Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 597276102 - Original comment: Reverted to Oct 17, 2016 9:06 pm** |
Wikispaces>TallKite **Imported revision 597276572 - 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>2016-10-27 03: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-27 03:55:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>597276572</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
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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 except for 8-edo, this notation is __**not recommended**__ except as an alternate, composer-oriented notation. | 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 except for 8-edo, this notation is __**not recommended for edos**__ except as an alternate, composer-oriented notation. | ||
For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 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. | For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 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. | ||
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To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning. | To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a **keyspan** (always +1) but also a **genspan**, which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning. | ||
Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the __right__ (fifthward) side of the 7\12 kite, we know that 12 __fifths__ add up to 1\17. Because it's on the __left__ (fourthward) side of the 3\5 kite, 5 __fourths__ add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up). | Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the __right__ (fifthward) side of the 7\12 kite, we know that 12 __fifths__ add up to 1\17. Because it's on the __left__ (fourthward) side of the 3\5 kite, 5 __fourths__ add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up). | ||
17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C | 17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C | ||
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Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of "up" here is "a half-octave minus a fourth", with the understanding that both the octave and the fourth may be tempered. Normally, the 12-tone half-octave would be A4 or d5, no ups or downs needed. But to notate this tuning, the half-octave must be written ^4 or v5. | Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of "up" here is "a half-octave minus a fourth", with the understanding that both the octave and the fourth may be tempered. Normally, the 12-tone half-octave would be A4 or d5, no ups or downs needed. But to notate this tuning, the half-octave must be written ^4 or v5. | ||
If "up" means "a half-octave minus a fourth" as well as "up one key or fret", a 4th must be one key or fret less than | If "up" means "a half-octave minus a fourth" as well as "up one key or fret", a 4th must be one key or fret less than a half-octave, which only holds for frameworks 10, 12, 14, 16 and 18b. For other frameworks, the period must be named differently. The general rule is, **__the period's name must have at least one up or down, and the generator's name must have none, or vice versa__.** This allows ups and downs to serve "double-duty" as genchain indicators. For 20-tone, the period is ^^4 or vv5. For 22-tone, it's vA4 or ^d5. For 24, 26 and 28, it's ^^4 or vv5. But for now, let's assume the period is ^4 or v5. | ||
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth: | It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth: | ||
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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 /> | ||
<br /> | <br /> | ||
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 except for 8-edo, this notation is <u><strong>not recommended</strong></u> except as an alternate, composer-oriented notation.<br /> | 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 except for 8-edo, this notation is <u><strong>not recommended for edos</strong></u> except as an alternate, composer-oriented notation.<br /> | ||
<br /> | <br /> | ||
For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 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 /> | For all EDOs with chroma 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all chroma-2 and chroma-5 edos, the natural generator is a 3rd. For chroma-3 and chroma-4, it's a 2nd. For chromas 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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To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.<br /> | To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a <strong>keyspan</strong> (always +1) but also a <strong>genspan</strong>, which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.<br /> | ||
<br /> | <br /> | ||
Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).<br /> | Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).<br /> | ||
<br /> | <br /> | ||
17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br /> | 17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C<br /> | ||
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Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &quot;up&quot; here is &quot;a half-octave minus a fourth&quot;, with the understanding that both the octave and the fourth may be tempered. Normally, the 12-tone half-octave would be A4 or d5, no ups or downs needed. But to notate this tuning, the half-octave must be written ^4 or v5.<br /> | Moving up from C to F^ moves up a half-octave. Ups and downs are used (F^ not F#) because F# is on the wrong genchain. It's two steps to the right of E. The exact meaning of &quot;up&quot; here is &quot;a half-octave minus a fourth&quot;, with the understanding that both the octave and the fourth may be tempered. Normally, the 12-tone half-octave would be A4 or d5, no ups or downs needed. But to notate this tuning, the half-octave must be written ^4 or v5.<br /> | ||
<br /> | <br /> | ||
If &quot;up&quot; means &quot;a half-octave minus a fourth&quot; as well as &quot;up one key or fret&quot;, a 4th must be one key or fret less than | If &quot;up&quot; means &quot;a half-octave minus a fourth&quot; as well as &quot;up one key or fret&quot;, a 4th must be one key or fret less than a half-octave, which only holds for frameworks 10, 12, 14, 16 and 18b. For other frameworks, the period must be named differently. The general rule is, <strong><u>the period's name must have at least one up or down, and the generator's name must have none, or vice versa</u>.</strong> This allows ups and downs to serve &quot;double-duty&quot; as genchain indicators. For 20-tone, the period is ^^4 or vv5. For 22-tone, it's vA4 or ^d5. For 24, 26 and 28, it's ^^4 or vv5. But for now, let's assume the period is ^4 or v5.<br /> | ||
<br /> | <br /> | ||
It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth:<br /> | It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth:<br /> | ||