Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584791911 - Original comment: ** |
Wikispaces>TallKite **Imported revision 587606199 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-07-22 21:56:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>587606199</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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12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C | 12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C | ||
12-tone genchain Ab to C#: C C# D Eb E F F# G Ab A Bb B C | 12-tone genchain Ab to C#: C C# D Eb E F F# G **Ab** A Bb B C | ||
12-tone genchain C to E#: C C# D D# E E# F# G G# A A# B C | 12-tone genchain C to E#: C C# D **D#** E **E#** F# G G# A **A#** B C | ||
In __12edo__, C# and Db are identical, but in __12-tone__, they may not be, and usually aren't. | In __12edo__, C# and Db are identical, but in __12-tone__, they may not be, and usually aren't. | ||
19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C | 19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C | ||
19-tone genchain Fb to A#: C C# Db D D# Eb E Fb F F# Gb G G# Ab A A# Bb B Cb C | 19-tone genchain Fb to A#: C C# Db D D# Eb E **Fb** F F# Gb G G# Ab A A# Bb B **Cb** C | ||
19-tone genchain F to Ax: C C# Cx D D# Dx E E# F F# Fx G G# Gx A A# Ax B B# C | 19-tone genchain F to Ax: C C# **Cx** D D# **Dx** E E# F F# **Fx** G G# **Gx** A A# **Ax** B B# C | ||
Fourthward frameworks use a genchain with fourthward sharps: | Fourthward frameworks use a genchain with fourthward sharps: | ||
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16-tone genchain D# to Ab: C Db D D# E E# Fb F Gb G Ab A A# B B# Cb C | 16-tone genchain D# to Ab: C Db D D# E E# Fb F Gb G Ab A A# B B# Cb C | ||
16-tone genchain F# to Cb: C C# D D# E E# Fb F F# G G# A A# B B# Cb C | 16-tone genchain F# to Cb: C **C#** D D# E E# Fb F **F#** G **G#** A A# B B# Cb C | ||
Alternatively, the meanings of # and b can be reversed, so that # has a positive genspan but a negative keyspan: | Alternatively, the meanings of # and b can be reversed, so that # has a positive genspan but a negative keyspan: | ||
16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C | 16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C | ||
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C | 16-tone genchain Fb to C#: C **Cb** D Db E Eb F# F **Fb** G **Gb** A Ab B Bb C# C | ||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24). | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24). | ||
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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 generator chain, or **genchain**, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain. | 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 generator chain, or **genchain**, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain. | ||
For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone. | Every node on the Tree of Kites, other than the spinal nodes, 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 13\22 node is on the side of the 10\17 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the __right__ side of the 10\17 kite, we know that 17 __fifths__ add up to 1\22. Because it's on the __left__ side of the 3\5 kite, 5 __fourths__ add up to 1\22. Between the two, choose the interval with smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone. | ||
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D. | The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D. | ||
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<br /> | <br /> | ||
12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C<br /> | 12-tone genchain Eb to G#: C C# D Eb E F F# G G# A Bb B C<br /> | ||
12-tone genchain Ab to C#: C C# D Eb E F F# G Ab A Bb B C<br /> | 12-tone genchain Ab to C#: C C# D Eb E F F# G <strong>Ab</strong> A Bb B C<br /> | ||
12-tone genchain C to E#: C C# D D# E E# F# G G# A A# B C<br /> | 12-tone genchain C to E#: C C# D <strong>D#</strong> E <strong>E#</strong> F# G G# A <strong>A#</strong> B C<br /> | ||
<br /> | <br /> | ||
In <u>12edo</u>, C# and Db are identical, but in <u>12-tone</u>, they may not be, and usually aren't.<br /> | In <u>12edo</u>, C# and Db are identical, but in <u>12-tone</u>, they may not be, and usually aren't.<br /> | ||
<br /> | <br /> | ||
19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C<br /> | 19-tone genchain Gb to B#: C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B B# C<br /> | ||
19-tone genchain Fb to A#: C C# Db D D# Eb E Fb F F# Gb G G# Ab A A# Bb B Cb C<br /> | 19-tone genchain Fb to A#: C C# Db D D# Eb E <strong>Fb</strong> F F# Gb G G# Ab A A# Bb B <strong>Cb</strong> C<br /> | ||
19-tone genchain F to Ax: C C# Cx D D# Dx E E# F F# Fx G G# Gx A A# Ax B B# C<br /> | 19-tone genchain F to Ax: C C# <strong>Cx</strong> D D# <strong>Dx</strong> E E# F F# <strong>Fx</strong> G G# <strong>Gx</strong> A A# <strong>Ax</strong> B B# C<br /> | ||
<br /> | <br /> | ||
Fourthward frameworks use a genchain with fourthward sharps:<br /> | Fourthward frameworks use a genchain with fourthward sharps:<br /> | ||
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<br /> | <br /> | ||
16-tone genchain D# to Ab: C Db D D# E E# Fb F Gb G Ab A A# B B# Cb C<br /> | 16-tone genchain D# to Ab: C Db D D# E E# Fb F Gb G Ab A A# B B# Cb C<br /> | ||
16-tone genchain F# to Cb: C C# D D# E E# Fb F F# G G# A A# B B# Cb C<br /> | 16-tone genchain F# to Cb: C <strong>C#</strong> D D# E E# Fb F <strong>F#</strong> G <strong>G#</strong> A A# B B# Cb C<br /> | ||
<br /> | <br /> | ||
Alternatively, the meanings of # and b can be reversed, so that # has a positive genspan but a negative keyspan:<br /> | Alternatively, the meanings of # and b can be reversed, so that # has a positive genspan but a negative keyspan:<br /> | ||
16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C<br /> | 16-tone genchain Db to A#: C D# D Db E Eb F# F G# G A# A Ab B Bb C# C<br /> | ||
16-tone genchain Fb to C#: C Cb D Db E Eb F# F Fb G Gb A Ab B Bb C# C<br /> | 16-tone genchain Fb to C#: C <strong>Cb</strong> D Db E Eb F# F <strong>Fb</strong> G <strong>Gb</strong> A Ab B Bb C# C<br /> | ||
<br /> | <br /> | ||
For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).<br /> | For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. These correspond to open EDOs. Each node in the Stern-Brocot chart is formed by these two keyspans, thus this node must not be on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone because the fifth's keyspan is 7, and 7 is coprime with 12. But neither are compatible with 15-tone, because the fifth's keyspan becomes 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone (3 or 4 isn't coprime with 12), but compatible with 24-tone (7 is coprime with 24).<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 generator chain, or <strong>genchain</strong>, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain.<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 generator chain, or <strong>genchain</strong>, one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding up the genspans of the sharps, flats, ups and/or downs attached to a note, we can determine the exact location of the note on the genchain.<br /> | ||
<br /> | <br /> | ||
For example, in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.<br /> | Every node on the Tree of Kites, other than the spinal nodes, 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 13\22 node is on the side of the 10\17 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> side of the 10\17 kite, we know that 17 <u>fifths</u> add up to 1\22. Because it's on the <u>left</u> side of the 3\5 kite, 5 <u>fourths</u> add up to 1\22. Between the two, choose the interval with smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 22-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. And C^^ = C^ + m2 = (C + m2)^, exactly equivalent to Db^. However, C^^ is not equivalent to Dvv, even though they occuy the same key on the keyboard, just as C# may not equal Db in 12-tone.<br /> | ||
<br /> | <br /> | ||
The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br /> | The usual genchain note names will run out of order when mapped to the 22-tone framework. For example, we might have C Db B# C# D. So ups and downs are used to provide alternate names for each note. It becomes C C^ C#v C# D, or equivalently C Db Dvv Dv D. The B# might instead be tuned Ebb, giving us C Db Ebb C# D. This could be written either C Db Db^ Dv D or C C^ C^^ C# D.<br /> | ||