.
+-edo contains 3 rings of 5-edo: an up-ring, a down-ring, and a plain-ring. Enharmonic intervals: v<sup>3</sup>A1 and m2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[15-edo]]'''
@@ Line 242: / Line 254: @@
 |  P8
 |}
--edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it.
+-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;"
 | rowspan="4" |'''[[16-edo]]'''
@@ Line 322: / Line 334: @@
 | P8
 |}
--edo is sharp-2 and has mid intervals.
+-edo is sharp-2 and has mid intervals. Enharmonic intervals: vvA1 and vm2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[17edo|17-edo]]'''
@@ Line 452: / Line 464: @@
 |  P8
 |}
--edo is sharp-1, thus doesn't need ups and downs.
+-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: dd2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[19-edo]]'''
@@ Line 498: / Line 510: @@
 |  P8
 |}
--edo contains 4 rings of 5-edo: an up-ring, a down-ring, a double-up/down-ring, and a plain-ring.
+-edo contains 4 rings of 5-edo: an up-ring, a down-ring, a double-up/down-ring, and a plain-ring. Enharmonic intervals: v<sup>4</sup>A1 and m2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[20-edo]]'''
@@ Line 546: / Line 558: @@
 |  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.
+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.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[21-edo]]'''
@@ Line 596: / Line 608: @@
 |  8
 |}
--edo is sharp-3.
+-edo is sharp-3. Enharmonic intervals: v<sup>3</sup>A1 and vm2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" |'''[[22-edo]]'''
@@ Line 740: / Line 752: @@
 | P8
 |}
--edo contains 2 rings of 12-edo: an up/down-ring and a plain-ring.
+-edo contains 2 rings of 12-edo: an up/down-ring and a plain-ring. Enharmonic intervals: vvA1 and d2.
 {| class="wikitable" style="text-align:center;"
 | rowspan="2" | '''[[24-edo]]'''
@@ Line 789: / Line 801: @@
 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.
-There's one exception: it's pointless to apply a global down to a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.
+Every conventional chord can accept a global up or down, with one exception: it's pointless to apply a global down to a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.
 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, never use lower case roman numerals for minor chords. Write either vIIm or VIIm, never vii.
@@ Line 1,021: / Line 1,033: @@
 <u>'''Ninth chords:'''</u>
-In '''bolded''' chords, the comma is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk '''*''', only the first comma is spoken as "add".
+In '''bolded''' chords, the comma is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk '''*''', only the first comma is spoken as "add".
 {| class="wikitable" style="text-align:center;"
 |+
@@ Line 1,223: / Line 1,235: @@
 The 2nd vowel is optionally separated from the first by an "h", a "w", or a "y"
-Thus vC# is Deo, pronounced as Deho or Dewo or Deyo.
+Thus vC# is Deo, pronounced as Deo or Deho or Dewo or Deyo.
 This suffices for many but not all EDOs, as some require triple sharps or quadruple ups.