Pergen names: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-21 17:08:36 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-21 23:38:04 UTC</tt>.

: The original revision id was <tt>~~622142511~~</tt>.

: The original revision id was <tt>622159045</tt>.

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

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

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||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g4T ||

||= {P8, P12/5} ||= fifth-twelfth ||= (-10,-1,5) ||= magic ||= large quintuple yellow ||= Ly5T ||

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one "W" per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.

[//Question: at what point is// "//W" actually needed?//]

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The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.

[//This 13b part needs clarification//]

[//This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.//]

(table is under construction)

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||~ halves ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/2, P5} ||= P8/2 = vA4 = ^d5

(if 5th > 700¢) ||= ^^d2 ||= C^^ = B# ||= C - F#v=Gb^ - C ||= srutal

^1 = 81/80 ||= 12, 14, 16, 18b, 20*,

22, 24*, 26, 28*, 30* ||

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vvA1,

^^\\d2,

vv\\M2 ||= C// = Db,

vv\\M2 ||= C //= Db,//

C^^ = C#

//C^^ = C#//

C^^// = D ||= C - D/=Eb\ - F,

//C^^// = D ||= C - D/=Eb\ - F,

C - Eb^=Ev - G,

C - F#v/=Gb^\ - C,

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\\m2,

vv\\A1 ||= C^^ = B#

C// = Db,

C //= Db,//

C^^// = C# ||= C - F#v=Gb^ - C,

//C^^// = C# ||= C - F#v=Gb^ - C,

C - D/=Eb\ - F,

C - Eb^/=Ev\ - G ||= sgg&bbT

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\\A1,

^^\\m2 ||= C^^ = B#

C// = C#

C //= C#//

C^^// = B## ||= C - F#v=Gb^ - C,

//C^^// = B## ||= C - F#v=Gb^ - C,

C - Eb/=E\ - G,

C - Dv/=Eb^\ - F ||= sgg&aaT

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/1 = 33/32 ||= " ||

||~ thirds ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 ||= ^3d2 ||= C^3 = B# ||= C - Ev - Ab^ - C ||= ||= 12, 15, 18b*, 21,

||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 ||= ^3d2 ||= C^3

=B#= || C - Ev - Ab^ - C ||= ||= 12, 15, 18b*, 21,

24*, 27, 30* ||

||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v3A1 ||= C^3 = C# ||= C - Dv - Eb^ - F ||= porcupine ||= 13b, 14*, 15, 21*,

||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v3A1 ||= C^3

=C#= || C - Dv - Eb^ - F ||= porcupine ||= 13b, 14*, 15, 21*,

22, 28*, 29, 30* ||

||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v3m2 ||= C^3 = Db ||= C - D^ - Fv - G ||= ||= 15*, 16, 20*, 21,

||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v3m2 ||= C^3

=Db= || C - D^ - Fv - G ||= ||= 15*, 16, 20*, 21,

25*, 26, 30*, 31 ||

||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^3dd2 ||= C^3 = B## ||= C - F#v - Cb^ - F ||= ||= ||

||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^3dd2 ||= C^3

||= " ||= P11/3 = ^P4 = vvP5 ||= v3M2 ||= C^3 = D ||= C F^ Cv F ||= ||= " ||

=B##= || C - F#v - Cb^ - F ||= ||= ||

||= " ||= P11/3 = ^P4 = vvP5 ||= v3M2 ||= C^3

=D= || C F^ Cv F ||= ||= " ||

||= {P8/3, P4/2} ||= ||= ||= ||= ||= ||= 15, 18b*, 24, 30 ||

||= {P8/3, P5/2} ||= ||= ||= ||= ||= ||= 18b, 24, 30 ||

||= {P8/2, P4/3} ||= ||= ||= ||= ||= ||= 14, 22, 28*, 30* ||

||= {P8/2, P5/3} ||= P8/2 = v3AA4 = ^3dd5

P5/3 = vvA2 = ^4dd3 ||= ^6d32 ||= C^6 = B#3 ||= C - Fxv3=Gbb^3 C

P5/3 = vvA2 = ^4dd3 ||= ^6d32 ||= C^6

C - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* ||

=B#3= || C - Fxv3=Gbb^3 CC - D#vv - Fb^^ - G ||= ||= 16, 20*, 26, 30* ||

||= " ||= P8/2 = vA4 = ^d5

P5/3 = /M2 = \\m3 ||= ^^d2,

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||= {P8/3, P4/3} ||= ||= ||= ||= ||= ||= 15, 21, 30* ||

||~ quarters ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/4, P5} ||= P8/4 = vm3 = ^3A2 ||= ^4d2 ||= C^4 = B# ||= C Ebv Gbvv A^ C ||= diminished,

||= {P8/4, P5} ||= P8/4 = vm3 = ^3A2 ||= ^4d2 ||= C^4

=B#= || C Ebv Gbvv A^ C ||= diminished,

^1 = 81/80 ||= 12, 16, 20, 24*, 28 ||

||= {P8, P4/4} ||= P4/4 = ^m2 = v3AA1 ||= ^4dd2 ||= C^4 = B## ||= C Db^ Ebb^^ Ev F ||= ||= ||

||= {P8, P4/4} ||= P4/4 = ^m2 = v3AA1 ||= ^4dd2 ||= C^4

||= {P8, P5/4} ||= P5/4 = vM2 = ^3m2 ||= v4A1 ||= C^4 = C# ||= C Dv Evv F^ G^ ||= ||= ||

=B##= || C Db^ Ebb^^ Ev F ||= ||= ||

||= {P8, P11/4} ||= P11/4 = ^M3 = v3dd5 ||= v4dd3 ||= C^4 = Eb3 ||= C E^ G#^^ Dbv F ||= ||= ||

||= {P8, P5/4} ||= P5/4 = vM2 = ^3m2 ||= v4A1 ||= C^4

||= {P8, P12/4} ||= P12/4 = vP4 = ^3M3 ||= v4m2 ||= C^4 = Db ||= C Fv Bbvv D^ G ||= ||= ||

=C#= || C Dv Evv F^ G^ ||= ||= ||

||= {P8, P11/4} ||= P11/4 = ^M3 = v3dd5 ||= v4dd3 ||= C^4

=Eb3= || C E^ G#^^ Dbv F ||= ||= ||

||= {P8, P12/4} ||= P12/4 = vP4 = ^3M3 ||= v4m2 ||= C^4

=Db= || C Fv Bbvv D^ G ||= ||= ||

||= {P8/4, P4/2} ||= ||= ||= ||= ||= ||= ||

||= {P8/2, P4/4} ||= ||= ||= ||= ||= ||= ||

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||~ fifths ||~ ||~ ||~ ||~ ||~ ||~ ||

||= {P8/5, P5} ||= P8/5 = ||= m2 ||= ||= ||= ||= ||

||= {P8, P5/5} ||= P5/5 = ^m2 = v4A31 ||= ^5d32 ||= C^5 = B#3 ||= C - Db^ - Ebb^ -

||= {P8, P5/5} ||= P5/5 = ^m2 = v4A31 ||= ^5d32 ||= C^5

- E#vv - F#v - G ||= ||= ||

=B#3= || C - Db^ - Ebb^ -- E#vv - F#v - G ||= ||= ||

||= " ||= ||= v5dd3 ||= ||= ||= ||= ||

[//Question: how to find all possible pergens?//]

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, ~~the first~~ enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to ~~make~~ the enharmonic interval ~~vanish~~. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

[//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is 646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

||||~ enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||

||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||

||= M2 ||= C - D ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||

||= m3 ||= C - Eb ||= 3-edo ||= 800¢ ||= none ||= all ||= downed ||

||= m2 ||= C - Db ||= 5-edo ||= 720¢ ||= none ||= all ||= downed ||

||= A1 ||= C - C# ||= 7-edo ||= 686¢ ||= 600-686¢ ||= 686¢-720¢ ||= downed ||

||= A1 ||= C - C# ||= 7-edo ||= ~686¢ ||= 600-686¢ ||= 686¢-720¢ ||= downed ||

||= d2 ||= C - Dbb ||= 12-edo ||= 700¢ ||= 700-720¢ ||= 600-700¢ ||= upped ||

||= dd3 ||= C - Eb3 ||= 17-edo ||= 706¢ ||= 706-720¢ ||= 600-706¢ ||= downed ||

||= dd3 ||= C - Eb3 ||= 17-edo ||= ~706¢ ||= 706-720¢ ||= 600-706¢ ||= downed ||

||= dd2 ||= C - Db3 ||= 19-edo ||= 695¢ ||= 695-720¢ ||= 600-695¢ ||= upped ||

||= dd2 ||= C - Db3 ||= 19-edo ||= ~695¢ ||= 695-720¢ ||= 600-695¢ ||= upped ||

||= d32 ||= C - Db4 ||= 26-edo ||= 692¢ ||= 692-720¢ ||= 600-692¢ ||= upped ||

||= d32 ||= C - Db4 ||= 26-edo ||= ~692¢ ||= 692-720¢ ||= 600-692¢ ||= upped ||

||= d44 ||= C - Fb4 ||= 29-edo ||= 703¢ ||= 703-720¢ ||= 600-703¢ ||= downed ||

||= d44 ||= C - Fb4 ||= 29-edo ||= ~703¢ ||= 703-720¢ ||= 600-703¢ ||= downed ||

||= d43 ||= C - Eb5 ||= 31-edo ||= 697¢ ||= 697-720¢ ||= 600-697¢ ||= upped ||

||= d43 ||= C - Eb5 ||= 31-edo ||= ~697¢ ||= 697-720¢ ||= 600-697¢ ||= upped ||

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

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As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.

The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. __The enharmonic interval's number of ups or downs is equal to the height__. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

Not all enharmonics work with all pergens. ~~__The~~ implied edo ~~is always~~ a multiple of the octave ~~fraction__~~. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d32. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.

[//Question: what if there are highs and lows?//]

Not all enharmonics work with all pergens. The implied edo must be a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d32. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.

Every rank-2 interval has a **genspan**, which is the number of generators needed to create the interval. It's also the position of the interval on the relative genchain. For conventional (un-upped) intervals, the genspan is the interval's position on the relative chain of 5ths, which runs ...d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... It equals the 3-factor of the interval's monzo.

If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.

~~[//Check this last paragraph!/~~/]

G (enharmonic) = N * G (gen) - G(multi-gen)

~~[//Question: what if there are highs and lows?/~~/]

P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2

P5/2 = M3, enh = 2(4) - 1 = A1

To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.

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(to be continued)</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a> | <a href="#x^/P4"> ^/P4 </a> | <a href="#B# ||"> B# ||</a> | <a href="#C# ||"> C# ||</a> | <a href="#Db ||"> Db ||</a> | <a href="#B## ||"> B## ||</a> | <a href="#D ||"> D ||</a> | <a href="#B#3 ||"> B#3 ||</a> | <a href="#B# ||"> B# ||</a> | <a href="#B## ||"> B## ||</a> | <a href="#C# ||"> C# ||</a> | <a href="#Eb3 ||"> Eb3 ||</a> | <a href="#Db ||"> Db ||</a> | <a href="#B#3 ||"> B#3 ||</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>pergen names</title></head><body><a href="#Definition">Definition</a> | <a href="#Derivation">Derivation</a> | <a href="#Applications">Applications</a> | <a href="#x^/P4"> ^/P4 </a> | <a href="#B#">B#</a> | <a href="#C#">C#</a> | <a href="#Db">Db</a> | <a href="#B##">B##</a> | <a href="#D">D</a> | <a href="#B#3">B#3</a> | <a href="#B#">B#</a> | <a href="#B##">B##</a> | <a href="#C#">C#</a> | <a href="#Eb3">Eb3</a> | <a href="#Db">Db</a> | <a href="#B#3">B#3</a>

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

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</table>

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.

The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &quot;W&quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.

[Question: at what point is &quot;W&quot; actually needed?]

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The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.

An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.

[This 13b part needs clarification]

[This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.]

(table is under construction)

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<td style="text-align: center;">C - F#v=Gb^ - C

</td>

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

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

^1 = 81/80

</td>

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vv\\M2

</td>

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

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

C^^ = C#

C^^ = C#

C^^ = D

C^^ = D

</td>

<td style="text-align: center;">C - D/=Eb\ - F,

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</td>

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

C = Db,

C = Db,

C^^ = C#

C^^ = C#

</td>

<td style="text-align: center;">C - F#v=Gb^ - C,

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</td>

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

C = C#

C = C#

C^^ = B##

C^^ = B##

</td>

<td style="text-align: center;">C - F#v=Gb^ - C,

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<td style="text-align: center;">^3d2

</td>

<td style="text-align: center;">C^3 <h1 id="toc4"><a name="B# ||"></a> B#

<td style="text-align: center;">C^3

<h1 id="toc4"><a name="B#"></a>B#</h1>

</td>

<td>C - Ev - Ab^ - C

</td>

~~<td></h1>~~

~~C - Ev - Ab^ - C</td>~~

</td>

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<td style="text-align: center;">v3A1

</td>

<td style="text-align: center;">C^3 <h1 id="toc5"><a name="C# ||"></a> C#

<td style="text-align: center;">C^3

<h1 id="toc5"><a name="C#"></a>C#</h1>

</td>

<td>C - Dv - Eb^ - F

</td>

~~<td></h1>~~

~~C - Dv - Eb^ - F</td>~~

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

</td>

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<td style="text-align: center;">v3m2

</td>

<td style="text-align: center;">C^3 <h1 id="toc6"><a name="Db ||"></a> Db

<td style="text-align: center;">C^3

<h1 id="toc6"><a name="Db"></a>Db</h1>

</td>

<td>C - D^ - Fv - G

</td>

~~<td></h1>~~

~~C - D^ - Fv - G</td>~~

</td>

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<td style="text-align: center;">^3dd2

</td>

<td style="text-align: center;">C^3 <h1 id="toc7"><a name="B## ||"></a> B##

<td style="text-align: center;">C^3

<h1 id="toc7"><a name="B##"></a>B##</h1>

</td>

<td>C - F#v - Cb^ - F

</td>

~~<td></h1>~~

~~C - F#v - Cb^ - F</td>~~

</td>

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<td style="text-align: center;">v3M2

</td>

<td style="text-align: center;">C^3 <h1 id="toc8"><a name="D ||"></a> D

<td style="text-align: center;">C^3

<h1 id="toc8"><a name="D"></a>D</h1>

</td>

<td>C F^ Cv F

</td>

~~<td></h1>~~

~~C F^ Cv F</td>~~

</td>

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<td style="text-align: center;">^6d32

</td>

<td style="text-align: center;">C^6 <h1 id="toc9"><a name="B#3 ||"></a> B#3

<td style="text-align: center;">C^6

<h1 id="toc9"><a name="B#3"></a>B#3</h1>

</td>

<td>~~</h1>~~

<td>C - Fxv3=Gbb^3 CC - D#vv - Fb^^ - G

C - Fxv3=Gbb^3 CC - D#vv - Fb^^ - G

</td>

Line 1,304:

Line 1,327:

<td style="text-align: center;">^4d2

</td>

<td style="text-align: center;">C^4 <h1 id="toc10"><a name="B# ||"></a> B#

<td style="text-align: center;">C^4

<h1 id="toc10"><a name="B#"></a>B#</h1>

</td>

<td>C Ebv Gbvv A^ C

</td>

~~<td></h1>~~

~~C Ebv Gbvv A^ C</td>~~

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

^1 = 81/80

Line 1,321:

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<td style="text-align: center;">^4dd2

</td>

<td style="text-align: center;">C^4 <h1 id="toc11"><a name="B## ||"></a> B##

<td style="text-align: center;">C^4

<h1 id="toc11"><a name="B##"></a>B##</h1>

</td>

<td>C Db^ Ebb^^ Ev F

</td>

~~<td></h1>~~

~~C Db^ Ebb^^ Ev F</td>~~

</td>

Line 1,337:

Line 1,362:

<td style="text-align: center;">v4A1

</td>

<td style="text-align: center;">C^4 <h1 id="toc12"><a name="C# ||"></a> C#

<td style="text-align: center;">C^4

<h1 id="toc12"><a name="C#"></a>C#</h1>

</td>

<td>C Dv Evv F^ G^

</td>

~~<td></h1>~~

~~C Dv Evv F^ G^</td>~~

</td>

Line 1,353:

Line 1,379:

<td style="text-align: center;">v4dd3

</td>

<td style="text-align: center;">C^4 <h1 id="toc13"><a name="Eb3 ||"></a> Eb3

<td style="text-align: center;">C^4

<h1 id="toc13"><a name="Eb3"></a>Eb3</h1>

</td>

<td>C E^ G#^^ Dbv F

</td>

~~<td></h1>~~

~~C E^ G#^^ Dbv F</td>~~

</td>

Line 1,369:

Line 1,396:

<td style="text-align: center;">v4m2

</td>

<td style="text-align: center;">C^4 <h1 id="toc14"><a name="Db ||"></a> Db

<td style="text-align: center;">C^4

<h1 id="toc14"><a name="Db"></a>Db</h1>

</td>

<td>C Fv Bbvv D^ G

</td>

~~<td></h1>~~

~~C Fv Bbvv D^ G</td>~~

</td>

Line 1,481:

Line 1,509:

<td style="text-align: center;">^5d32

</td>

<td style="text-align: center;">C^5 <h1 id="toc15"><a name="B#3 ||"></a> B#3

<td style="text-align: center;">C^5

<h1 id="toc15"><a name="B#3"></a>B#3</h1>

</td>

<td>~~</h1>~~

<td>C - Db^ - Ebb^ -- E#vv - F#v - G

C - Db^ - Ebb^ -- E#vv - F#v - G

</td>

Line 1,512:

Line 1,540:

[Question: how to find all possible pergens?]

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, ~~the first~~ enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to ~~make~~ the enharmonic interval ~~vanish~~. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus ups and downs may need to be swapped, depending on the size of the 5th in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &quot;sweet spot&quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus ups and downs may need to be swapped, depending on the size of the 5th in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.

[Question: What to do if the edo's 5th falls in the sweet spot? Example?]

[Question: What to do if the edo's 5th falls in the sweet spot? Example?]

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is 646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.

<tr>

<th colspan="2">enharmonic interval

<th colspan="2">bare enharmonic interval

</th>

<th>implied edo

Line 1,589:

Line 1,617:

<td style="text-align: center;">7-edo

</td>

<td style="text-align: center;">686¢

<td style="text-align: center;">~686¢

</td>

<td style="text-align: center;">600-686¢

Line 1,621:

Line 1,649:

<td style="text-align: center;">17-edo

</td>

<td style="text-align: center;">706¢

<td style="text-align: center;">~706¢

</td>

<td style="text-align: center;">706-720¢

Line 1,637:

Line 1,665:

<td style="text-align: center;">19-edo

</td>

<td style="text-align: center;">695¢

<td style="text-align: center;">~695¢

</td>

<td style="text-align: center;">695-720¢

Line 1,653:

Line 1,681:

<td style="text-align: center;">26-edo

</td>

<td style="text-align: center;">692¢

<td style="text-align: center;">~692¢

</td>

<td style="text-align: center;">692-720¢

Line 1,669:

Line 1,697:

<td style="text-align: center;">29-edo

</td>

<td style="text-align: center;">703¢

<td style="text-align: center;">~703¢

</td>

<td style="text-align: center;">703-720¢

Line 1,685:

Line 1,713:

<td style="text-align: center;">31-edo

</td>

<td style="text-align: center;">697¢

<td style="text-align: center;">~697¢

</td>

<td style="text-align: center;">697-720¢

Line 1,717:

Line 1,745:

As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.

The LCM of the pergen's two splitting fractions is called the height of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

The LCM of the pergen's two splitting fractions is called the height of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. The enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.

~~Not all enharmonics work with all pergens.~~ ~~The implied edo is always a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo,~~ and ~~its enharmonic can only be those that imply even edos: M2, d2, or d3~~</~~span~~>~~2. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.~~

[Question: what if there are highs and lows?]

[~~Check this last paragraph!~~]

Not all enharmonics work with all pergens. The implied edo must be a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d32. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.

[~~Question: what if there are highs and lows?~~]

Every rank-2 interval has a genspan, which is the number of generators needed to create the interval. It's also the position of the interval on the relative genchain. For conventional (un-upped) intervals, the genspan is the interval's position on the relative chain of 5ths, which runs ...d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... It equals the 3-factor of the interval's monzo.

If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.

G (enharmonic) = N * G (gen) - G(multi-gen)

P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2

P5/2 = M3, enh = 2(4) - 1 = A1

To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.

@@ 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>2017-11-21 17:08:36 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-21 23:38:04 UTC</tt>.<br>
-: The original revision id was <tt>622142511</tt>.<br>
+: The original revision id was <tt>622159045</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 39: / Line 39: @@
 ||= {P8/4, P5} ||= quarter-octave ||= (3,4,-4) ||= diminished ||= quadruple green ||= g&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;T ||
 ||= {P8, P12/5} ||= fifth-twelfth ||= (-10,-1,5) ||= magic ||= large quintuple yellow ||= Ly&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;T ||
 The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one "W" per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.
 [//Question: at what point is// "//W" actually needed?//]
@@ Line 122: / Line 122: @@
 The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.
 An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.
-[//This 13b part needs clarification//]
+[//This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.//]
 (table is under construction)
@@ Line 138: / Line 138: @@
 ||~ halves ||~   ||~   ||~   ||~   ||~   ||~   ||
 ||= {P8/2, P5} ||= P8/2 = vA4 = ^d5
 (if 5th &gt; 700¢) ||= ^^d2 ||= C^^ = B# ||= C - F#v=Gb^ - C ||= srutal
 ^1 = 81/80 ||= 12, 14, 16, 18b, 20*,
 , 24*, 26, 28*, 30* ||
@@ Line 158: / Line 158: @@
 vvA1,
 ^^\\d2,
-vv\\M2 ||= C// = Db,
+vv\\M2 ||= C //= Db,//
-C^^ = C#
+//C^^ = C#//
-C^^// = D ||= C - D/=Eb\ - F,
+//C^^// = D ||= C - D/=Eb\ - F,
 C - Eb^=Ev - G,
 C - F#v/=Gb^\ - C,
@@ Line 172: / Line 172: @@
 \\m2,
 vv\\A1 ||= C^^ = B#
-C// = Db,
+C //= Db,//
-C^^// = C# ||= C - F#v=Gb^ - C,
+//C^^// = C# ||= C - F#v=Gb^ - C,
 C - D/=Eb\ - F,
 C - Eb^/=Ev\ - G ||= sgg&amp;bbT
@@ Line 183: / Line 183: @@
 \\A1,
 ^^\\m2 ||= C^^ = B#
-C// = C#
+C //= C#//
-C^^// = B## ||= C - F#v=Gb^ - C,
+//C^^// = B## ||= C - F#v=Gb^ - C,
 C - Eb/=E\ - G,
 C - Dv/=Eb^\ - F ||= sgg&amp;aaT
@@ Line 190: / Line 190: @@
 /1 = 33/32 ||= " ||
 ||~ thirds ||~   ||~   ||~   ||~   ||~   ||~   ||
-||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = B# ||= &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;C - Ev - Ab^ - C&lt;/span&gt; ||=   ||= 12, 15, 18b*, 21,
+||= {P8/3, P5} ||= P8/3 = vM3 = ^^d4 ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;
+=B#= || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;C - Ev - Ab^ - C&lt;/span&gt; ||=   ||= 12, 15, 18b*, 21,
 *, 27, 30* ||
-||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = C# ||= C - Dv - Eb^ - F ||= porcupine ||= 13b, 14*, 15, 21*,
+||= {P8, P4/3} ||= P4/3 = ^^m2 = vM2 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;
+=C#= || C - Dv - Eb^ - F ||= porcupine ||= 13b, 14*, 15, 21*,
 , 28*, 29, 30* ||
-||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = Db ||= C - D^ - Fv - G ||=   ||= 15*, 16, 20*, 21,
+||= {P8, P5/3} ||= P5/3 = ^M2 = vvm3 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;
+=Db= || C - D^ - Fv - G ||=   ||= 15*, 16, 20*, 21,
 *, 26, 30*, 31 ||
-||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = B## ||= C - F#v - Cb^ - F ||=   ||=   ||
+||= {P8, P11/3} ||= P11/3 = vA4 = ^^dd5 ||= ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;
-||= " ||= P11/3 = ^P4 = vvP5 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; = D ||= C F^ Cv F ||=   ||= " ||
+=B##= || C - F#v - Cb^ - F ||=   ||=   ||
+||= " ||= P11/3 = ^P4 = vvP5 ||= v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2 ||= C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;
+=D= || C F^ Cv F ||=   ||= " ||
 ||= {P8/3, P4/2} ||=   ||=   ||=   ||=   ||=   ||= 15, 18b*, 24, 30 ||
 ||= {P8/3, P5/2} ||=   ||=   ||=   ||=   ||=   ||= 18b, 24, 30 ||
 ||= {P8/2, P4/3} ||=   ||=   ||=   ||=   ||=   ||= 14, 22, 28*, 30* ||
 ||= {P8/2, P5/3} ||= P8/2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5
-P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt; = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; C
+P5/3 = vvA2 = ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= ^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;
-C - D#vv - Fb^^ - G ||=   ||= 16, 20*, 26, 30* ||
+=B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;= || C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; CC - D#vv - Fb^^ - G ||=   ||= 16, 20*, 26, 30* ||
 ||= " ||= P8/2 = vA4 = ^d5
 P5/3 = /M2 = \\m3 ||= ^^d2,
@@ Line 212: / Line 217: @@
 ||= {P8/3, P4/3} ||=   ||=   ||=   ||=   ||=   ||= 15, 21, 30* ||
 ||~ quarters ||~   ||~   ||~   ||~   ||~   ||~   ||
-||= {P8/4, P5} ||= P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2 ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; = B# ||= C Ebv Gbvv A^ C ||= diminished,
+||= {P8/4, P5} ||= P8/4 = vm3 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A2 ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;
+=B#= || C Ebv Gbvv A^ C ||= diminished,
 ^1 = 81/80 ||= 12, 16, 20, 24*, 28 ||
-||= {P8, P4/4} ||= P4/4 = ^m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA1 ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; = B## ||= C Db^ Ebb^^ Ev F ||=   ||=   ||
+||= {P8, P4/4} ||= P4/4 = ^m2 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;AA1 ||= ^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;
-||= {P8, P5/4} ||= P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; = C# ||= C Dv Evv F^ G^ ||=   ||=   ||
+=B##= || C Db^ Ebb^^ Ev F ||=   ||=   ||
-||= {P8, P11/4} ||= P11/4 = ^M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; = Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= C E^ G#^^ Dbv F ||=   ||=   ||
+||= {P8, P5/4} ||= P5/4 = vM2 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;
-||= {P8, P12/4} ||= P12/4 = vP4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; = Db ||= C Fv Bbvv D^ G ||=   ||=   ||
+=C#= || C Dv Evv F^ G^ ||=   ||=   ||
+||= {P8, P11/4} ||= P11/4 = ^M3 = v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd5 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;
+=Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;= || C E^ G#^^ Dbv F ||=   ||=   ||
+||= {P8, P12/4} ||= P12/4 = vP4 = ^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M3 ||= v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2 ||= C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;
+=Db= || C Fv Bbvv D^ G ||=   ||=   ||
 ||= {P8/4, P4/2} ||=   ||=   ||=   ||=   ||=   ||=   ||
 ||= {P8/2, P4/4} ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 224: / Line 234: @@
 ||~ fifths ||~   ||~   ||~   ||~   ||~   ||~   ||
 ||= {P8/5, P5} ||= P8/5 = ||= m2 ||=   ||=   ||=   ||=   ||
-||= {P8, P5/5} ||= P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1 ||= ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; = B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= C - Db^ - Ebb^ -
+||= {P8, P5/5} ||= P5/5 = ^m2 = v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;1 ||= ^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;
-- E#vv - F#v - G ||=   ||=   ||
+=B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;= || C - Db^ - Ebb^ -- E#vv - F#v - G ||=   ||=   ||
 ||= " ||=   ||= v&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;dd3 ||=   ||=   ||=   ||=   ||
 [//Question: how to find all possible pergens?//]
-Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.
+Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a "sweet spot" for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus __**ups and downs may need to be swapped, depending on the size of the 5th**__ in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.
 [//Question: What to do if the edo's 5th falls in the sweet spot? Example?//]
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is 646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.
-||||~ enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||
+||||~ bare enharmonic interval ||~ implied edo ||~ edo's 5th ||~ upping range ||~ downing range ||~ if 5th is just ||
 ||= M2 ||= C - D ||= 2-edo ||= 600¢ ||= none ||= all ||= downed ||
 ||= m3 ||= C - Eb ||= 3-edo ||= 800¢ ||= none ||= all ||= downed ||
 ||= m2 ||= C - Db ||= 5-edo ||= 720¢ ||= none ||= all ||= downed ||
-||= A1 ||= C - C# ||= 7-edo ||= 686¢ ||= 600-686¢ ||= 686¢-720¢ ||= downed ||
+||= A1 ||= C - C# ||= 7-edo ||= ~686¢ ||= 600-686¢ ||= 686¢-720¢ ||= downed ||
 ||= d2 ||= C - Dbb ||= 12-edo ||= 700¢ ||= 700-720¢ ||= 600-700¢ ||= upped ||
-||= dd3 ||= C - Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= 17-edo ||= 706¢ ||= 706-720¢ ||= 600-706¢ ||= downed ||
+||= dd3 ||= C - Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= 17-edo ||= ~706¢ ||= 706-720¢ ||= 600-706¢ ||= downed ||
-||= dd2 ||= C - Db&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= 19-edo ||= 695¢ ||= 695-720¢ ||= 600-695¢ ||= upped ||
+||= dd2 ||= C - Db&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; ||= 19-edo ||= ~695¢ ||= 695-720¢ ||= 600-695¢ ||= upped ||
-||= d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C - Db&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= 26-edo ||= 692¢ ||= 692-720¢ ||= 600-692¢ ||= upped ||
+||= d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2 ||= C - Db&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= 26-edo ||= ~692¢ ||= 692-720¢ ||= 600-692¢ ||= upped ||
-||= d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 ||= C - Fb&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= 29-edo ||= 703¢ ||= 703-720¢ ||= 600-703¢ ||= downed ||
+||= d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;4 ||= C - Fb&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt; ||= 29-edo ||= ~703¢ ||= 703-720¢ ||= 600-703¢ ||= downed ||
-||= d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;3 ||= C - Eb&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ||= 31-edo ||= 697¢ ||= 697-720¢ ||= 600-697¢ ||= upped ||
+||= d&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;3 ||= C - Eb&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt; ||= 31-edo ||= ~697¢ ||= 697-720¢ ||= 600-697¢ ||= upped ||
 ||= etc. ||=   ||=   ||=   ||=   ||=   ||=   ||
@@ Line 252: / Line 262: @@
 As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.
 The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. __The enharmonic interval's number of ups or downs is equal to the height__. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.
-Not all enharmonics work with all pergens. __The implied edo is always a multiple of the octave fraction__. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.
+[//Question: what if there are highs and lows?//]
+Not all enharmonics work with all pergens. The implied edo must be a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.
+Every rank-2 interval has a **genspan**, which is the number of generators needed to create the interval. It's also the position of the interval on the relative genchain. For conventional (un-upped) intervals, the genspan is the interval's position on the relative chain of 5ths, which runs ...d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... It equals the 3-factor of the interval's monzo.
+If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.
-[//Check this last paragraph!//]
+G (enharmonic) = N * G (gen) - G(multi-gen)
-[//Question: what if there are highs and lows?//]
+P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2
+P5/2 = M3, enh = 2(4) - 1 = A1
 To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.
@@ Line 266: / Line 284: @@
 (to be continued)</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:32:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#x^/P4"&gt; ^/P4 &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#B# ||"&gt; B# ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt; | &lt;a href="#C# ||"&gt; C# ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt; | &lt;a href="#Db ||"&gt; Db ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt; | &lt;a href="#B## ||"&gt; B## ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt; | &lt;a href="#D ||"&gt; D ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#B#3 ||"&gt; B#3 ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt; | &lt;a href="#B# ||"&gt; B# ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt; | &lt;a href="#B## ||"&gt; B## ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt; | &lt;a href="#C# ||"&gt; C# ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt; | &lt;a href="#Eb3 ||"&gt; Eb3 ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Db ||"&gt; Db ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt; | &lt;a href="#B#3 ||"&gt; B#3 ||&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:32:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt; | &lt;a href="#x^/P4"&gt; ^/P4 &lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt; | &lt;a href="#B#"&gt;B#&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt; | &lt;a href="#C#"&gt;C#&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt; | &lt;a href="#Db"&gt;Db&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt; | &lt;a href="#B##"&gt;B##&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt; | &lt;a href="#D"&gt;D&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#B#3"&gt;B#3&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt; | &lt;a href="#B#"&gt;B#&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt; | &lt;a href="#B##"&gt;B##&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt; | &lt;a href="#C#"&gt;C#&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt; | &lt;a href="#Eb3"&gt;Eb3&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt; | &lt;a href="#Db"&gt;Db&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt; | &lt;a href="#B#3"&gt;B#3&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;
 &lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
@@ Line 502: / Line 520: @@
 &lt;/table&gt;
-The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc. &lt;br /&gt;
+The color names indicate the amount of splitting: deep splits something into two parts, triple into three parts, etc. For quadruple colors, the multi-gen may be the major 2nd 9/8, a whole tone. For example, large quadruple jade tempers out (-17,2,0,0,4), and is {P8/2, M2/4} = half-octave, quarter-tone. For nine-fold colors, the multi-gen may be the minor 3rd 32/27. These intervals may also be voiced wider, as 3/1, 8/3, etc. To avoid cumbersome degree names like 16th or 18th, for degrees above 12, the widening is indicated with one &amp;quot;W&amp;quot; per octave. Thus 27/8 = WM6, 9/2 = WWM2, etc.&lt;br /&gt;
 &lt;br /&gt;
 [&lt;em&gt;Question: at what point is&lt;/em&gt; &amp;quot;&lt;em&gt;W&amp;quot; actually needed?&lt;/em&gt;]&lt;br /&gt;
@@ Line 814: / Line 832: @@
 The genchain shown is a short section of the full genchain. C - G implies ...Eb Bb F C G D A E B F# C#... And C - Eb^=Ev - G implies ...F - Ab^=Av - C - Eb^=Ev - G - Bb^=Bv - D - F^=F#v - A - C^=C#v - E... If the octave is split, the genchain shows the octave: In C - F#v=Gb^ - C, the last C is an octave above the first one.&lt;br /&gt;
 &lt;br /&gt;
-An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation. &lt;br /&gt;
+An edo is incompatible with a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. An edo is somewhat incompatible with a pergen if the period and generator can only generate a subset of the edo. For example, 15-edo is somewhat incompatible with {P8, P5}, because any chain-of-5ths scale could only make a 5-edo subset. Such edos are marked with asterisks. 13b is incompatible with {P8, P5/2}, but 13 isn't. However, 13 is incompatible with heptatonic notation.&lt;br /&gt;
 &lt;br /&gt;
-[&lt;em&gt;This 13b part needs clarification&lt;/em&gt;]&lt;br /&gt;
+[&lt;em&gt;This 13b part needs clarification. 5ths wider than 720¢ can be played, but can't be notated as perfect 5ths.&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
 (table is under construction)&lt;br /&gt;
@@ Line 889: / Line 907: @@
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;srutal &lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;srutal&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
 &lt;/td&gt;
@@ Line 995: / Line 1,013: @@
 vv\\M2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C&lt;em&gt; = Db,&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C &lt;em&gt;= Db,&lt;/em&gt;&lt;br /&gt;
-C^^ = C#&lt;br /&gt;
+&lt;em&gt;C^^ = C#&lt;/em&gt;&lt;br /&gt;
-C^^&lt;/em&gt; = D&lt;br /&gt;
+&lt;em&gt;C^^&lt;/em&gt; = D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D/=Eb\ - F,&lt;br /&gt;
@@ Line 1,024: / Line 1,042: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;em&gt; = Db,&lt;br /&gt;
+C &lt;em&gt;= Db,&lt;/em&gt;&lt;br /&gt;
-C^^&lt;/em&gt; = C#&lt;br /&gt;
+&lt;em&gt;C^^&lt;/em&gt; = C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C,&lt;br /&gt;
@@ Line 1,050: / Line 1,068: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;em&gt; = C#&lt;br /&gt;
+C &lt;em&gt;= C#&lt;/em&gt;&lt;br /&gt;
-C^^&lt;/em&gt; = B##&lt;br /&gt;
+&lt;em&gt;C^^&lt;/em&gt; = B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C,&lt;br /&gt;
@@ Line 1,087: / Line 1,105: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;d2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="B# ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt; B#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="B#"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;B#&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;C - Ev - Ab^ - C&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;C - Ev - Ab^ - C&lt;/span&gt;&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,104: / Line 1,123: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;A1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="C# ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt; C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="C#"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;C#&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C - Dv - Eb^ - F&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C - Dv - Eb^ - F&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;porcupine&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,121: / Line 1,141: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Db ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt; Db&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Db"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Db&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C - D^ - Fv - G&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C - D^ - Fv - G&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,138: / Line 1,159: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;dd2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="B## ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt; B##&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="B##"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;B##&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C - F#v - Cb^ - F&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C - F#v - Cb^ - F&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,154: / Line 1,176: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;M2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="D ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt; D&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="D"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;D&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C F^ Cv F&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C F^ Cv F&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,219: / Line 1,242: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="B#3 ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;6&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="B#3"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;/h1&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;/h1&gt;
+         &lt;td&gt;C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; CC - D#vv - Fb^^ - G&lt;br /&gt;
- C - F&lt;span style="vertical-align: super;"&gt;x&lt;/span&gt;v&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;=Gbb^&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt; CC - D#vv - Fb^^ - G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,304: / Line 1,327: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;d2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="B# ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt; B#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="B#"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;B#&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Ebv Gbvv A^ C&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C Ebv Gbvv A^ C&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;diminished,&lt;br /&gt;
 ^1 = 81/80&lt;br /&gt;
@@ Line 1,321: / Line 1,345: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="B## ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt; B##&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="B##"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;B##&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Db^ Ebb^^ Ev F&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C Db^ Ebb^^ Ev F&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,337: / Line 1,362: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;A1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="C# ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt; C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="C#"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;C#&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Dv Evv F^ G^&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C Dv Evv F^ G^&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,353: / Line 1,379: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;dd3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Eb3 ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt; Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc13"&gt;&lt;a name="Eb3"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C E^ G#^^ Dbv F&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C E^ G#^^ Dbv F&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,369: / Line 1,396: @@
          &lt;td style="text-align: center;"&gt;v&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;m2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Db ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt; Db&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;4&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc14"&gt;&lt;a name="Db"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Db&lt;/h1&gt;
+&lt;/td&gt;
+        &lt;td&gt;C Fv Bbvv D^ G&lt;br /&gt;
 &lt;/td&gt;
-        &lt;td&gt;&lt;/h1&gt;
- C Fv Bbvv D^ G&lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 1,481: / Line 1,509: @@
          &lt;td style="text-align: center;"&gt;^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;  &lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="B#3 ||"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;5&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc15"&gt;&lt;a name="B#3"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;/h1&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;/h1&gt;
+         &lt;td&gt;C - Db^ - Ebb^ -- E#vv - F#v - G&lt;br /&gt;
- C - Db^ - Ebb^ -- E#vv - F#v - G&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 1,512: / Line 1,540: @@
 [&lt;em&gt;Question: how to find all possible pergens?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, the first enharmonic interval is ^^d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to make the enharmonic interval vanish. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
+Removing the ups and downs from an enharmonic interval makes a conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic interval (BEI) is d2, and d2 vanishes in 12-edo. Every rank-2 temperament has a &amp;quot;sweet spot&amp;quot; for tuning the 5th, usually a narrow range of about 5-10¢. If the temperament's 5th is flatter than 12-edo's, d2 is ascending, and if it's sharper, it's descending. The ups and downs are meant to indicate that the enharmonic interval vanishes. Thus if d2 is ascending, it should be downed, and if it's descending, upped. Thus &lt;u&gt;&lt;strong&gt;ups and downs may need to be swapped, depending on the size of the 5th&lt;/strong&gt;&lt;/u&gt; in the particular rank-2 tuning you are using. In the table, this is shown explicitly for {P8/2, P5}, and implied for all the other pergens. The other pergens' enharmonic intervals are upped or downed as if the 5th were just.&lt;br /&gt;
 &lt;br /&gt;
-[&lt;em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?&lt;/em&gt;] &lt;br /&gt;
+[&lt;em&gt;Question: What to do if the edo's 5th falls in the sweet spot? Example?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is 646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.&lt;br /&gt;
+Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. In practice, the lower limit of this range is ~646¢, for 13b-edo. For every enharmonic interval, the following table shows in what parts of this range the interval should be upped or downed.&lt;br /&gt;
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;enharmonic interval&lt;br /&gt;
+         &lt;th colspan="2"&gt;bare enharmonic interval&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;implied edo&lt;br /&gt;
@@ Line 1,589: / Line 1,617: @@
          &lt;td style="text-align: center;"&gt;7-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;686¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~686¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;600-686¢&lt;br /&gt;
@@ Line 1,621: / Line 1,649: @@
          &lt;td style="text-align: center;"&gt;17-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;706¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~706¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;706-720¢&lt;br /&gt;
@@ Line 1,637: / Line 1,665: @@
          &lt;td style="text-align: center;"&gt;19-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;695¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~695¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;695-720¢&lt;br /&gt;
@@ Line 1,653: / Line 1,681: @@
          &lt;td style="text-align: center;"&gt;26-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;692¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~692¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;692-720¢&lt;br /&gt;
@@ Line 1,669: / Line 1,697: @@
          &lt;td style="text-align: center;"&gt;29-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;703¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~703¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;703-720¢&lt;br /&gt;
@@ Line 1,685: / Line 1,713: @@
          &lt;td style="text-align: center;"&gt;31-edo&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;697¢&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;~697¢&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;697-720¢&lt;br /&gt;
@@ Line 1,717: / Line 1,745: @@
 As a corollary, every comma implies an edo, except for those that map to P1: notational ones, and those that are the sum or difference of notational ones.&lt;br /&gt;
 &lt;br /&gt;
-The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. &lt;u&gt;The enharmonic interval's number of ups or downs is equal to the height&lt;/u&gt;. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed. &lt;br /&gt;
+The LCM of the pergen's two splitting fractions is called the &lt;strong&gt;height&lt;/strong&gt; of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. &lt;u&gt;The enharmonic interval's number of ups or downs is equal to the height&lt;/u&gt;. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.&lt;br /&gt;
 &lt;br /&gt;
-Not all enharmonics work with all pergens. &lt;u&gt;The implied edo is always a multiple of the octave fraction&lt;/u&gt;. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.&lt;br /&gt;
+[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;br /&gt;
 &lt;br /&gt;
-[&lt;em&gt;Check this last paragraph!&lt;/em&gt;]&lt;br /&gt;
+Not all enharmonics work with all pergens. The implied edo must be a multiple of the octave fraction. Thus a half-octave pergen can never imply an odd-numbered edo, and its enharmonic can only be those that imply even edos: M2, d2, or d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. A quarter-octave pergen must imply 12-edo, and its enharmonic must be a d2.&lt;br /&gt;
-[&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;br /&gt;
+&lt;br /&gt;
+Every rank-2 interval has a &lt;strong&gt;genspan&lt;/strong&gt;, which is the number of generators needed to create the interval. It's also the position of the interval on the relative genchain. For conventional (un-upped) intervals, the genspan is the interval's position on the relative chain of 5ths, which runs ...d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... It equals the 3-factor of the interval's monzo.&lt;br /&gt;
+&lt;br /&gt;
+If the multi-gen's fraction is N, the enharmonic interval's genspan is N times the genspan of the gen minus the genspan of the multi-gen.&lt;br /&gt;
+&lt;br /&gt;
+G (enharmonic) = N * G (gen) - G(multi-gen)&lt;br /&gt;
+&lt;br /&gt;
+P5/2 = m3, G(enh) = 2 (-3) - 1 = -5 = m2&lt;br /&gt;
+P5/2 = M3, enh = 2(4) - 1 = A1&lt;br /&gt;
 &lt;br /&gt;
 To notate a single-comma rank-2 temperament, first find the temper's pergen. Then find the enharmonic interval, which is the comma's mapping. Then look up the pergen / enharmonic combination in the main table.&lt;br /&gt;