Kite's ups and downs notation: Difference between revisions

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<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>2016-06-04 02:00:22 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 03:56:09 UTC</tt>.

: The original revision id was <tt>~~584785603~~</tt>.

: The original revision id was <tt>584786725</tt>.

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||= 43edo ||= sweet ||= ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||= ||= ||

||= 44ddo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||= ||

||= 45edo ||= sweet ||= ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||= ||= ||

||= 45edo ||= sweet ||= ||= C ||= * ||= C# ||= * ||= * ||= Db ||= * ||= D ||= ||= ||

||= 46edo ||= sweet ||= ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||= ||

||= 47edo ||= sweet ||= ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||= ||= ||

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==__Rank-2 Notation__==

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. Such instruments use a 12-tone framework. 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.)

Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. 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.)

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# ...

...Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx...

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

Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either 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

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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:~~

Fourthward frameworks use a genchain with fourthward sharps:

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

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

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 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. Each node in the Stern-Brocot EDO 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 15edo, because the fifth'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. These two are easily notated without ups and downs:

All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks are incompatible with fifth-generated rank-2 tunings, except for 5-tone and 7-tone. These two are easily notated without ups and downs:

5-tone genchain C to E: C D E G A C

5-tone genchain F to A: C D F G A C

5-tone genchain Eb to G: C Eb F G Bb C

7-tone genchain C to F#: C D E F# G A B C

7-tone genchain Bb to E: C D E F G A Bb C

~~All fifthless frameworks~~ are ~~incompatible with fifth~~-~~generated heptatonic notation~~, ~~since the minor 2nd~~ is ~~a descending interval~~.

The notes selected from the genchain needn't make a continuous chain. Such scales are easily notated:

7-tone: Eb * F C G D A * B = C D Eb F G A B C

5-tone: Bb * C G D * E = C D E G Bb C

12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C

The notes selected DO need to have a 1-to-1 mapping onto the keys of the keyboard, so that every key is tuned to exactly one note.

If the sharp's keyspan is 1 or -1, as with 12-tone, 19-tone, and all fourthward frameworks, ups and downs aren't needed to notate rank-2. They also aren't needed for 5-tone and 7-tone. Since perfect, pentatonic and fifthless frameworks are incompatible, 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.

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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.

~~Here's part of the~~ 22-tone genchain~~. There are more than 22 notes because the genchain is theoretically infinite~~:

The 22-tone genchain:

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

||= genspan from C ||= keyspan from C || || || ||= ||= ||

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

||= -13 ||= 7 || || || Gbb ||= Fb^ ||= Eb^^ ||

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

||= -12 ||= 20 || || || Dbb ||= Cb^ ||= Bb^^ ||

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

||= -11 ||= 11 || || || Abb ||= Gb^ ||= F^^ ||

||= ||= ||= Ab ||= G^ ||= F#^^ ||

||= -10 ||= 2 || || || Ebb ||= Db^ ||= C^^ ||

||= ||= Fbv ||= Eb ||= D^ ||= C#^^ ||

||= -9 ||= 15 || || || Bbb ||= Ab^ ||= G^^ ||

||= ||= Cbv ||= Bb ||= A^ ||= G#^^ ||

||= -8 ||= 6 ||= ||= ||= Fb ||= Eb^ ||= D^^ ||

||= ||= Gbv ||= F ||= E^ ||= A#^^ ||

||= -7 ||= 19 ||= ||= ||= Cb ||= Bb^ ||= A^^ ||

||= ||= Dbv ||= C ||= B^ ||= E#^^ ||

||= -6 ||= 10 ||= ||= ||= Gb ||= F^ ||= E^^ ||

||= ||= Abv ||= G ||= F#^ ||= B#^^ ||

||= -5 ||= 1 ||= ||= ||= Db ||= C^ ||= B^^ ||

||= Fbvv ||= Ebv ||= D ||= C#^ ||= ||

||= -4 ||= 14 ||= ||= ||= Ab ||= G^ ||= F#^^ ||

||= Cbvv ||= Bbv ||= A ||= G#^ ||= ||

||= -3 ||= 5 ||= ||= Fbv ||= Eb ||= D^ ||= C#^^ ||

||= Gbvv ||= Fv ||= E ||= D#^ ||= ||

||= -2 ||= 18 ||= ||= Cbv ||= Bb ||= A^ ||= G#^^ ||

||= Dbvv ||= Cv ||= B ||= A#^ ||= ||

||= -1 ||= 9 ||= ||= Gbv ||= F ||= E^ ||= D#^^ ||

||= Abvv ||= Gv ||= F# ||= E#^ ||= ||

||= 0 ||= 0 ||= ||= Dbv ||= C ||= B^ ||= A#^^ ||

||= Ebvv ||= Dv ||= C# ||= B#^ ||= ||

||= 1 ||= 13 ||= ||= Abv ||= G ||= F#^ ||= E#^^ ||

||= Bbvv ||= Av ||= G# ||= ||= ||

||= 2 ||= 4 ||= Fbvv ||= Ebv ||= D ||= C#^ ||= B#^^ ||

||= Fvv ||= Ev ||= D# ||= ||= ||

||= 3 ||= 17 ||= Cbvv ||= Bbv ||= A ||= G#^ ||= ||

||= Cvv ||= Bv ||= A# ||= ||= ||

||= 4 ||= 8 ||= Gbvv ||= Fv ||= E ||= D#^ ||= ||

||= Gvv ||= F#v ||= E# ||= ||= ||

||= 5 ||= 21 ||= Dbvv ||= Cv ||= B ||= A#^ ||= ||

||= Dvv ||= C#v ||= B# ||= ||= ||

||= 6 ||= 12 ||= Abvv ||= Gv ||= F# ||= E#^ ||= ||

||= Avv ||= G#v ||= Fx ||= ||= ||

||= 7 ||= 3 ||= Ebvv ||= Dv ||= C# ||= B#^ ||= ||

||= Evv ||= D#v ||= Cx ||= ||= ||

||= 8 ||= 16 ||= Bbvv ||= Av ||= G# ||= ||= ||

||= Bvv ||= A#v ||= Gx ||= ||= ||

||= 9 ||= 7 ||= Fvv ||= Ev ||= D# ||= ||= ||

||= F#vv ||= E#v ||= Dx ||= ||= ||

||= 10 ||= 20 ||= Cvv ||= Bv ||= A# ||= ||= ||

||= C#vv ||= B#v ||= Ax ||= ||= ||

||= 11 ||= 11 ||= Gvv ||= F#v ||= E# ||= ||= ||

|| || || etc. || || ||

||= 12 ||= 2 ||= Dvv ||= C#v ||= B# ||= ||= ||

||= 13 ||= 15 ||= Avv ||= G#v ||= Fx ||= ||= ||

||= 14 ||= 6 ||= Evv ||= D#v ||= Cx ||= ||= ||

||= 15 ||= 19 ||= Bvv ||= A#v ||= Gx ||= ||= ||

||= 16 ||= 10 ||= F#vv ||= E#v ||= Dx ||= ||= ||

||= 17 ||= 1 ||= C#vv ||= B#v ||= Ax ||= ||= ||

||= ||= || || || etc. ||= ||= ||

~~Here's each note~~, with alternate tunings for the black keys~~, with the keyspan and genspan both measured from C~~:

Each of the 22 keys, with alternate tunings for the black keys:

||= keyspan ||= genspan ||= note ||= genspan ||= note ||

||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note ||

||= 0 ||= 0 ||= C ||= ||= ||

||= 1 ||= -5 ||= Db = C^ ||= +17 ||= C#vv = Dv3 ||

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||= 22 ||= 0 ||= C ||= ||= ||

"^3" means ~~three ups~~. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

"^3" means triple-up. Positive genspans, which lie on the fifthward part of the genchain, create sharps and downs. Negative genspans, from the fourthwards part of the genchain, create flats and ups.

The genspan for the up symbol in 22-tone is calculated from the keyspans:

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

K(#) = X, K(b) = -X (X = keyspan of ~~the~~ sharp ~~symbol~~, i.e., how many keys wide it is. For 22-tone, X = 3)

K(#) = X, K(b) = -X (X = keyspan of a sharp, i.e., how many keys wide it is. For 22-tone, X = 3)

K(#vX) = K(#) + X * K(v) = 0 (going up X keys using a sharp, then going down X keys using X downs, must cancel out)

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|| genspan from C || 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 ||

|| 12-tone keyspan from C || 0 || 7 || 2 || 9 || 4 || 11 || 6 || 1 || 8 || 3 || 10 || 5 || 0 ||

|| 29-tone || 0 || 17 || 5 || 22 || 10 || 27 || 15 || 3 || 8 || || || || ||

B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down X downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.

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7 - X * G(^) = i * N

G(^) = - (i * N - 7) / X

Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx

29-tone

C * Db C# * D * Eb D# * E * F * Gb F# * G * Ab G# * A * Bb A# * B * C

For 22-tone, X = 3 and N = 22. We choose i to be the smallest (least absolute value) number that avoids fractions. Thus i = 1, G(^) = -5, and ^ = min 2nd.

||= ||= Keyspan of # ||= value of i ||= genspan of ^ ||= stepspan &

||= ||= Keyspan of # ||= value of i ||= genspan of ^ ||= ||= stepspan &

quality of ^ ||

||= 17-tone ||= 2 ||= 1 ||= -5 ||= min 2nd ||

||= 17-tone ||= 2 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 22-tone ||= 3 ||= 1 ||= -5 ||= min 2nd ||

||= 22-tone ||= 3 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 27-tone ||= 4 ||= ||= ||= ||

||= 27-tone ||= 4 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 29-tone ||= 3 ||= ||= ||= ||

||= 29-tone ||= 3 ||= 2, -1 ||= -17, +12 ||= C-Ebbb, C-B# ||= dbl-dim 3rd, desc dim 2nd ||

||= 31-tone ||= 2 ||= 1 ||= -12 ||= dim 2nd ||

||= 31-tone ||= 2 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||

||= 32-tone ||= 5 ||= ||= ||= ||

||= 32-tone ||= 5 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 37-tone ||= 6 ||= ||= ||= ||

||= 37-tone ||= 6 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 39-tone ||= 5 ||= ||= ||= ||

||= 39-tone ||= 5 ||= 3, -2 ||= -22, +17 ||= C-Fbbb, C-Ax ||= triple-dim 4th ||

||= 41-tone ||= 4 ||= ||= ||= ||

||= 41-tone ||= 4 ||= 3, -1 ||= -29, +12 ||= C-Fbbbb, C-B# ||= ||

||= 42-tone ||= 7 ||= ||= ||= ||

||= 42-tone ||= 7 ||= 1 ||= -5 ||= C-Db ||= min 2nd ||

||= 43-tone ||= 3 ||= ||= ||= ||

||= 43-tone ||= 3 ||= 1 ||= -12 ||= C-Dbb ||= dim 2nd ||

||= 45-tone ||= 5 ||= ||= ||= ||

||= 45-tone ||= 2 ||= 1 ||= -19 || C-Dbbb

||= 49-tone ||= 7 ||= ||= ||= ||

||= double-dim 2nd ||

||= 50-tone ||= 3 ||= ||= ||= ||

||= 49-tone ||= 7 ||= 0 ||= -7 ||= C-Cb ||= desc aug unison ||

||= 53-tone ||= 5 ||= ||= ||= ||</pre></div>

||= 50-tone ||= 3 ||= ||= ||= ||= ||

||= 53-tone ||= 5 ||= ||= ||= ||= ||</pre></div>

<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><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a>&quot;Ups and Downs&quot; Notation</h1>

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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.

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<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>

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<td style="text-align: center;">*

</td>

<td style="text-align: center;">Db

<td style="text-align: center;">C#

</td>

<td style="text-align: center;">*

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<td style="text-align: center;">*

</td>

<td style="text-align: center;">C#

<td style="text-align: center;">Db

</td>

<td style="text-align: center;">*

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<h2 id="toc18"><a name="Summary of EDO notation-Rank-2 Notation"></a>Rank-2 Notation</h2>

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. Such instruments use a 12-tone framework. 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.)

Ups and downs can be used to notate 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. Such instruments use a 12-tone framework. 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.)

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# ...

...Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx...

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

Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either 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

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