Kite's ups and downs notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-01 03:05:52 UTC</tt>.

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All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

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

Edos 11 and 13 and problematic. See "Fifthless EDOs" below for alternate notations for them.

**__9edo__:** C/D# Cb/D (# = v)

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(8, 11b, 13 and 18)

There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth. This doesn't work for 8edo.

There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.

Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A look at the scale fragments reveals why 29-tone's up is a descending interval:

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Line 1,140:

29-tone: C * Db C# * D

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. Descending ups are not a problem:

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. Descending ups are not a problem.

The 29-tone keyboard, with alternate tunings for the black keys:

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

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

The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:

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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. In theory, 39-tone's C^ could be an ~~octuply~~-diminished 9th. Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.

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 octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.

=__Rank-2 Scales: Non-8ve Periods__=

Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different "height"; one is up, another is down, etc. See [[xenharmonic/Naming Rank-2 Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods|xenharmonic.wikispaces.com/Naming+Rank-2+Scales]]

Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.

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Line 1,168:

F --- C --- G --- D --- A --- E --- B

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, [[Diaschismic family|Srutal]] [10] might look like this:

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.

Ups and downs can be used to indicate the genchain. This is a completely separate use of ups and downs than in the last section! Rather than a genspan, the up has a "period-span" of 1. Each genchain has a different "height"; one is up, another is down, etc.

For example, [[Diaschismic family|Srutal]] [10] might look like this:

F^3 --- C^4 --- G^4 --- D^5 --- A^5

C3 ---- G3 ----- D4 ---- A4 ---- E5

Line 1,180:

C1 ---- G1 ----- D2 ---- A2 ---- E2

~~which~~ octave-reduces to two genchains:

This octave-reduces to two genchains:

F^ --- C^ --- G^ --- D^ --- A^

C ---- G ----- D ---- A ---- E

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. ~~F^ is a fourth plus an up~~, ~~which works out~~ to ~~be exactly a~~ half-octave.

Scale: C - C^ - D - D^ - E - F^ - G - G^ - A - A^ - C

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.

It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.

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 an 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:

Gv --- Dv --- Av --- Ev --- Bv

C ----- G ----- D ---- A ---- E

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Line 1,198:

Gv --- Dv --- Av --- Ev --- Bv

~~Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.~~

Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same scales:

In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.

~~If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator.~~ Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same ~~mode numbers and thus the same scale names~~:

F^ -- G --- G^ -- A --- A^

C --- C^ -- D --- D^ -- E

Another alternative is to use [[Kite's color notation|color notation]]. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:

~~yF# --- yC# --- yG# --- yD# --- yA#~~

~~wC ---- wG ---- wD ---- wA ---- wE~~

~~As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:~~

~~wC ---- wG ---- wD ----- wA ---- wE~~

~~gGb --- gDb --- gAb --- gEb --- gBb~~

~~All five Srutal [10] modes, using ups and downs. Every other scale note has an up.~~

~~|| scale name || sL pattern || example in C || 1st genchain || 2nd genchain ||~~

~~|| 1st Srutal [10] || ssssL-ssssL || C C^ D D^ E F^ G G^ A A^ C || __**C**__ G D A E || F^ C^ G^ D^ A^ ||~~

~~|| 2nd Srutal [10] || sssLs-sssLs || C C^ D D^ F F^ G G^ A Bb^ C || F __**C**__ G D A || Bb^ F^ C^ G^ D^ ||~~

~~|| 3rd Srutal [10] || ssLss-ssLss || C C^ D Eb^ F F^ G G^ Bb Bb^ C || Bb F __**C**__ G D || Eb^ Bb^ F^ C^ G^ ||~~

~~|| 4th Srutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ ||~~

~~|| 5th Srutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ ||~~

The [[Octatonic scale|Diminished]] [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.

~~Gb^^~~ ----- ~~Db^^~~

G#vv ----- D#vv

~~Eb^~~ ------- ~~Bb^~~

Ev --------- Bv

C ---------- G

Av -------~~-- Ev~~

Ab^ ------- Eb^

~~The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from~~ Eb ~~= 32/27 = 294¢. "Up" means "a quarter-octave minus a ~32/27".~~

~~Using ~25/24 as the generator yields the same scales and mode numbers:~~

~~Gb^^ ----- G~~

~~Eb^ ------- Ev~~

~~C ---------- Db^^~~

~~Av --------- Bb~~^

~~In color notation, the diminished comma 648/625 is g42. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].~~

~~ggGb ----- ggDb~~

~~gEb ------- gBb~~

~~wC -------- wG~~

~~yA --------- yE~~

~~Both Diminished [8] modes, using ups and downs~~:

Scale: C - D#vv - Ev - Eb^ - G#vv - G - Ab^ - Bv - C

~~|| scale name || sL pattern || example in~~ C ~~|| 1st chain || 2nd chain || 3rd chain || 4th chain ||~~

~~|| 1st Diminished[ 8] || sLsL sLsL || C Db^^~~ Eb^ ~~Ev Gb^^~~ G ~~Av Bb^ C ||= __**C**__~~ G ~~|| Eb^ Bb^ || Gb^^ Db^^ || Av Ev ||~~

~~|| 2nd Diminished [8] || LsLs LsLs || C Dv Eb^ F Gb^^~~ Ab^ ~~Av Cb^^~~ C ~~||= F __**C**__ || Ab^ Eb^ || Cb^^ Gb^^ || Dv Av ||~~

The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, "up" means "a major 3rd (~81/64) minus a quarter-octave". Using ~25/24 as the generator yields the same scale.

There are only two [[Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":

~~E^^ ------- G#^^~~

~~D^ -------- F#^~~

~~C ---------- E~~

~~Bbv ------- Fv~~

~~Gvv ------- Dvv~~

~~Ups~~ and downs ~~could~~ indicate the generator instead of the period:

[[Blackwood]] [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. "Up" means "2/5 of an octave minus ~5/4":

F ------ Av

D ------ F#v

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Line 1,220:

G ------ Bv

~~Assuming octave equivalence~~, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.

Since octaves are equivalent, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.

F ------ Av

C ------ Ev

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Line 1,227:

A ------ C#v

~~In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.~~

The scale: C C#v D Ev F F#v G Av A Bv C

~~wF ------ yA~~

~~wC ------ yE~~

~~wG ------ yB~~

~~wD ------ yF#~~

~~wA ------ yC#~~

~~Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4"~~:

~~|| scale name || sL pattern || example in C || genchains ||~~

~~|| 1st Blackwood [10] || Ls-Ls-Ls-Ls-Ls ||~~ C C#v D Ev F F#v G Av A Bv C ~~||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v ||~~

~~|| 2nd Blackwood [10] || sL-sL-sL-sL-sL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^-__**C**__, Bb^-D, C^-E, Eb^-G, F^-A ||~~

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The porcupine genchain:

||= genspan from D ||= 22-tone keyspan from D ||= note ||

~~||= -18 ||= 12 ||= G### ||~~

||= -17 ||= 15 ||= Ax ||

||= -16 ||= 18 ||= Bx ||

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Line 1,278:

||= 16 ||= 4 ||= Fbb ||

||= 17 ||= 7 ||= Gbb ||

~~||= 18 ||= 10 ||= Abbb ||~~

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

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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 1,682:

Line 1,632:

<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 3,462:

Line 3,412:

All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.

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

Edos 11 and 13 and problematic. See &quot;Fifthless EDOs&quot; below for alternate notations for them.

9edo: C/D# Cb/D (# = v)

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Line 3,508:

(8, 11b, 13 and 18)

There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth. This doesn't work for 8edo.

There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.

Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.

Line 4,780:

Line 4,731:

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

</td>

<td style="text-align: center;">desc dim 2nd

<td style="text-align: center;">desc dim 2nd

</td>

<td style="text-align: center;">^d2

Line 4,844:

Line 4,795:

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

</td>

<td style="text-align: center;">desc double-dim 3rd

<td style="text-align: center;">desc double-dim 3rd

</td>

<td style="text-align: center;">^dd3

Line 4,925:

Line 4,876:

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

</td>

<td style="text-align: center;">desc triple-dim 4th

<td style="text-align: center;">desc triple-dim 4th

</td>

<td style="text-align: center;">^ddd4

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Line 4,892:

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

</td>

<td style="text-align: center;">desc double-dim 2nd

<td style="text-align: center;">desc double-dim 2nd

</td>

<td style="text-align: center;">^dd2

Line 4,970:

Line 4,921:

29-tone: C * Db C# * D

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. Descending ups are not a problem:

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. Descending ups are not a problem.

The 29-tone keyboard, with alternate tunings for the black keys:

Line 5,062:

Line 5,013:

</table>

~~ ~~

The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:

Line 5,068:

Line 5,018:

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. In theory, 39-tone's C^ could be an ~~octuply~~-diminished 9th. Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.

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 octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.

<h1 id="toc15"><a name="Rank-2 Scales: Non-8ve Periods"></a>Rank-2 Scales: Non-8ve Periods</h1>

Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &quot;height&quot;; one is up, another is down, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods">xenharmonic.wikispaces.com/Naming+Rank-2+Scales</a>

~~ ~~

Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &quot;genweb&quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.

F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5

Line 5,085:

Line 5,031:

F --- C --- G --- D --- A --- E --- B

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, <a class="wiki_link" href="/Diaschismic%20family">Srutal</a> [10] might look like this:

But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.

Ups and downs can be used to indicate the genchain. This is a completely separate use of ups and downs than in the last section! Rather than a genspan, the up has a &quot;period-span&quot; of 1. Each genchain has a different &quot;height&quot;; one is up, another is down, etc.

For example, <a class="wiki_link" href="/Diaschismic%20family">Srutal</a> [10] might look like this:

F^3 --- C^4 --- G^4 --- D^5 --- A^5

C3 ---- G3 ----- D4 ---- A4 ---- E5

Line 5,093:

Line 5,043:

C1 ---- G1 ----- D2 ---- A2 ---- E2

~~which~~ octave-reduces to two genchains:

This octave-reduces to two genchains:

F^ --- C^ --- G^ --- D^ --- A^

C ---- G ----- D ---- A ---- E

~~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.~~ F^ ~~is a fourth plus an up, which works out to be exactly a half~~-~~octave.~~

Scale: C - C^ - D - D^ - E - F^ - G - G^ - A - A^ - C

It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.

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.

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 an 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 &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.

It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth:

Gv --- Dv --- Av --- Ev --- Bv

C ----- G ----- D ---- A ---- E

Line 5,107:

Line 5,061:

Gv --- Dv --- Av --- Ev --- Bv

~~Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8. ~~

Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same scales:

~~ ~~

In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.

~~ ~~

~~If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator.~~ Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same ~~mode numbers and thus the same scale names~~:

F^ -- G --- G^ -- A --- A^

C --- C^ -- D --- D^ -- E

~~Another alternative is to use~~ <a class="wiki_link" href="/~~Kite%27s~~%~~20color%20notation~~">~~color notation~~</a>. The ~~srutal comma~~ is ~~2048/2025~~ = ~~sgg2, and the temperament's color name~~ is ~~sggT [10]~~. ~~This comma makes the half~~-~~octave either ~45~~/~~32 = Ty4 or ~64~~/~~45 = Tg5, which from~~ C ~~would be yF~~# ~~or gGb. Here's 1st sggT [10]:~~

The <a class="wiki_link" href="/Octatonic%20scale">Diminished</a> [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.

G#vv ----- D#vv

Ev --------- Bv

C ---------- G

Ab^ ------- Eb^

Scale: C - D#vv - Ev - Eb^ - G#vv - G - Ab^ - Bv - C

~~yF#~~ --- ~~yC#~~ -~~-- yG# --- yD# --- yA#~~

The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, &quot;up&quot; means &quot;a major 3rd (~81/64) minus a quarter-octave&quot;. Using ~25/24 as the generator yields the same scale.

~~wC ---- wG ---- wD ---- wA ---- wE~~

~~As always, y means &quot;81/80 below w&quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period: ~~

~~wC ---- wG ---- wD ----- wA ---- wE ~~

~~gGb --- gDb --- gAb --- gEb --- gBb ~~

~~All five Srutal [10] modes, using ups and downs. Every other scale note has an up. ~~

<a class="wiki_link" href="/Blackwood">Blackwood</a> [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. &quot;Up&quot; means &quot;2/5 of an octave minus ~5/4&quot;:

F ------ Av

D ------ F#v

~~<table class="wiki_table">~~

C ------ Ev

~~<tr>~~

A ------ C#v

~~<td>scale name ~~

G ------ Bv

~~</td>~~

~~<td>sL pattern ~~

Since octaves are equivalent, the lattice rows can be reordered to make a &quot;pseudo-period&quot; of 3\5 = ~3/2.

~~</td>~~

F ------ Av

~~<td>example in C ~~

C ------ Ev

~~</td>~~

~~<td>1st genchain ~~

~~</td>~~

~~<td>2nd genchain ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>1st Srutal [10] ~~

~~</td>~~

~~<td>ssssL-ssssL ~~

~~</td>~~

~~<td>C C^ D D^ E F^ G G^ A A^ C ~~

~~</td>~~

~~<td>C G D A E ~~

~~</td>~~

~~<td>F^ C^ G^ D^ A^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>2nd Srutal [10] ~~

~~</td>~~

~~<td>sssLs-sssLs ~~

~~</td>~~

~~<td>C C^ D D^ F F^ G G^ A Bb^ C ~~

~~</td>~~

~~<td>F C G D A ~~

~~</td>~~

~~<td>Bb^ F^ C^ G^ D^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>3rd Srutal [10] ~~

~~</td>~~

~~<td>ssLss-ssLss ~~

~~</td>~~

~~<td>C C^ D Eb^ F F^ G G^ Bb Bb^ C ~~

~~</td>~~

~~<td>Bb F C G D ~~

~~</td>~~

~~<td>Eb^ Bb^ F^ C^ G^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>4th Srutal [10] ~~

~~</td>~~

~~<td>sLsss-sLsss ~~

~~</td>~~

~~<td>C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C ~~

~~</td>~~

~~<td>Eb Bb F C G ~~

~~</td>~~

~~<td>Ab^ Eb^ Bb^ F^ C^ ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>5th Srutal [10] ~~

~~</td>~~

~~<td>Lssss-Lssss ~~

~~</td>~~

~~<td>C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C ~~

~~</td>~~

~~<td>Ab Eb Bb F C ~~

~~</td>~~

~~<td>Db^ Ab^ Eb^ Bb^ F^ ~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~ ~~

~~The~~ <a class="wiki_link" href="/~~Octatonic%20scale~~">~~Diminished~~</a> [8] ~~scale has only two modes. The~~ period is a ~~quarter~~-octave ~~= 300¢. The~~ generator is ~3/2. ~~There are four very short genchains. ~~

~~Gb^^~~ -~~---- Db^^ ~~

~~Eb^ ------- Bb^ ~~

~~C ---------- G ~~

~~Av --------- Ev ~~

~~The choice~~ of ~~up or down is rather arbitrary, Eb^ could be Ebv. However if~~ the ~~3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢~~. &quot;Up&quot; means &quot~~;a quarter-octave minus a ~32/27&quot;. ~~

~~ ~~

~~Using ~25/24 as the generator yields the same scales and mode numbers: ~~

~~Gb^^ ----- G ~~

~~Eb^ ------- Ev ~~

~~C ---------- Db^^ ~~

~~Av --------- Bb^ ~~

~~In color notation, the diminished comma 648/625 is g4</span&gt~~;2~~. The period is ~6~~/5 ~~= Tg3. The color name is 4-EDO+y [8]. ~~

~~ggGb ----- ggDb ~~

~~gEb ------- gBb ~~

~~wC -------- wG ~~

~~yA --------- yE ~~

~~ ~~

~~Both Diminished [8] modes, using ups and downs: ~~

~~<table class="wiki_table">~~

~~<tr>~~

~~<td>scale name ~~

~~</td>~~

~~<td>sL pattern ~~

~~</td>~~

~~<td>example in C ~~

~~</td>~~

~~<td>1st chain ~~

~~</td>~~

~~<td>2nd chain ~~

~~</td>~~

~~<td>3rd chain ~~

~~</td>~~

~~<td>4th chain ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>1st Diminished[ 8] ~~

~~</td>~~

~~<td>sLsL sLsL ~~

~~</td>~~

~~<td>C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ~~

~~</td>~~

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

~~</td>~~

~~<td>Eb^ Bb^ ~~

~~</td>~~

~~<td>Gb^^ Db^^ ~~

~~</td>~~

~~<td>Av Ev ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>2nd Diminished [8] ~~

~~</td>~~

~~<td>LsLs LsLs ~~

~~</td>~~

~~<td>C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C ~~

~~</td>~~

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

~~</td>~~

~~<td>Ab^ Eb^ ~~

~~</td>~~

~~<td>Cb^^ Gb^^ ~~

~~</td>~~

~~<td>Dv Av ~~

~~</td>~~

~~</tr>~~

~~</table>~~

~~ ~~

~~There are only two <a class="wiki_link" href="/Blackwood">Blackwood</a> [10] modes. The period is a fifth-~~octave ~~= 240¢. The generator is a just~~ 5/4 ~~= 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &quot;height~~&quot;~~: ~~

~~E^^ ------- G#^^ ~~

~~D^ -------- F#^ ~~

~~C ---------- E ~~

~~Bbv ------- Fv ~~

~~Gvv ------- Dvv ~~

~~ ~~

~~Ups and downs could indicate the generator instead of the period~~:

F ------ Av

D ------ F#v

C ------ Ev

A ------ C#v

G ------ Bv

~~Assuming octave equivalence~~, the lattice rows can be reordered to make a &quot;pseudo-period&quot; of 3\5 = ~3/2.

F ------ Av

C ------ Ev

G ------ Bv

D ------ F#v

A ------ C#v

~~In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y. ~~

The scale: C C#v D Ev F F#v G Av A Bv C

~~wF ------ yA ~~

~~wC ------ yE ~~

~~wG ------ yB ~~

~~wD ------ yF# ~~

~~wA ------ yC# ~~

~~ ~~

~~Both Blackwood modes, using ups and downs to mean &quot;raised/lowered by 2/5 of an octave minus ~5/4&quot;~~:~~ ~~

~~<table class="wiki_table">~~

~~<tr>~~

~~<td>scale name ~~

~~</td>~~

~~<td>sL pattern ~~

~~</td>~~

~~<td>example in C ~~

~~</td>~~

~~<td>genchains ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>1st Blackwood [10] ~~

~~</td>~~

~~<td>Ls-Ls-Ls-Ls-Ls ~~

~~</td>~~

~~<td>~~C C#v D Ev F F#v G Av A Bv C

~~</td>~~

~~<td style="text-align: center;">C-Ev, D-F#v, F-Av, G-Bv, A-C#v ~~

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td>2nd Blackwood [10] ~~

~~</td>~~

~~<td>sL-sL-sL-sL-sL ~~

~~</td>~~

~~<td>C C^ D Eb^ E F^ G Ab^ A Bb^ C ~~

~~</td>~~

~~<td style="text-align: center;">Ab^-C, Bb^-D, C^-E, Eb^-G, F^-A ~~

~~</td>~~

~~</tr>~~

~~</table>~~

Line 5,365:

Line 5,114:

</td>

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">-18 ~~

~~</td>~~

~~<td style="text-align: center;">12 ~~

~~</td>~~

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

</td>

</tr>

Line 5,653:

Line 5,394:

</td>

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

~~</td>~~

~~</tr>~~

~~<tr>~~

~~<td style="text-align: center;">18 ~~

~~</td>~~

~~<td style="text-align: center;">10 ~~

~~</td>~~

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

</td>

</tr>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2016-09-30 22:56:19 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-10-01 03:05:52 UTC</tt>.<br>
-: The original revision id was <tt>593744268</tt>.<br>
+: The original revision id was <tt>593748500</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>
@@ Line 767: / Line 767: @@
 All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
 Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
+Edos 11 and 13 and problematic. See "Fifthless EDOs" below for alternate notations for them.
 **__9edo__:** C/D# Cb/D (# = v)
@@ Line 862: / Line 863: @@
 (8, 11b, 13 and 18)
-There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth. This doesn't work for 8edo.
+There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.
 Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.
@@ Line 1,120: / Line 1,121: @@
 ||= 22-tone ||= 3 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||
 ||= 27-tone ||= 4 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||
-||= 29-tone ||= 3 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= ^d2 ||
+||= 29-tone ||= 3 ||= -1 ||= +12 ||= C^ = B# ||= **desc** dim 2nd ||= ^d2 ||
 ||= 31-tone ||= 2 ||= 1 ||= -12 ||= C^ = Dbb ||= dim 2nd ||= vd2 ||
 ||= 32-tone ||= 5 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||
 ||= 37-tone ||= 6 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||
-||= **39-tone** ||= 5 ||= -2 ||= +17 ||= C^ = Ax ||= desc double-dim 3rd ||= **^dd3** ||
+||= **39-tone** ||= 5 ||= -2 ||= +17 ||= C^ = Ax ||= **desc** double-dim **3rd** ||= **^dd3** ||
 ||= 41-tone ||= 4 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= ^d2 ||
 ||= 42-tone ||= 7 ||= 1 ||= -5 ||= C^ = Db ||= min 2nd ||= vm2 ||
@@ Line 1,130: / Line 1,131: @@
 ||= 45-tone ||= 2 ||= 1 ||= -19 || &lt;span style="display: block; text-align: center;"&gt;C^ = Dbbb
 &lt;/span&gt; ||= double-dim 2nd ||= vdd2 ||
-||= **49-tone** ||= 7 ||= -3 ||= +22 ||= C^ = G### ||= desc triple-dim 4th ||= **^ddd4** ||
+||= **49-tone** ||= 7 ||= -3 ||= +22 ||= C^ = G### ||= **desc** triple-dim **4th** ||= **^ddd4** ||
-||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= desc double-dim 2nd ||= ^dd2 ||
+||= 50-tone ||= 3 ||= -1 ||= +19 ||= C^ = Bx ||= **desc** double-dim 2nd ||= ^dd2 ||
 ||= 53-tone ||= 5 ||= -1 ||= +12 ||= C^ = B# ||= desc dim 2nd ||= vd2 ||
 A look at the scale fragments reveals why 29-tone's up is a descending interval:
@@ Line 1,139: / Line 1,140: @@
 -tone: C * Db C# * D
-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. Descending ups are not a problem:
+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. Descending ups are not a problem.
 The 29-tone keyboard, with alternate tunings for the black keys:
@@ Line 1,149: / Line 1,150: @@
 ||= 4 ||= -10 ||= Dv ||= +19 ||= C#^ = Db^^ ||
 ||= 5 ||= +2 ||= D ||=   ||=   ||
 The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:
@@ Line 1,155: / Line 1,155: @@
 -tone: C * Db * * * * C# * D
-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. Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.
+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 octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.
 =__Rank-2 Scales: Non-8ve Periods__=
-Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different "height"; one is up, another is down, etc. See [[xenharmonic/Naming Rank-2 Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods|xenharmonic.wikispaces.com/Naming+Rank-2+Scales]]
 Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional "genweb", running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.
@@ Line 1,172: / Line 1,168: @@
 F --- C --- G --- D --- A --- E --- B
-But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, [[Diaschismic family|Srutal]] [10] might look like this:
+But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth.
+Ups and downs can be used to indicate the genchain. This is a completely separate use of ups and downs than in the last section! Rather than a genspan, the up has a "period-span" of 1. Each genchain has a different "height"; one is up, another is down, etc.
+For example, [[Diaschismic family|Srutal]] [10] might look like this:
 F^3 --- C^4 --- G^4 --- D^5 --- A^5
 C3 ---- G3 ----- D4 ---- A4 ---- E5
@@ Line 1,180: / Line 1,180: @@
 C1 ---- G1 ----- D2 ---- A2 ---- E2
-which octave-reduces to two genchains:
+This octave-reduces to two genchains:
 F^ --- C^ --- G^ --- D^ --- A^
 C ---- G ----- D ---- A ---- E
-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. F^ is a fourth plus an up, which works out to be exactly a half-octave.
+Scale: C - C^ - D - D^ - E - F^ - G - G^ - A - A^ - C
+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.
-It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.
+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 an 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:
 Gv --- Dv --- Av --- Ev --- Bv
 C ----- G ----- D ---- A ---- E
@@ Line 1,194: / Line 1,198: @@
 Gv --- Dv --- Av --- Ev --- Bv
-Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.
+Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same scales:
-In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.
-If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:
 F^ -- G --- G^ -- A --- A^
 C --- C^ -- D --- D^ -- E
-Another alternative is to use [[Kite's color notation|color notation]]. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:
-yF# --- yC# --- yG# --- yD# --- yA#
-wC ---- wG ---- wD ---- wA ---- wE
-As always, y means "81/80 below w". TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:
-wC ---- wG ---- wD ----- wA ---- wE
-gGb --- gDb --- gAb --- gEb --- gBb
-All five Srutal [10] modes, using ups and downs. Every other scale note has an up.
-|| scale name || sL pattern || example in C || 1st genchain || 2nd genchain ||
-|| 1st Srutal [10] || ssssL-ssssL || C C^ D D^ E F^ G G^ A A^ C || __**C**__ G D A E || F^ C^ G^ D^ A^ ||
-|| 2nd Srutal [10] || sssLs-sssLs || C C^ D D^ F F^ G G^ A Bb^ C || F __**C**__ G D A || Bb^ F^ C^ G^ D^ ||
-|| 3rd Srutal [10] || ssLss-ssLss || C C^ D Eb^ F F^ G G^ Bb Bb^ C || Bb F __**C**__ G D || Eb^ Bb^ F^ C^ G^ ||
-|| 4th Srutal [10] || sLsss-sLsss || C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C || Eb Bb F __**C**__ G || Ab^ Eb^ Bb^ F^ C^ ||
-|| 5th Srutal [10] || Lssss-Lssss || C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C || Ab Eb Bb F __**C**__ || Db^ Ab^ Eb^ Bb^ F^ ||
 The [[Octatonic scale|Diminished]] [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.
-Gb^^ ----- Db^^
+G#vv ----- D#vv
-Eb^ ------- Bb^
+Ev --------- Bv
 C ---------- G
-Av --------- Ev
+Ab^ ------- Eb^
-The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. "Up" means "a quarter-octave minus a ~32/27".
-Using ~25/24 as the generator yields the same scales and mode numbers:
-Gb^^ ----- G
-Eb^ ------- Ev
-C ---------- Db^^
-Av --------- Bb^
-In color notation, the diminished comma 648/625 is g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].
-ggGb ----- ggDb
-gEb ------- gBb
-wC -------- wG
-yA --------- yE
-Both Diminished [8] modes, using ups and downs:
+Scale: C - D#vv - Ev - Eb^ - G#vv - G - Ab^ - Bv - C
-|| scale name || sL pattern || example in C || 1st chain || 2nd chain || 3rd chain || 4th chain ||
-|| 1st Diminished[ 8] || sLsL sLsL || C Db^^ Eb^ Ev Gb^^ G Av Bb^ C ||= __**C**__ G || Eb^ Bb^ || Gb^^ Db^^ || Av Ev ||
-|| 2nd Diminished [8] || LsLs LsLs || C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C ||= F __**C**__ || Ab^ Eb^ || Cb^^ Gb^^ || Dv Av ||
+The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, "up" means "a major 3rd (~81/64) minus a quarter-octave". Using ~25/24 as the generator yields the same scale.
-There are only two [[Blackwood]] [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different "height":
-E^^ ------- G#^^
-D^ -------- F#^
-C ---------- E
-Bbv ------- Fv
-Gvv ------- Dvv
-Ups and downs could indicate the generator instead of the period:
+[[Blackwood]] [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. "Up" means "2/5 of an octave minus ~5/4":
 F ------ Av
 D ------ F#v
@@ Line 1,258: / Line 1,220: @@
 G ------ Bv
-Assuming octave equivalence, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.
+Since octaves are equivalent, the lattice rows can be reordered to make a "pseudo-period" of 3\5 = ~3/2.
 F ------ Av
 C ------ Ev
@@ Line 1,265: / Line 1,227: @@
 A ------ C#v
-In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.
+The scale: C C#v D Ev F F#v G Av A Bv C
-wF ------ yA
-wC ------ yE
-wG ------ yB
-wD ------ yF#
-wA ------ yC#
-Both Blackwood modes, using ups and downs to mean "raised/lowered by 2/5 of an octave minus ~5/4":
-|| scale name || sL pattern || example in C || genchains ||
-|| 1st Blackwood [10] || Ls-Ls-Ls-Ls-Ls || C C#v D Ev F F#v G Av A Bv C ||= __**C**__-Ev, D-F#v, F-Av, G-Bv, A-C#v ||
-|| 2nd Blackwood [10] || sL-sL-sL-sL-sL || C C^ D Eb^ E F^ G Ab^ A Bb^ C ||= Ab^-__**C**__, Bb^-D, C^-E, Eb^-G, F^-A ||
@@ Line 1,291: / Line 1,243: @@
 The porcupine genchain:
 ||= genspan from D ||= 22-tone keyspan from D ||= note ||
-||= -18 ||= 12 ||= G### ||
 ||= -17 ||= 15 ||= Ax ||
 ||= -16 ||= 18 ||= Bx ||
@@ Line 1,327: / Line 1,278: @@
 ||= 16 ||= 4 ||= Fbb ||
 ||= 17 ||= 7 ||= Gbb ||
-||= 18 ||= 10 ||= Abbb ||
 ||=   ||= etc. ||=   ||
@@ Line 1,465: / Line 1,415: @@
 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.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3475:&amp;lt;img src=&amp;quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3475 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3303:&amp;lt;img src=&amp;quot;/file/view/The%20Scale%20Tree.png/623953169/800x1002/The%20Scale%20Tree.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1002px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3303 --&gt;&lt;br /&gt;
 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 &amp;quot;generation&amp;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 &lt;strong&gt;kites&lt;/strong&gt;, 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 &lt;strong&gt;spinal&lt;/strong&gt; node. Every non-spinal node is part of three kites. It's the head of one kite and on the side of two others.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 1,682: / Line 1,632: @@
 &lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:3476:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3476 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:3304:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 1035px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3304 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:3477:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3477 --&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;&lt;!-- ws:start:WikiTextLocalImageRule:3305:&amp;lt;img src=&amp;quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 957px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;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;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:3305 --&gt;&lt;/h2&gt;
   &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Chord names in other EDOs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;&lt;u&gt;Chord names in other EDOs&lt;/u&gt;&lt;/h1&gt;
@@ Line 3,462: / Line 3,412: @@
 All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
 Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
+Edos 11 and 13 and problematic. See &amp;quot;Fifthless EDOs&amp;quot; below for alternate notations for them.&lt;br /&gt;
 &lt;br /&gt;
 &lt;strong&gt;&lt;u&gt;9edo&lt;/u&gt;:&lt;/strong&gt; C/D# Cb/D (# = v)&lt;br /&gt;
@@ Line 3,557: / Line 3,508: @@
   (8, 11b, 13 and 18)&lt;br /&gt;
 &lt;br /&gt;
-There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth. This doesn't work for 8edo.&lt;br /&gt;
+There are three strategies for notating these EDOs. One is to convert them to fourthwards EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.&lt;br /&gt;
 &lt;br /&gt;
 Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.&lt;br /&gt;
@@ Line 4,780: / Line 4,731: @@
          &lt;td style="text-align: center;"&gt;C^ = B#&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;desc dim 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;desc&lt;/strong&gt; dim 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^d2&lt;br /&gt;
@@ Line 4,844: / Line 4,795: @@
          &lt;td style="text-align: center;"&gt;C^ = Ax&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;desc double-dim 3rd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;desc&lt;/strong&gt; double-dim &lt;strong&gt;3rd&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;strong&gt;^dd3&lt;/strong&gt;&lt;br /&gt;
@@ Line 4,925: / Line 4,876: @@
          &lt;td style="text-align: center;"&gt;C^ = G###&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;desc triple-dim 4th&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;desc&lt;/strong&gt; triple-dim &lt;strong&gt;4th&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;strong&gt;^ddd4&lt;/strong&gt;&lt;br /&gt;
@@ Line 4,941: / Line 4,892: @@
          &lt;td style="text-align: center;"&gt;C^ = Bx&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;desc double-dim 2nd&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;desc&lt;/strong&gt; double-dim 2nd&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;^dd2&lt;br /&gt;
@@ Line 4,970: / Line 4,921: @@
 -tone: C * Db C# * D&lt;br /&gt;
 &lt;br /&gt;
-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 &lt;u&gt;below&lt;/u&gt; C. Thus Cv = Dbb, and C^ = B#, ^ is a descending dim 2nd, and the unison is an up-dim 2nd, ^d2. Descending ups are not a problem:&lt;br /&gt;
+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 &lt;u&gt;below&lt;/u&gt; C. Thus Cv = Dbb, and C^ = B#, ^ is a descending dim 2nd, and the unison is an up-dim 2nd, ^d2. Descending ups are not a problem.&lt;br /&gt;
 &lt;br /&gt;
 The 29-tone keyboard, with alternate tunings for the black keys:&lt;br /&gt;
@@ Line 5,062: / Line 5,013: @@
 &lt;/table&gt;
-&lt;br /&gt;
 &lt;br /&gt;
 The value of i equals the stepspan of the up interval. 39-tone and 49-tone are problematic:&lt;br /&gt;
@@ Line 5,068: / Line 5,018: @@
 -tone: C * Db * * * * C# * D&lt;br /&gt;
 &lt;br /&gt;
-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. Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.&lt;br /&gt;
+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 octuple-diminished 9th.) Notating rank-2 fifth-generated tunings in these two frameworks requires out-of-order notes.&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="Rank-2 Scales: Non-8ve Periods"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;u&gt;Rank-2 Scales: Non-8ve Periods&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-Ups and downs can also be used when naming fractional octave rank-2 tunings. These tunings have multiple genchains. Each genchain has a different &amp;quot;height&amp;quot;; one is up, another is down, etc. See &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Naming%20Rank-2%20Scales#Kite%20Giedraitis%20method-Fractional-octave%20periods"&gt;xenharmonic.wikispaces.com/Naming+Rank-2+Scales&lt;/a&gt;&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
 Fractional-period rank-2 temperaments have multiple genchains running in parallel. Multiple genchains occur because a rank-2 genchain is really a 2 dimensional &amp;quot;genweb&amp;quot;, running in octaves (or whatever the period is) vertically and fifths (or whatever the generator is) horizontally.&lt;br /&gt;
 F2 --- C3 --- G3 --- D4 --- A4 --- E5 --- B5&lt;br /&gt;
@@ Line 5,085: / Line 5,031: @@
 F --- C --- G --- D --- A --- E --- B&lt;br /&gt;
 &lt;br /&gt;
-But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. For example, &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Srutal&lt;/a&gt; [10] might look like this:&lt;br /&gt;
+But if the period is a half-octave, the genweb has vertical half-octaves, which octave-reduces to two parallel genchains. Temperaments with third-octave periods reduce to a triple-genchain, and so forth. &lt;br /&gt;
+&lt;br /&gt;
+Ups and downs can be used to indicate the genchain. This is a completely separate use of ups and downs than in the last section! Rather than a genspan, the up has a &amp;quot;period-span&amp;quot; of 1. Each genchain has a different &amp;quot;height&amp;quot;; one is up, another is down, etc. &lt;br /&gt;
+&lt;br /&gt;
+For example, &lt;a class="wiki_link" href="/Diaschismic%20family"&gt;Srutal&lt;/a&gt; [10] might look like this:&lt;br /&gt;
 F^3 --- C^4 --- G^4 --- D^5 --- A^5&lt;br /&gt;
 C3 ---- G3 ----- D4 ---- A4 ---- E5&lt;br /&gt;
@@ Line 5,093: / Line 5,043: @@
 C1 ---- G1 ----- D2 ---- A2 ---- E2&lt;br /&gt;
 &lt;br /&gt;
-which octave-reduces to two genchains:&lt;br /&gt;
+This octave-reduces to two genchains:&lt;br /&gt;
 F^ --- C^ --- G^ --- D^ --- A^&lt;br /&gt;
 C ---- G ----- D ---- A ---- E&lt;br /&gt;
 &lt;br /&gt;
-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 &amp;quot;up&amp;quot; here is &amp;quot;a half-octave minus a fourth&amp;quot;, with the understanding that both the octave and the fourth may be tempered. F^ is a fourth plus an up, which works out to be exactly a half-octave.&lt;br /&gt;
+Scale: C - C^ - D - D^ - E - F^ - G - G^ - A - A^ - C&lt;br /&gt;
 &lt;br /&gt;
-It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth.&lt;br /&gt;
+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 &amp;quot;up&amp;quot; here is &amp;quot;a half-octave minus a fourth&amp;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.&lt;br /&gt;
+&lt;br /&gt;
+If &amp;quot;up&amp;quot; means &amp;quot;a half-octave minus a fourth&amp;quot; as well as &amp;quot;up one key or fret&amp;quot;, a 4th must be one key or fret less than an 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, &lt;strong&gt;&lt;u&gt;the period's name must have at least one up or down, and the generator's name must have none, or vice versa&lt;/u&gt;.&lt;/strong&gt; This allows ups and downs to serve &amp;quot;double-duty&amp;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.&lt;br /&gt;
+&lt;br /&gt;
+It would be equally valid to write the half-octave not as an up-fourth but as a down-fifth:&lt;br /&gt;
 Gv --- Dv --- Av --- Ev --- Bv&lt;br /&gt;
 C ----- G ----- D ---- A ---- E&lt;br /&gt;
@@ Line 5,107: / Line 5,061: @@
 Gv --- Dv --- Av --- Ev --- Bv&lt;br /&gt;
 &lt;br /&gt;
-Gv is a fifth minus an up, which again works out to be a half-octave. Thus F^ = Gv, F^^ = G, and ^^ = ~9/8.&lt;br /&gt;
+Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same scales:&lt;br /&gt;
-&lt;br /&gt;
-In order to be a MOS scale, the parallel genchains must of course be the right length, and without any gaps. But they must also line up exactly, so that each note has a neighbor immediately above and/or below. In other words, every column of the genweb must be complete.&lt;br /&gt;
-&lt;br /&gt;
-If the period is a fraction of an octave, 3/2 is still preferred over 4/3, even though that makes the generator larger than the period. A generator plus or minus a period is still a generator. Srutal's generator could be thought of as either ~3/2 or ~16/15, because ~16/15 would still create the same mode numbers and thus the same scale names:&lt;br /&gt;
 F^ -- G --- G^ -- A --- A^&lt;br /&gt;
 C --- C^ -- D --- D^ -- E&lt;br /&gt;
 &lt;br /&gt;
-Another alternative is to use &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;color notation&lt;/a&gt;. The srutal comma is 2048/2025 = sgg2, and the temperament's color name is sggT [10]. This comma makes the half-octave either ~45/32 = Ty4 or ~64/45 = Tg5, which from C would be yF# or gGb. Here's 1st sggT [10]:&lt;br /&gt;
+The &lt;a class="wiki_link" href="/Octatonic%20scale"&gt;Diminished&lt;/a&gt; [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.&lt;br /&gt;
+G#vv ----- D#vv&lt;br /&gt;
+Ev --------- Bv&lt;br /&gt;
+C ---------- G&lt;br /&gt;
+Ab^ ------- Eb^&lt;br /&gt;
+&lt;br /&gt;
+Scale: C - D#vv - Ev - Eb^ - G#vv - G - Ab^ - Bv - C&lt;br /&gt;
 &lt;br /&gt;
-yF# --- yC# --- yG# --- yD# --- yA#&lt;br /&gt;
+The period interval is named as a 3rd with at least one up or down. Again, it varies by framework. For 12-tone, it's a downmajor 3rd, as above. For 16-tone, vm3, for 20-tone, ^m3, for 24-tone, vvM3. For a vM3, &amp;quot;up&amp;quot; means &amp;quot;a major 3rd (~81/64) minus a quarter-octave&amp;quot;. Using ~25/24 as the generator yields the same scale.&lt;br /&gt;
-wC ---- wG ---- wD ---- wA ---- wE&lt;br /&gt;
 &lt;br /&gt;
-As always, y means &amp;quot;81/80 below w&amp;quot;. TyF# = TgGb because the interval between them, sgg2, is tempered out. Using Tg5 instead of Ty4 as the period:&lt;br /&gt;
-wC ---- wG ---- wD ----- wA ---- wE&lt;br /&gt;
-gGb --- gDb --- gAb --- gEb --- gBb&lt;br /&gt;
 &lt;br /&gt;
-All five Srutal [10] modes, using ups and downs. Every other scale note has an up.&lt;br /&gt;
+&lt;a class="wiki_link" href="/Blackwood"&gt;Blackwood&lt;/a&gt; [10]'s period is a fifth-octave, and the generator is ~5/4. Here it's better if ups and downs indicate the generator-span instead of the period-span. &amp;quot;Up&amp;quot; means &amp;quot;2/5 of an octave minus ~5/4&amp;quot;:&lt;br /&gt;
+F ------ Av&lt;br /&gt;
+D ------ F#v&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
+C ------ Ev&lt;br /&gt;
-    &lt;tr&gt;
+A ------ C#v&lt;br /&gt;
-        &lt;td&gt;scale name&lt;br /&gt;
+G ------ Bv&lt;br /&gt;
-&lt;/td&gt;
+&lt;br /&gt;
-        &lt;td&gt;sL pattern&lt;br /&gt;
+Since octaves are equivalent, the lattice rows can be reordered to make a &amp;quot;pseudo-period&amp;quot; of 3\5 = ~3/2.&lt;br /&gt;
-&lt;/td&gt;
+F ------ Av&lt;br /&gt;
-        &lt;td&gt;example in C&lt;br /&gt;
+C ------ Ev&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1st genchain&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2nd genchain&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1st Srutal [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;ssssL-ssssL&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C^ D D^ E F^ G G^ A A^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A E&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F^ C^ G^ D^ A^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;2nd Srutal [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sssLs-sssLs&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C^ D D^ F F^ G G^ A Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D A&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb^ F^ C^ G^ D^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;3rd Srutal [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;ssLss-ssLss&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C^ D Eb^ F F^ G G^ Bb Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G D&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb^ Bb^ F^ C^ G^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;4th Srutal [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sLsss-sLsss&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C^ Eb Eb^ F F^ G Ab^ Bb Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab^ Eb^ Bb^ F^ C^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;5th Srutal [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Lssss-Lssss&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Db^ Eb Eb^ F F^ Ab Ab^ Bb Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab Eb Bb F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Db^ Ab^ Eb^ Bb^ F^&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-The &lt;a class="wiki_link" href="/Octatonic%20scale"&gt;Diminished&lt;/a&gt; [8] scale has only two modes. The period is a quarter-octave = 300¢. The generator is ~3/2. There are four very short genchains.&lt;br /&gt;
-Gb^^ ----- Db^^&lt;br /&gt;
-Eb^ ------- Bb^&lt;br /&gt;
-C ---------- G&lt;br /&gt;
-Av --------- Ev&lt;br /&gt;
-The choice of up or down is rather arbitrary, Eb^ could be Ebv. However if the 3/2 is tuned justly, Eb^ = 300¢ would indeed be up from Eb = 32/27 = 294¢. &amp;quot;Up&amp;quot; means &amp;quot;a quarter-octave minus a ~32/27&amp;quot;.&lt;br /&gt;
-&lt;br /&gt;
-Using ~25/24 as the generator yields the same scales and mode numbers:&lt;br /&gt;
-Gb^^ ----- G&lt;br /&gt;
-Eb^ ------- Ev&lt;br /&gt;
-C ---------- Db^^&lt;br /&gt;
-Av --------- Bb^&lt;br /&gt;
-In color notation, the diminished comma 648/625 is g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;2. The period is ~6/5 = Tg3. The color name is 4-EDO+y [8].&lt;br /&gt;
-ggGb ----- ggDb&lt;br /&gt;
-gEb ------- gBb&lt;br /&gt;
-wC -------- wG&lt;br /&gt;
-yA --------- yE&lt;br /&gt;
-&lt;br /&gt;
-Both Diminished [8] modes, using ups and downs:&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;scale name&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sL pattern&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;example in C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;1st chain&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2nd chain&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;3rd chain&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;4th chain&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1st Diminished[ 8]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sLsL sLsL&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Db^^ Eb^ Ev Gb^^ G Av Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt; G&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Eb^ Bb^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Gb^^ Db^^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Av Ev&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;2nd Diminished [8]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;LsLs LsLs&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C Dv Eb^ F Gb^^ Ab^ Av Cb^^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;F &lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ab^ Eb^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Cb^^ Gb^^&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Dv Av&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-There are only two &lt;a class="wiki_link" href="/Blackwood"&gt;Blackwood&lt;/a&gt; [10] modes. The period is a fifth-octave = 240¢. The generator is a just 5/4 = 386¢. L = 146¢ and s = 94¢. The lattice can be expressed using a 3\5 period Using ups and downs as before with each genchain at a different &amp;quot;height&amp;quot;:&lt;br /&gt;
-E^^ ------- G#^^&lt;br /&gt;
-D^ -------- F#^&lt;br /&gt;
-C ---------- E&lt;br /&gt;
-Bbv ------- Fv&lt;br /&gt;
-Gvv ------- Dvv&lt;br /&gt;
-&lt;br /&gt;
-Ups and downs could indicate the generator instead of the period:&lt;br /&gt;
-F ------ Av&lt;br /&gt;
-D ------ F#v&lt;br /&gt;
-C ------ Ev&lt;br /&gt;
-A ------ C#v&lt;br /&gt;
-G ------ Bv&lt;br /&gt;
-&lt;br /&gt;
-Assuming octave equivalence, the lattice rows can be reordered to make a &amp;quot;pseudo-period&amp;quot; of 3\5 = ~3/2.&lt;br /&gt;
-F ------ Av&lt;br /&gt;
-C ------ Ev&lt;br /&gt;
 G ------ Bv&lt;br /&gt;
 D ------ F#v&lt;br /&gt;
 A ------ C#v&lt;br /&gt;
 &lt;br /&gt;
-In color notation, the comma is 256/243 = sw2, the generator is ~5/4 = Ty3, and the color name is 5-EDO+y.&lt;br /&gt;
+The scale: C C#v D Ev F F#v G Av A Bv C&lt;br /&gt;
-wF ------ yA&lt;br /&gt;
-wC ------ yE&lt;br /&gt;
-wG ------ yB&lt;br /&gt;
-wD ------ yF#&lt;br /&gt;
-wA ------ yC#&lt;br /&gt;
-&lt;br /&gt;
-Both Blackwood modes, using ups and downs to mean &amp;quot;raised/lowered by 2/5 of an octave minus ~5/4&amp;quot;:&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;scale name&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sL pattern&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;example in C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;genchains&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1st Blackwood [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;Ls-Ls-Ls-Ls-Ls&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C#v D Ev F F#v G Av A Bv C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;-Ev, D-F#v, F-Av, G-Bv, A-C#v&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;2nd Blackwood [10]&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;sL-sL-sL-sL-sL&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;C C^ D Eb^ E F^ G Ab^ A Bb^ C&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Ab^-&lt;u&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;/u&gt;, Bb^-D, C^-E, Eb^-G, F^-A&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
 &lt;br /&gt;
@@ Line 5,365: / Line 5,114: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;note&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;-18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;G###&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 5,653: / Line 5,394: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Gbb&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td style="text-align: center;"&gt;18&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td style="text-align: center;"&gt;Abbb&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;