Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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=__Rank-2 Scales: 8ve Periods__=

Ups and downs can be used to notate rank-2 scales as well. ~~Let's start with fifth~~-~~generated tunings. For large frameworks~~, we'll ~~need~~ a ~~long~~ genchain:

Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about **frameworks** like 12-tone. The generator chain is called a **genchain**. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (chroma 1 or -1):

~~...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 notated without ups and downs in any framework on either side of the 4\7 kite:

12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C

12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B 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 single-ring 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, but not with 15~~-tone~~. 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).~~

When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, 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

~~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:~~

For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.

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

All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with chroma > 1 or < -1. If these are notated without ups and downs, the notes run out of order:

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

17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C

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

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

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.

~~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~~

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 regular frameworks, excluding those~~ that ~~lie~~ on the side of the ~~heptatonic~~ kite ~~and those that lie on~~ the ~~spine~~ of any ~~kite~~.

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).

~~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.

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

Every node on the scale tree, 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__ (fifthward) side of the 10\17 kite, we know that 17 __fifths__ add up to 1\22. Because it's on the __left__ (fourthward) 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 tempered 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 occupy the same key on the keyboard, just as C# ~~may not equal~~ Db in 12-tone.

Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 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.

22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:

The 22-tone genchain:

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

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

Line 1,084:

Line 1,076:

In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups.

The three black keys between C and D each have two names, one some version of C and the other some version of D. You can choose which one you want to keep the notes in order. Here are four possible tunings of these 3 keys, each written out in four ways:

C C#vv C#v C# D = C C#vv C#v Dv D = C C#vv Dvv Dv D = C Dv3 Dvv Dv D

C C^ C#v C# D = C C^ C#v Dv D = C C^ Dvv Dv D = C Db Dvv Dv D

C C^ C^^ C# D = C C^ C^^ Dv D = C C^ Db^ Dv D = C Db Db^ Dv D

C C^ C^^ C^3 D = C C^ C^^ Db^^ D = C C^ Db^ Db^^ D = C Db Db^ Db^^ D

All four tunings could be part of a MOS. Here's one that requires a MODMOS:

C C^ C#v C^3 D = C C^ C#v Db^^ D = C C^ Dvv Db^^ D = C Db Dvv Db^^ D

There are hundreds of possibilities, and ups and downs can notate all of them.

__**Finding the up's genspan**__

The genspan for the up symbol in 22-tone can be found from the scale tree. Or it can be derived more rigorously if calculated from the keyspans:

Line 1,103:

Line 1,108:

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

For 22-tone, N = 22 and c = 3. We choose i to be the smallest (least absolute value) number that avoids fractions, and produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd. However as we'll see, this isn't always possible.

For 22-tone, N = 22 and c = 3. We choose i to be the smallest (least absolute value) number that avoids fractions, and produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd, or a descending 2nd which is a 7th. This means the up's genspan modulo 7 should be 2 or 5. However as we'll see, this isn't always possible.

All relevant frameworks of size 53 or less:

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

quality of ^ ||

quality of ^ ||= stepspan &

||= 11-tone ||= 2 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd ||

quality of unison ||

||= 13b-tone ||= 3 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd ||

||= 11-tone ||= -2 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd ||= vM2 ||

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

||= 13b-tone ||= -3 ||= 1 ||= 2 ||= C^ = D ||= maj 2nd ||= vM2 ||

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

||= 17-tone ||= 2 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

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

||= 22-tone ||= 3 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

||= 29-tone ||= 3 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||

||= 27-tone ||= 4 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

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

||= 29-tone ||= 3 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= ^d2 ||

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

||= 31-tone ||= 2 ||= 1 ||= -12 ||= C^ = Dbb ||= dim 2nd ||= vd2 ||

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

||= 32-tone ||= 5 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

||= 39-tone ||= 5 ||= -2 ||= +17 ||= C^ = Ax ||= desc double-dim 3rd ||

||= 37-tone ||= 6 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||

||= **39-tone** ||= 5 ||= -2 ||= +17 ||= C^ = Ax ||= desc double-dim 3rd ||= **^dd3** ||

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

||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= ^d2 ||

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

||= 42-tone ||= 7 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||

||= 43-tone ||= 3 ||= 1 ||= -12 ||= C^ = Dbb ||= dim 2nd ||= vd2 ||

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

||= double-dim 2nd ||

||= double-dim 2nd ||= vdd2 ||

||= 49-tone ||= 7 ||= -3 ||= +22 ||= C^ = G### ||= desc triple-dim 4th ||

||= **49-tone** ||= 7 ||= -3 ||= +22 ||= C^ = G### ||= desc triple-dim 4th ||= **^ddd4** ||

||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd ||

||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd ||= ^dd2 ||

||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||

||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= vd2 ||

The value of i equals the stepspan of the up interval. A look at the scale fragments reveals why 29-tone ~~has~~ a ~~negative i~~:

The value of i equals the stepspan of the up interval. A look at the scale fragments reveals why 29-tone's up is a descending interval:

~~17-tone: C Db C# D~~

22-tone: C Db * C# D

27-tone: C Db * * C# D

29-tone: C * Db C# * D

The ~~17-tone,~~ 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key __below__ C. Thus Cv = Dbb, and C^ = B#, ~~and~~ ^ is a descending dim 2nd. In 41-tone, 50-tone and 53-tone~~, the up is~~ also a descending ~~2nd~~. ~~The descending nature~~ is not ~~an issue, as the following charts show.~~

The 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key __below__ C. Thus Cv = Dbb, and C^ = B#, ^ is a descending dim 2nd, and the unison is an up-dim 2nd, ^d2. 41-tone, 50-tone and 53-tone also have ups that are descending. This is not a problem:

The 29-tone genchain:

Line 1,175:

Line 1,182:

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

||= 1 ||= -17 ||= Dbv = C#vv ||= +12 ||= C^ ||

||= 2 ||= -5 ||= Db = C#v ||= +24 ||= C^^ = Dbb^3 ||

||= 2 ||= -5 ||= Db = C#v ||= +24 ||= C^^ = Dbb^3 ||

||= 3 ||= -22 ||= Dvv = ~~Cxv3~~ ||= +7 ||= C# = Db^ ||

||= 3 ||= -22 ||= Dvv = Cxv3 ||= +7 ||= C# = Db^ ||

||= 4 ||= -10 ||= Dv ||= +19 ||= C#^ = Db^^ ||

||= 5 ||= +2 ||= D ||= ||= ||

Line 1,209:

Line 1,216:

49-tone: C * Db * * * * C# * D

There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for rank-2 fifth-generated tunings in these two frameworks.

There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. In theory, 39-tone's C^ could be an octuply-diminished 9th. Some other method of notation must be used for rank-2 fifth-generated tunings in these two frameworks.

Line 1,519:

Line 1,526:

This is in addition to the trivial EDOs, 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.

<img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" />

<img src="/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png" alt="The Scale Tree.png" title="The Scale Tree.png" style="height: 1002px; width: 800px;" />

The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. This version of the Stern-Brocot tree is the scale tree. The colored regions of the tree are what I call kites, and The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side on the right (7\12, 11\19, etc.) and a fourthward side on the left (5\9, 9\16, etc.). Every node on a spine is a spinal node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.

Line 1,736:

Line 1,743:

<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>

<h1 id="toc4"><a name="Chord names in other EDOs"></a>Chord names in other EDOs</h1>

Line 3,739:

Line 3,746:

<h1 id="toc14"><a name="Rank-2 Scales: 8ve Periods"></a>Rank-2 Scales: 8ve Periods</h1>

Ups and downs can be used to notate rank-2 scales as well. ~~Let's start with fifth~~-~~generated tunings. For large frameworks~~, we'll ~~need a long genchain:~~

Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about frameworks like 12-tone. The generator chain is called a genchain. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (chroma 1 or -1):

~~...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..~~.<~~br /~~>

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 Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C

12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B 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 single-ring 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, but not with 15~~-tone~~. 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).~~

When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, 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

~~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:~~

For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.

5-tone ~~genchain C to E: C D E G A C~~<~~br /&~~gt;

All fifthless frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring regular frameworks with chroma &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:

~~5-tone genchain F to A: C D F G A C~~<~~br />~~

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~~ ~~

17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C

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

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

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.

~~7-tone: Eb * F C G D A * B = C D Eb F G A B C~~<~~br /~~>

~~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~~

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 regular frameworks, excluding those~~ that ~~lie~~ on the side of the ~~heptatonic~~ kite ~~and those that lie on~~ the ~~spine~~ of any ~~kite~~.

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).

~~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.

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

Every node on the scale tree, 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 (fifthward) side of the 10\17 kite, we know that 17 fifths add up to 1\22. Because it's on the left (fourthward) 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 tempered 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 occupy the same key on the keyboard, just as C# ~~may not equal~~ Db in 12-tone.

Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 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.~~ ~~

22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:

~~ ~~

The 22-tone genchain:

Line 4,602:

Line 4,601: