Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 627987423 - Original comment: ** |
Wikispaces>TallKite **Imported revision 628164927 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-31 04:32:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>628164927</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | ||
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 B#v or even B^^ ("B double-up"). However avoid using both C# and Db, as the ascending Db-C# | 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 B#v or even B^^ ("B double-up"). However avoid using both C# and Db, as the ascending Db-C# interval appears descending. | ||
__**Interval arithmetic**__ | __**Interval arithmetic**__ | ||
22-EDO interval arithmetic works out very neatly. Ups and downs are just added in: | 22-EDO interval arithmetic works out very neatly. Ups and downs are just added in: | ||
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=__**Other EDOs**__= | =__**Other EDOs**__= | ||
The up symbol means "sharpened by one | The up symbol means "sharpened by one edo-step" in any edo that uses them. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47edo. | ||
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | ||
**supersharp** = EDOs, with fifths wider than 720¢ | |||
**pentatonic** = EDOs, with a fifth = 720¢ | |||
**regular** = EDOs, with a fifth that hits the "sweet spot" between 720¢ and 686¢ | |||
**perfect** = EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | |||
**superflat** = EDOs, with a fifth less than 686¢ | |||
This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | This is in addition to the **trivial** EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | ||
[[image:The Scale Tree.png width="800" height="1023"]] | [[image:The Scale Tree.png width="800" height="1023"]] | ||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a **spinal** node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others. | ||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs. | Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs. | ||
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As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs. | ||
22-EDO is a sharp-3 edo because a sharp equals 3 ups. 17-EDO is sharp-2. | |||
The mid symbol "~" means "exactly midway between major and minor", hence neutral. In other words, mid is a quality like major or perfect. This only applies to EDOs in which the sharpness is an even number. For example, every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.) is a sharp-2 EDO, upminor equals downmajor, and "mid" replaces both terms. 20edo, 27edo, 34edo, 41edo, etc., are sharp-4 EDOs, and mid replaces both double-upminor and double-downmajor. 11-edo and 18b-edo are flat-2 EDOs, and mid replaces upmajor and downminor. | |||
The chain of fifths in | There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) These five edos are sharp-0 edos, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals. | ||
The chain of fifths in perfect EDOs (3/2 maps to 4\7) is a circle of 7 notes: | |||
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc. | P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc. | ||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc. | F - C - G - D - A - E - B - F - C - G - D - A - E - B etc. | ||
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21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# would look like a perfect fifth. | ||
The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth. | The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth. | ||
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There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, double-ups and double-downs would be necessary. | There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, double-ups and double-downs would be necessary. | ||
0-8-13 in C has C E & G, and is written | 0-8-13 in C has C E & G, and is written C and pronounced "C" or "C major". | ||
0-7-13 = C Ev G is written | 0-7-13 = C Ev G is written C.v, spoken as "C downmajor" or "C dot down". | ||
The period is needed because | The period is needed because Cv, spoken as "C down", is either a note, or a major chord Cv Ev Gv. | ||
0-6-13 = C Eb^ G | 0-6-13 = C Eb^ G = C.^m, "C upminor" | ||
0-5-13 = C Eb G | 0-5-13 = C Eb G = Cm, "C minor" | ||
The period isn't needed in the last chord name because there's no ups or downs immediately after the note name. | The period isn't needed in the last chord name because there's no ups or downs immediately after the note name. | ||
0-8-13-18 = C E G Bb | 0-8-13-18 = C E G Bb = C7, "C seven", a standard C7 chord with a M3 and a m7. | ||
0-7-13-18 = C Ev G Bb | 0-7-13-18 = C Ev G Bb = C7(v3), "C seven, down third". The altered note or notes are in parentheses. | ||
0-8-13-21 = C E G B | 0-8-13-21 = C E G B = CM7, "C major seven". | ||
0-7-13-20 = C Ev G Bv | 0-7-13-20 = C Ev G Bv = C.vM7, "C downmajor seven". The down symbol affects both the 3rd and the 7th. | ||
Often the root of a chord will not be a mid note. The root in the next two examples is Cv. | Often the root of a chord will not be a mid note. The root in the next two examples is Cv. | ||
0-8-13-21 = Cv Ev Gv Bv | 0-8-13-21 = Cv Ev Gv Bv = Cv.M7, "C down, major seven" | ||
To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause. | To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause. | ||
0-7-13-20 = Cv Evv Gv Bvv | 0-7-13-20 = Cv Evv Gv Bvv = Cv.vM7, "C down, downmajor seven". | ||
Sus chords: as in conventional notation, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Certain larger edos might have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception. | Sus chords: as in conventional notation, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Certain larger edos might have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception. | ||
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A7(v3) = "A-seven down-three" | A7(v3) = "A-seven down-three" | ||
Here are all possible 22edo chord roots in relative notation: | |||
I | I | ||
^I or bII | ^I or bII | ||
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=__Chord names in other EDOs__= | =__Chord names in other EDOs__= | ||
In chord names, the mid symbol "~" means "exactly midway between major and minor", hence neutral. This only applies to even- | In chord names, the mid symbol "~" means "exactly midway between major and minor", hence neutral. This only applies to even-sharpness edos. In sharp-2 edos (10, 17, 24, etc.), upminor equals downmajor, and "mid" replaces both terms. In sharp-4 edos (20, 27, 34, etc.), mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces both upmajor and downminor. | ||
Alterations are enclosed in parentheses, additions never are. | Alterations are enclosed in parentheses, additions never are. | ||
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=__**Summary of EDO notation**__= | =__**Summary of EDO notation**__= | ||
== | ==__Regular EDOs__== | ||
(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher) | (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher) | ||
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
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== | ==__Perfect EDOs__== | ||
(7, 14, 21, 28 and 35) | (7, 14, 21, 28 and 35) | ||
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc. | All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc. | ||
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== | ==__Superflat EDOs__== | ||
(9, 11, 13b, 16, 18b and 23) | (9, 11, 13b, 16, 18b and 23) | ||
If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
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== | ==__Pentatonic EDOs__== | ||
(5, 10, 15, 20, 25 and 30) | (5, 10, 15, 20, 25 and 30) | ||
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc. | ||
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==__** | ==__**Supersharp EDOs**__== | ||
(8, 11b, 13 and 18) | (8, 11b, 13 and 18) | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | ||
Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So | Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So this notation is __**not recommended for edos**__ except as an alternate, composer-oriented notation. | ||
For all EDOs with | For all EDOs with sharpness 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all sharp-2 and sharp-5 edos, the natural generator is a 3rd. For sharp-3 and sharp-4, it's a 2nd. For sharp-6, -7 or -8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72. | ||
__** | __**Sharp-2 edos:**__ | ||
genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc. | genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc. | ||
Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C# | Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C# | ||
| Line 1,046: | Line 1,035: | ||
etc. | etc. | ||
__** | __**Sharp-3 edos**__**:** | ||
genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc. | genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc. | ||
D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc. | D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc. | ||
| Line 1,060: | Line 1,049: | ||
etc. | etc. | ||
__** | __**Sharp-4 edos**__**:** | ||
genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc. | genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc. | ||
Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc. | Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc. | ||
| Line 1,074: | Line 1,063: | ||
etc. | etc. | ||
__** | __**Sharp-5 edos:**__ | ||
genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc. | genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc. | ||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb | ||
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<span style="font-size: 150%;">This section is somewhat obsolete, see the [[pergen|pergens]] page instead.</span> | <span style="font-size: 150%;">This section is somewhat obsolete, see the [[pergen|pergens]] page instead.</span> | ||
Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about **frameworks** like 12-tone. The generator chain is called a **genchain**. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite ( | Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about **frameworks** like 12-tone. The generator chain is called a **genchain**. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1): | ||
12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C | 12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C | ||
| Line 1,102: | Line 1,091: | ||
For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't. | For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't. | ||
All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with | All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with sharpness > 1 or < -1. If these are notated without ups and downs, the notes run out of order: | ||
17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C | 17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C | ||
| Line 1,149: | Line 1,138: | ||
||= ||= || || || etc. ||= ||= || | ||= ||= || || || etc. ||= ||= || | ||
The 22-tone keyboard, with alternate tunings for the black keys | The 22-tone keyboard, with alternate tunings for the black keys: | ||
||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||
||= 0 ||= 0 ||= C ||= ||= || | ||= 0 ||= 0 ||= C ||= ||= || | ||
| Line 1,193: | Line 1,182: | ||
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1) | ||
K(#) = c, K(b) = -c (c = | K(#) = c, K(b) = -c (c = sharpness = keyspan of a sharp = how many keys wide aug1 is. For 22-tone, c = 3) | ||
K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out) | K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out) | ||
| Line 1,330: | Line 1,319: | ||
The scale: C C#v D Ev F F#v G Av A Bv C | The scale: C C#v D Ev F F#v G Av A Bv C | ||
| Line 1,437: | Line 1,351: | ||
etc.</pre></div> | etc.</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextTocRule:36:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 1em;"><a href="#A 22edo example">A 22edo example</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 1em;"><a href="#Other EDOs">Other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#EDOs, with a fifth"> EDOs, with a fifth </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#EDOs, with a fifth"> EDOs, with a fifth </a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#x22edo Chord Names">22edo Chord Names</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#Chord names in other EDOs">Chord names in other EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 1em;"><a href="#Cross-EDO considerations">Cross-EDO considerations</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 1em;"><a href="#Scale Fragments">Scale Fragments</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 1em;"><a href="#Summary of EDO notation">Summary of EDO notation</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Regular EDOs">Regular EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Perfect EDOs">Perfect EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Superflat EDOs">Superflat EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Pentatonic EDOs">Pentatonic EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Summary of EDO notation-Supersharp EDOs">Supersharp EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 1em;"><a href="#Natural Generators">Natural Generators</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: 8ve Periods">Rank-2 Scales: 8ve Periods</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 1em;"><a href="#Rank-2 Scales: Non-8ve Periods">Rank-2 Scales: Non-8ve Periods</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 1em;"><a href="#Ups and downs solfege">Ups and downs solfege</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --></div> | ||
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="A 22edo example"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>A 22edo example</u></h1> | |||
<br /> | <br /> | ||
Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs. When extended with highs and lows, it works with almost all rank 2 tunings (see the <a class="wiki_link" href="/pergen">pergens</a> page). It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;.<br /> | Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works with almost all EDOs. When extended with highs and lows, it works with almost all rank 2 tunings (see the <a class="wiki_link" href="/pergen">pergens</a> page). It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;.<br /> | ||
| Line 1,473: | Line 1,388: | ||
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | ||
<br /> | <br /> | ||
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 B#v or even B^^ (&quot;B double-up&quot;). However avoid using both C# and Db, as the ascending Db-C# | 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 B#v or even B^^ (&quot;B double-up&quot;). However avoid using both C# and Db, as the ascending Db-C# interval appears descending.<br /> | ||
<br /> | <br /> | ||
<u><strong>Interval arithmetic</strong></u><br /> | <u><strong>Interval arithmetic</strong></u><br /> | ||
<br /> | <br /> | ||
22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:<br /> | 22-EDO interval arithmetic works out very neatly. Ups and downs are just added in:<br /> | ||
| Line 1,537: | Line 1,440: | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Other EDOs</strong></u></h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Other EDOs</strong></u></h1> | ||
<br /> | <br /> | ||
The up symbol means &quot;sharpened by one | The up symbol means &quot;sharpened by one edo-step&quot; in any edo that uses them. The size in cents of the up changes greatly depending on the edo, from 120¢ in 10-edo to ~17¢ in 72-edo. The sharp symbol's cents size also depends on the edo, ranging from 240¢ in 5-edo to ~26¢ in 47edo.<br /> | ||
<br /> | <br /> | ||
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | ||
& | <strong>supersharp</strong> = EDOs, with fifths wider than 720¢<br /> | ||
& | <strong>pentatonic</strong> <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="EDOs, with a fifth"></a><!-- ws:end:WikiTextHeadingRule:4 --> EDOs, with a fifth </h1> | ||
& | 720¢<br /> | ||
& | <strong>regular</strong> = EDOs, with a fifth that hits the &quot;sweet spot&quot; between 720¢ and 686¢<br /> | ||
& | <strong>perfect</strong> <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="EDOs, with a fifth"></a><!-- ws:end:WikiTextHeadingRule:6 --> EDOs, with a fifth </h1> | ||
four sevenths of an octave = 4\7 = 686¢<br /> | |||
<strong>superflat</strong> = EDOs, with a fifth less than 686¢<br /> | |||
<br /> | <br /> | ||
This is in addition to the trivial EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | This is in addition to the <strong>trivial</strong> EDOs, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:2882:&lt;img src=&quot;/file/view/The%20Scale%20Tree.png/623953169/800x1023/The%20Scale%20Tree.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1023px; width: 800px;&quot; /&gt; --><img src="/file/view/The%20Scale%20Tree.png/623953169/800x1023/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1023px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2882 --><br /> | ||
<br /> | <br /> | ||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are <strong>kites</strong>. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a <strong>spinal</strong> node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
<br /> | <br /> | ||
Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.<br /> | Every EDO with a node on the head or either side of the heptatonic kite (7, 9, 12, 16, 19, 23, etc.) can be notated heptatonically without using ups and downs. All others require ups and downs. Likewise the pentatonic kite, minus the spine, contains the only EDOs that can be notated pentatonically without ups and downs.<br /> | ||
| Line 1,560: | Line 1,465: | ||
As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | As we've seen, 19-EDO doesn't require ups and downs. Let the keyspan of the octave in an EDO be K1 and the keyspan of the fifth be K2. For example, in 12-EDO, K1 = 12 and K2 = 7. The stepspan is one less than the degree. For our usual heptatonic framework, the stepspan of the octave S1 is 7 and the stepspan of the fifth S2 is 4. In order for ups and downs to be unnecessary, S1 * K2 - S2 * K1 = +/-1. Examples of EDOs that don't need ups and downs are 5, 12, 19, 26, 33, 40, etc. (every 7th EDO). There are 4 other such EDOs, 7, 9, 16 and 23. All other EDOs need ups and downs.<br /> | ||
<br /> | <br /> | ||
22-EDO is a sharp-3 edo because a sharp equals 3 ups. 17-EDO is sharp-2.<br /> | |||
<br /> | <br /> | ||
The mid symbol &quot;~&quot; means &quot;exactly midway between major and minor&quot;, hence neutral. In other words, mid is a quality like major or perfect. This only applies to EDOs in which the sharpness is an even number. For example, every seventh EDO (10edo, 17edo, 24edo, 31edo, etc.) is a sharp-2 EDO, upminor equals downmajor, and &quot;mid&quot; replaces both terms. 20edo, 27edo, 34edo, 41edo, etc., are sharp-4 EDOs, and mid replaces both double-upminor and double-downmajor. 11-edo and 18b-edo are flat-2 EDOs, and mid replaces upmajor and downminor.<br /> | |||
<br /> | <br /> | ||
The chain of fifths | There are three special cases to be addressed. The first special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) These five edos are sharp-0 edos, and all intervals are perfect. The scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.<br /> | ||
<br /> | |||
The chain of fifths in perfect EDOs (3/2 maps to 4\7) is a circle of 7 notes:<br /> | |||
P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | ||
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br /> | F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.<br /> | ||
| Line 1,571: | Line 1,478: | ||
21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing, because the upfifth B - F# would look like a perfect fifth.<br /> | ||
<br /> | <br /> | ||
The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.<br /> | The second special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo and 18edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. 13edo and 18edo can be notated by using the 2nd best fifth.<br /> | ||
| Line 1,793: | Line 1,700: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="x22edo Chord Names"></a><!-- ws:end:WikiTextHeadingRule:8 --><u>22edo Chord Names</u></h1> | ||
<br /> | <br /> | ||
Ups and downs allow us to name any chord easily. The quality of an interval (major, minor, perfect, etc.) is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | Ups and downs allow us to name any chord easily. The quality of an interval (major, minor, perfect, etc.) is defined by its position on the chain of 5ths. Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | ||
| Line 1,828: | Line 1,735: | ||
There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, double-ups and double-downs would be necessary.<br /> | There are some alternate names. However double-ups and double-downs are to be avoided in 22edo. Thus 7\22 would never be written ^^m3. In larger edos, double-ups and double-downs would be necessary.<br /> | ||
<br /> | <br /> | ||
0-8-13 in C has C E &amp; G, and is written | 0-8-13 in C has C E &amp; G, and is written C and pronounced &quot;C&quot; or &quot;C major&quot;.<br /> | ||
0-7-13 = C Ev G is written | 0-7-13 = C Ev G is written C.v, spoken as &quot;C downmajor&quot; or &quot;C dot down&quot;.<br /> | ||
The period is needed because | The period is needed because Cv, spoken as &quot;C down&quot;, is either a note, or a major chord Cv Ev Gv.<br /> | ||
0-6-13 = C Eb^ G | 0-6-13 = C Eb^ G = C.^m, &quot;C upminor&quot;<br /> | ||
0-5-13 = C Eb G | 0-5-13 = C Eb G = Cm, &quot;C minor&quot;<br /> | ||
The period isn't needed in the last chord name because there's no ups or downs immediately after the note name.<br /> | The period isn't needed in the last chord name because there's no ups or downs immediately after the note name.<br /> | ||
<br /> | <br /> | ||
0-8-13-18 = C E G Bb | 0-8-13-18 = C E G Bb = C7, &quot;C seven&quot;, a standard C7 chord with a M3 and a m7.<br /> | ||
0-7-13-18 = C Ev G Bb | 0-7-13-18 = C Ev G Bb = C7(v3), &quot;C seven, down third&quot;. The altered note or notes are in parentheses.<br /> | ||
<br /> | <br /> | ||
0-8-13-21 = C E G B | 0-8-13-21 = C E G B = CM7, &quot;C major seven&quot;.<br /> | ||
0-7-13-20 = C Ev G Bv | 0-7-13-20 = C Ev G Bv = C.vM7, &quot;C downmajor seven&quot;. The down symbol affects both the 3rd and the 7th.<br /> | ||
Often the root of a chord will not be a mid note. The root in the next two examples is Cv.<br /> | Often the root of a chord will not be a mid note. The root in the next two examples is Cv.<br /> | ||
0-8-13-21 = Cv Ev Gv Bv | 0-8-13-21 = Cv Ev Gv Bv = Cv.M7, &quot;C down, major seven&quot;<br /> | ||
To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.<br /> | To distinguish between C.vM7 and Cv.M7, one has to pronounce the period with a small pause.<br /> | ||
0-7-13-20 = Cv Evv Gv Bvv | 0-7-13-20 = Cv Evv Gv Bvv = Cv.vM7, &quot;C down, downmajor seven&quot;.<br /> | ||
<br /> | <br /> | ||
Sus chords: as in conventional notation, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Certain larger edos might have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception.<br /> | Sus chords: as in conventional notation, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Certain larger edos might have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the superflat EDOs below for an exception.<br /> | ||
| Line 1,892: | Line 1,799: | ||
A7(v3) = &quot;A-seven down-three&quot;<br /> | A7(v3) = &quot;A-seven down-three&quot;<br /> | ||
<br /> | <br /> | ||
Here are all possible 22edo chord roots in relative notation:<br /> | |||
<br /> | |||
I<br /> | I<br /> | ||
^I or bII<br /> | ^I or bII<br /> | ||
| Line 1,923: | Line 1,829: | ||
IV7(4) = &quot;four-seven sus-four&quot;<br /> | IV7(4) = &quot;four-seven sus-four&quot;<br /> | ||
II7(v3) = &quot;two-seven down-three&quot;<br /> | II7(v3) = &quot;two-seven down-three&quot;<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:10 --><u>Chord names in other EDOs</u></h1> | ||
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In chord names, the mid symbol &quot;~&quot; means &quot;exactly midway between major and minor&quot;, hence neutral. This only applies to even- | In chord names, the mid symbol &quot;~&quot; means &quot;exactly midway between major and minor&quot;, hence neutral. This only applies to even-sharpness edos. In sharp-2 edos (10, 17, 24, etc.), upminor equals downmajor, and &quot;mid&quot; replaces both terms. In sharp-4 edos (20, 27, 34, etc.), mid replaces both double-upminor and double-downmajor. In 11-edo and 18b-edo, mid replaces both upmajor and downminor.<br /> | ||
<br /> | <br /> | ||
Alterations are enclosed in parentheses, additions never are.<br /> | Alterations are enclosed in parentheses, additions never are.<br /> | ||
| Line 2,236: | Line 2,142: | ||
0-3-6-9 = D Fv A Cv = D.~7 = &quot;D dot mid seven&quot;, or D Fv A Bb = D~,b6 = &quot;D mid flat-six&quot;<br /> | 0-3-6-9 = D Fv A Cv = D.~7 = &quot;D dot mid seven&quot;, or D Fv A Bb = D~,b6 = &quot;D mid flat-six&quot;<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Cross-EDO considerations"></a><!-- ws:end:WikiTextHeadingRule:12 --><strong><u>Cross-EDO considerations</u></strong></h1> | ||
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In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.<br /> | In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="Scale Fragments"></a><!-- ws:end:WikiTextHeadingRule:14 --><u><strong>Scale Fragments</strong></u></h1> | ||
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To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:<br /> | To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h1&gt; --><h1 id="toc8"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:16 --><u><strong>Summary of EDO notation</strong></u></h1> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Summary of EDO notation-Regular EDOs"></a><!-- ws:end:WikiTextHeadingRule:18 --><u>Regular EDOs</u></h2> | ||
(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)<br /> | (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)<br /> | ||
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Summary of EDO notation-Perfect EDOs"></a><!-- ws:end:WikiTextHeadingRule:20 --><u>Perfect EDOs</u></h2> | ||
(7, 14, 21, 28 and 35)<br /> | (7, 14, 21, 28 and 35)<br /> | ||
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.<br /> | All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="Summary of EDO notation-Superflat EDOs"></a><!-- ws:end:WikiTextHeadingRule:22 --><u>Superflat EDOs</u></h2> | ||
(9, 11, 13b, 16, 18b and 23)<br /> | (9, 11, 13b, 16, 18b and 23)<br /> | ||
If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Summary of EDO notation-Pentatonic EDOs"></a><!-- ws:end:WikiTextHeadingRule:24 --><u>Pentatonic EDOs</u></h2> | ||
(5, 10, 15, 20, 25 and 30)<br /> | (5, 10, 15, 20, 25 and 30)<br /> | ||
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Summary of EDO notation-Supersharp EDOs"></a><!-- ws:end:WikiTextHeadingRule:26 --><u><strong>Supersharp EDOs</strong></u></h2> | ||
(8, 11b, 13 and 18)<br /> | (8, 11b, 13 and 18)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Natural Generators"></a><!-- ws:end:WikiTextHeadingRule:28 --><u>Natural Generators</u></h1> | ||
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Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth.<br /> | Ups and downs can be avoided entirely by using some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth.<br /> | ||
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E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | ||
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Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So | Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So this notation is <u><strong>not recommended for edos</strong></u> except as an alternate, composer-oriented notation.<br /> | ||
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For all EDOs with | For all EDOs with sharpness 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all sharp-2 and sharp-5 edos, the natural generator is a 3rd. For sharp-3 and sharp-4, it's a 2nd. For sharp-6, -7 or -8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.<br /> | ||
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<u><strong> | <u><strong>Sharp-2 edos:</strong></u><br /> | ||
genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.<br /> | genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.<br /> | ||
Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#<br /> | Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#<br /> | ||
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etc.<br /> | etc.<br /> | ||
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<u><strong> | <u><strong>Sharp-3 edos</strong></u><strong>:</strong><br /> | ||
genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.<br /> | genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.<br /> | ||
D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.<br /> | D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.<br /> | ||
| Line 3,976: | Line 3,882: | ||
etc.<br /> | etc.<br /> | ||
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<u><strong> | <u><strong>Sharp-4 edos</strong></u><strong>:</strong><br /> | ||
genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.<br /> | genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.<br /> | ||
Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.<br /> | Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.<br /> | ||
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etc.<br /> | etc.<br /> | ||
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<u><strong> | <u><strong>Sharp-5 edos:</strong></u><br /> | ||
genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.<br /> | genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.<br /> | ||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h1&gt; --><h1 id="toc15"><a name="Rank-2 Scales: 8ve Periods"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Rank-2 Scales: 8ve Periods</u></h1> | ||
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<span style="font-size: 150%;">This section is somewhat obsolete, see the <a class="wiki_link" href="/pergen">pergens</a> page instead.</span><br /> | <span style="font-size: 150%;">This section is somewhat obsolete, see the <a class="wiki_link" href="/pergen">pergens</a> page instead.</span><br /> | ||
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Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about <strong>frameworks</strong> like 12-tone. The generator chain is called a <strong>genchain</strong>. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite ( | Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about <strong>frameworks</strong> like 12-tone. The generator chain is called a <strong>genchain</strong>. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1):<br /> | ||
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12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C<br /> | 12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C<br /> | ||
| Line 4,018: | Line 3,924: | ||
For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.<br /> | For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.<br /> | ||
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All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with | All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:<br /> | ||
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17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C<br /> | 17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C<br /> | ||
| Line 4,565: | Line 4,471: | ||
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The 22-tone keyboard, with alternate tunings for the black keys | The 22-tone keyboard, with alternate tunings for the black keys:<br /> | ||
| Line 4,878: | Line 4,784: | ||
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K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)<br /> | ||
K(#) = c, K(b) = -c (c = | K(#) = c, K(b) = -c (c = sharpness = keyspan of a sharp = how many keys wide aug1 is. For 22-tone, c = 3)<br /> | ||
K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out)<br /> | K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out)<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h1&gt; --><h1 id="toc16"><a name="Rank-2 Scales: Non-8ve Periods"></a><!-- ws:end:WikiTextHeadingRule:32 --><u>Rank-2 Scales: Non-8ve Periods</u></h1> | ||
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<span style="font-size: 24px;">This section is somewhat obsolete, see the <a class="wiki_link" href="/pergen">pergens</a> page instead.</span><br /> | <span style="font-size: 24px;">This section is somewhat obsolete, see the <a class="wiki_link" href="/pergen">pergens</a> page instead.</span><br /> | ||
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The scale: C C#v D Ev F F#v G Av A Bv C<br /> | The scale: C C#v D Ev F F#v G Av A Bv C<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Ups and downs solfege"></a><!-- ws:end:WikiTextHeadingRule:34 --><u>Ups and downs solfege</u></h1> | ||
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Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | ||