Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 599966960 - Original comment: ** |
Wikispaces>TallKite **Imported revision 600699878 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2016-11- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-29 01:38:29 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>600699878</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> | ||
| Line 976: | Line 976: | ||
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. | 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. | ||
Every non-perfect EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", and ups and downs aren't needed. The generator is always perfect, so there's | Every non-perfect EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", major is one EDO-step wider than minor, and ups and downs aren't needed. | ||
The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic. | |||
For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7... | |||
__**Natural generators for 8edo, 11edo, 13edo and 18edo**__ | __**Natural generators for 8edo, 11edo, 13edo and 18edo**__ | ||
| Line 998: | Line 1,002: | ||
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D | D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D | ||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8 | ||
genchain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | genchain of seconds: d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc. | ||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G# | ||
| Line 3,849: | Line 3,853: | ||
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 /> | ||
<br /> | <br /> | ||
Every non-perfect EDO has a &quot;natural&quot; heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means &quot;sharpened by one EDO-step&quot;, and ups and downs aren't needed. The generator is always perfect, so there's | Every non-perfect EDO has a &quot;natural&quot; heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means &quot;sharpened by one EDO-step&quot;, major is one EDO-step wider than minor, and ups and downs aren't needed.<br /> | ||
<br /> | |||
The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic.<br /> | |||
<br /> | |||
For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7...<br /> | |||
<br /> | <br /> | ||
<u><strong>Natural generators for 8edo, 11edo, 13edo and 18edo</strong></u><br /> | <u><strong>Natural generators for 8edo, 11edo, 13edo and 18edo</strong></u><br /> | ||
| Line 3,871: | Line 3,879: | ||
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br /> | D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D<br /> | ||
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8<br /> | ||
genchain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | genchain of seconds: d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.<br /> | ||
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#<br /> | ||
<br /> | <br /> | ||