Kite's ups and downs notation: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 20:06:33 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-03 20:49:11 UTC</tt>.

: The original revision id was <tt>~~584779483~~</tt>.

: The original revision id was <tt>584780549</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 759:

==~~Ups~~ and downs ~~solfege~~==

==__Ups and downs solfege__==

Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:

The initial consonant remains as before: D, R, M, F, S, L and T

The first vowel indicates sharp/flat: a = natural, e = #, i = ##, o = b, u = bb

The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb

The vowels are pronounced as in Spanish or Italian

The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u

Line 793:

Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, 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 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:

...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...

This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^

Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:

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 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.

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 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 can be notated with the # sign meaning harmonically sharp but melodically flat:

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

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's 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 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone.

Line 799:

Line 821:

If the sharp's keyspan is 1 or -1, ups and downs aren't needed to notate rank-2. The keyspan is zero only for perfect EDOs. Thus we can omit fourthward frameworks, and assume for this discussion that K(#) > 1. We need only consider sweet frameworks, excluding those that lie on the side of the heptatonic kite and those that lie on the spine of any kite.

To extend ups and downs to rank-2 tunings, the up symbol is ~~given~~ 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. For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. ~~The interval~~ is ~~always a 2nd~~. The genspan is calculated from the keyspans:

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.

For example, in the 22-tone framework, up has a genspan of -5, corresponding to a pythagorean minor 2nd of 256/243. Thus C^ is exactly equivalent to Db, because C^ = C + m2 = Db. C^^ is 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. Here's the 22-tone genchain:

Fb = Eb^ = D^^

Cb = Bb^ = A^^

Gb = F^ = E^^

Db = C^ = B^^

Ab = G^ = F#^^

Eb = D^ = C#^^

Bb = A^ = Cbv

F = E^ = Gbv

C = B^ = Dbv

G = F#^ =

|| || || Fb || Eb^ || D^^ ||

|| || || Cb || Bb^ || A^^ ||

|| || || Gb || F^ || E^^ ||

|| || || Db || C^ || B^^ ||

|| || || Ab || G^ || ||

|| || || Eb || D^ || ||

|| || || Bb || A^ || ||

|| || || F || E^ || ||

|| || || C || B^ || ||

|| || || G || F#^ || ||

|| || || D || C#^ || ||

|| || || A || G#^ || ||

|| || || E || D#^ || ||

|| || || B || A#^ || ||

|| || || F# || E#^ || ||

|| || || C# || B#^ || ||

|| || || G# || || ||

|| || || D# || || ||

|| || || A# || || ||

|| || || E# || || ||

|| || || B# || || ||

|| || || Fx || || ||

|| || || Cx || || ||

|| || || Gx || || ||

|| || || Dx || || ||

The genspan is calculated from the keyspans:

K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)

Line 945:

Line 1,010:

fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" />

<img src="/file/view/The%20fifth%20of%20EDOs%205-53.png/570450231/800x1035/The%20fifth%20of%20EDOs%205-53.png" alt="The fifth of EDOs 5-53.png" title="The fifth of EDOs 5-53.png" style="height: 1035px; width: 800px;" />

This is in addition to the trivial EDOs, 1, 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.

Line 1,161:

Line 1,226:

<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" />

<img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" />

<h2 id="toc3"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc3"><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /></h2>

<h2 id="toc4"> </h2>

<h2 id="toc5"> </h2>

Line 2,909:

Line 2,974:

<h2 id="toc17"><a name="Summary of EDO notation-Ups and downs solfege"></a>Ups and downs solfege</h2>

<h2 id="toc17"><a name="Summary of EDO notation-Ups and downs solfege"></a>Ups and downs solfege</h2>

Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:

The initial consonant remains as before: D, R, M, F, S, L and T

The first vowel indicates sharp/flat: a = natural, e = #, i = ##, o = b, u = bb

The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb

The vowels are pronounced as in Spanish or Italian

The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u

Line 2,943:

Line 3,008:

Ups and downs can be extended to rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)

For rank-2 scales to work with a given framework, the keyspans of the generator and the period must be coprime. I.e. the framework's node must not be on the spine of a kite. For example, 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 15edo, because the 5th's keyspan is 9. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12edo (3 or 4 not coprime with 12), but compatible with 24edo (7 coprime with 24).

Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:

...Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...

This genchain suffices to name each note, but the note names will often run out of order. For example, in 22-tone, we might have C Db B# C# D, or C Db Ebb C# D. So ups and downs are needed to allow alternate names for each note. The first example becomes C C^

Fifth-generated rank-2 tunings can be written without ups and downs in any EDO on the side of the 4\7 kite:

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 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.