Pergen names: 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>2017-11-22 00:18:44 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 01:34:58 UTC</tt>.

: The original revision id was <tt>~~622159783~~</tt>.

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=__Derivation__=

For any comma containing primes 2 and 3, let N = the GCD of all its prime factors other than the 2-factor, and let N' = the GCD of all its higher-prime factors. The comma will split the octave into N parts, and if N' > N, it will split some other 3-limit interval into N' parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N > M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

[//Question: for multi-comma tempers, does the hermite-reduced (minimal prime subgroups) comma list always indicate all the splits?//]

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Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G.

~~Not all possible~~ combinations of periods and generators are ~~unique~~ pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.

Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.

The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.

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^^\\d2,

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

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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[//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, {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 above 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 "bare" conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 above 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. The implied edo is ~~just~~ the 3-factor of the bare enharmonic ~~interval~~.

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 bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.

||||~ 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 ||

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[//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.~~

Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:

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~~.

For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d32. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.

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~~.

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d32. For {P8, P12/5}, the implied edo is 5y ± 1.

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

For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.

Therefore there must be two accidental pairs, each and two implied edos, and

P5/2 = m3, ~~G(enh)~~ = ~~2 (-~~3) - ~~1 =~~ -~~7 = d8~~, ~~inverts to A1~~

For {P8/M, P5}, the implied edo = Mx

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

For {P8, multi-gen/N}, the implied edo = Ny ± 1

For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,

or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1

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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>

<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>

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

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

A pergen (pronounced &quot;peer-gen&quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

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The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: {P8, P5, 81/80, 64/63, ...}. These commas are called notational commas. They determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.

<h1 id="toc1"><a name="Derivation"></a>Derivation</h1>

<h1 id="toc1"><a name="Derivation"></a>Derivation</h1>

For any comma containing primes 2 and 3, let N = the GCD of all its prime factors other than the 2-factor, and let N' = the GCD of all its higher-prime factors. The comma will split the octave into N parts, and if N' &gt; N, it will split some other 3-limit interval into N' parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.

[Question: for multi-comma tempers, does the hermite-reduced (minimal prime subgroups) comma list always indicate all the splits?]

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Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, but only if the smaller prime is &gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.

<h1 id="toc2"><a name="Applications"></a>Applications</h1>

<h1 id="toc2"><a name="Applications"></a>Applications</h1>

Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, <a class="wiki_link" href="/Ups%20and%20Downs%20Notation">ups and downs</a>. And certain rank-2 temperaments require another additional pair. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.

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Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G.

~~Not all possible~~ combinations of periods and generators are ~~unique~~ pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.

Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.

The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.

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</td>

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

~~~~C^^ = C#~~~~

C^^ = C#

~~C^^~~<~~/em~~> = 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^^~~<~~/em~~> = 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#~~<~~/em~~>

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 = B#

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

</td>

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

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

</td>

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

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

</td>

<td style="text-align: center;">C - Dv - Eb^ - F

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

</td>

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

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

</td>

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

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

</td>

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

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

</td>

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

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

</td>

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

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

</td>

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

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

</td>

<td style="text-align: center;">C^6 = B#3

<td style="text-align: center;">C^6 = B#3

</td>

<td style="text-align: center;">C - Fxv3=Gbb^3 C

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</td>

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

C/// = Db

C/// = Db

</td>

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

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

</td>

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

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

</td>

<td style="text-align: center;">C Ebv Gbvv A^ C

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

</td>

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

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

</td>

<td style="text-align: center;">C Db^ Ebb^^ Ev F

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

</td>

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

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

</td>

<td style="text-align: center;">C Dv Evv F^ G

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

</td>

<td style="text-align: center;">C^4 = Eb3

<td style="text-align: center;">C^4 = Eb3

</td>

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

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

</td>

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

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

</td>

<td style="text-align: center;">C Fv Bbvv D^ G

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

</td>

<td style="text-align: center;">C^5 = B#3

<td style="text-align: center;">C^5 = B#3

</td>

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

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[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, {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 above 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 &quot;bare&quot; conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 above 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. The implied edo is ~~just~~ the 3-factor of the bare enharmonic ~~interval~~.

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 bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.

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[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.~~

Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:

~~Every rank-2 interval has a genspan<~~/~~strong>~~, ~~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~~.

For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d32. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.

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~~.

For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d32. For {P8, P12/5}, the implied edo is 5y ± 1.

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

For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.

Therefore there must be two accidental pairs, each and two implied edos, and

P5/2 = m3, ~~G(enh)~~ = ~~2 (-~~3) - ~~1 =~~ -~~7 = d8~~, ~~inverts to A1~~

For {P8/M, P5}, the implied edo = Mx

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

For {P8, multi-gen/N}, the implied edo = Ny ± 1

For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,

or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1

@@ 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>2017-11-22 00:18:44 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-22 01:34:58 UTC</tt>.<br>
-: The original revision id was <tt>622159783</tt>.<br>
+: The original revision id was <tt>622161181</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 55: / Line 55: @@
 =__Derivation__=
-For any comma containing primes 2 and 3, let N = the GCD of all its prime factors other than the 2-factor, and let N' = the GCD of all its higher-prime factors. The comma will split the octave into N parts, and if N' &gt; N, it will split some other 3-limit interval into N' parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.
+For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.
 [//Question: for multi-comma tempers, does the hermite-reduced (minimal prime subgroups) comma list always indicate all the splits?//]
@@ Line 116: / Line 116: @@
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G.
-Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.
+Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.
 The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.
@@ Line 161: / Line 161: @@
 ^^\\d2,
 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 175: / Line 175: @@
 vv\\A1 ||= C^^ = B#
 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 185: / Line 185: @@
 \\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 232: / Line 232: @@
 [//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, {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 above 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 "bare" conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 above 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. The implied edo is just the 3-factor of the bare enharmonic interval.
+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 bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.
 ||||~ 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 ||
@@ Line 258: / Line 258: @@
 [//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.
+Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:
-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.
+For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.
-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.
+For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8, P12/5}, the implied edo is 5y ± 1.
-G (enharmonic) = N * G (gen) - G(multi-gen)
+For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.
+Therefore there must be two accidental pairs, each and two implied edos, and
-P5/2 = m3, G(enh) = 2 (-3) - 1 = -7 = d8, inverts to A1
+For {P8/M, P5}, the implied edo = Mx
-P5/2 = M3, enh = 2(4) - 1 = A1
+For {P8, multi-gen/N}, the implied edo = Ny ± 1
+For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,
+or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1
@@ Line 277: / Line 280: @@
 (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:22:&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:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&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:25:&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:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Derivation"&gt;Derivation&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt; | &lt;a href="#Applications"&gt;Applications&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:19:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:19 --&gt;&lt;u&gt;&lt;strong&gt;Definition&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 A &lt;strong&gt;pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) is a way of identifying a rank-2 or rank-3 regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.&lt;br /&gt;
@@ Line 527: / Line 530: @@
 The preferred pergen for untempered just intonation is the octave, the fifth, and a list of commas, each containing only one higher prime: {P8, P5, 81/80, 64/63, ...}. These commas are called &lt;strong&gt;notational commas&lt;/strong&gt;. They determine how a higher prime is mapped to a 3-limit interval, and thus how ratios containing higher primes are notated on the staff. By definition, the only commas that map to P1 are notational ones, and those that are the sum or difference of notational ones. There is universal agreement that 5's notational comma is 81/80. But the choice of notational commas for other primes, especially 11 and 13, is somewhat arbitrary. For example, if 11's notational comma is 33/32, 11/8 is notated as a perfect 4th. But if it's 729/704, 11/8 is an augmented 4th. Note that the choice of 11's notational comma affects the mapping of other 11-limit commas.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:21:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Derivation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:21 --&gt;&lt;u&gt;Derivation&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
-For any comma containing primes 2 and 3, let N = the GCD of all its prime factors other than the 2-factor, and let N' = the GCD of all its higher-prime factors. The comma will split the octave into N parts, and if N' &amp;gt; N, it will split some other 3-limit interval into N' parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.&lt;br /&gt;
+For any comma containing primes 2 and 3, let M = the GCD of all its prime factors other than the 2-factor, and let N = the GCD of all its higher-prime factors. The comma will split the octave into M parts, and if N &amp;gt; M, it will split some other 3-limit interval into N parts. Therefore a comma with only two factors, one of which is the 2-factor, always splits the octave (unless the other factor is ±1, e.g. 32/31). And a comma with only one higher-prime factor will always split something, unless that factor is ±1.&lt;br /&gt;
 &lt;br /&gt;
 [&lt;em&gt;Question: for multi-comma tempers, does the hermite-reduced (minimal prime subgroups) comma list always indicate all the splits?&lt;/em&gt;]&lt;br /&gt;
@@ Line 813: / Line 816: @@
 Again, period = P8 and gen1 = P5/2. Gen2 = (-3, -1, 2)/2. To add gen1 to gen2, add a double gen1 to the 2nd multi-gen, the multi-gen2. A double half-fifth is a fifth = (-1, 1, 0), and this gives us (-4, 0, 2)/2 = 7/4. The fraction disappears, the multi-gen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7 = r2. Adding an octave and subtracting 4 half-fifths makes 64/63 = r1. The pergen is {P8, P5/2, r1} = half-fifth with red. This is far better than {P8, P5/2, gg7/4}. The pergen sometimes uses a larger prime in place of a smaller one, in order to avoid splitting gen2, but only if the smaller prime is &amp;gt; 3. In other words, the first priority is to have as few higher primes (colors) as possible, next to have as few fractions as possible, finally to have the higher primes be as small as possible.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:23:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Applications"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:23 --&gt;&lt;u&gt;Applications&lt;/u&gt;&lt;/h1&gt;
   &lt;br /&gt;
 Pergens allow a systematic exploration of notations for rank-2, rank-3, etc. regular temperaments, without having to examine each of the thousands of individual temperaments. For example, all unsplit temperaments are notated identically. They require only conventional notation: 7 nominals, plus sharps and flats. But most rank-2 temperaments require an additional pair of accidentals, &lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;ups and downs&lt;/a&gt;. And certain rank-2 temperaments require another additional pair. One possibility is highs and lows, written / and \. v\D is down-low D, and /P5 is high-five. Alternatively, color accidentals (y/g, r/b, j/a, etc.) could be used. However, this constrains a pergen to a specific temperament. For example, both mohajira and dicot are {P8, P5/2}. Using y/g implies dicot, using j/a implies mohajira, but using ^/v implies neither, and is a more general notation.&lt;br /&gt;
@@ Line 819: / Line 822: @@
 Analogous to 22-edo, sometimes additional accidentals aren't needed, but are desirable, to avoid misspelled chords. For example, schismic is unsplit and can be notated conventionally. But this causes 4:5:6 to be spelled as C Fb G. With ^1 = 81/80, the chord can be spelled properly as C Ev G.&lt;br /&gt;
 &lt;br /&gt;
-Not all possible combinations of periods and generators are unique pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.&lt;br /&gt;
+Some combinations of periods and generators are duplicates of other pergens. {P8/2, P5/2} is actually {P8/2, P4/2}, and {P8/2, M2/2} is actually {P8/2, P5}. Some combinations are impossible. There is no {P8, M2/2}. The following table lists all the rank-2 pergens that contain primes 2 and 3, grouped by the size of the larger splitting factor.&lt;br /&gt;
 &lt;br /&gt;
 The enharmonic interval can be added to or subtracted from any note or interval, renaming it, but not changing the pitch of the note (or width of the interval). It's analogous to the dim 2nd in 12-edo, which equates C# with Db, A4 with d5, etc. In a single-comma temperament, the comma maps to the enharmonic interval.&lt;br /&gt;
@@ Line 1,009: / Line 1,012: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C&lt;!-- ws:start:WikiTextRawRule:01:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:01 --&gt; = Db&lt;br /&gt;
-&lt;em&gt;C^^ = C#&lt;/em&gt;&lt;br /&gt;
+C^^ = C#&lt;br /&gt;
-&lt;em&gt;C^^&lt;/em&gt; = D&lt;br /&gt;
+C^^&lt;!-- ws:start:WikiTextRawRule:02:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:02 --&gt; = D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D/=Eb\ - F,&lt;br /&gt;
@@ Line 1,037: / Line 1,040: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;!-- ws:start:WikiTextRawRule:02:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:02 --&gt; = Db&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:03:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:03 --&gt; = Db&lt;br /&gt;
-&lt;em&gt;C^^&lt;/em&gt; = C#&lt;br /&gt;
+C^^&lt;!-- ws:start:WikiTextRawRule:04:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:04 --&gt; = C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C,&lt;br /&gt;
@@ Line 1,063: / Line 1,066: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C &lt;em&gt;= C#&lt;/em&gt;&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:05:``//`` --&gt;//&lt;!-- ws:end:WikiTextRawRule:05 --&gt; = C#&lt;br /&gt;
-&lt;em&gt;C^^&lt;/em&gt; = B##&lt;br /&gt;
+C^^\\ = B&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C,&lt;br /&gt;
@@ Line 1,100: / Line 1,103: @@
          &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:WikiTextRawRule:03:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:03 --&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;!-- ws:start:WikiTextRawRule:06:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:06 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Ev - Ab^ - C&lt;br /&gt;
@@ Line 1,117: / Line 1,120: @@
          &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;!-- ws:start:WikiTextRawRule:04:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:04 --&gt; &lt;/span&gt;C#&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;C^&lt;span style="vertical-align: super;"&gt;3 &lt;!-- ws:start:WikiTextRawRule:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&gt; &lt;/span&gt;C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Dv - Eb^ - F&lt;br /&gt;
@@ Line 1,134: / Line 1,137: @@
          &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:WikiTextRawRule:05:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:05 --&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;!-- ws:start:WikiTextRawRule:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - D^ - Fv - G&lt;br /&gt;
@@ Line 1,151: / Line 1,154: @@
          &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:WikiTextRawRule:06:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:06 --&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;!-- ws:start:WikiTextRawRule:09:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:09 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v - Cb^ - F&lt;br /&gt;
@@ Line 1,167: / Line 1,170: @@
          &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:WikiTextRawRule:07:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:07 --&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;!-- ws:start:WikiTextRawRule:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&gt; D&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C F^ Cv F&lt;br /&gt;
@@ Line 1,232: / Line 1,235: @@
          &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:WikiTextRawRule:08:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:08 --&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;!-- ws:start:WikiTextRawRule:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&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&lt;br /&gt;
@@ Line 1,252: / Line 1,255: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C^^ = B#&lt;br /&gt;
-C&lt;!-- ws:start:WikiTextRawRule:09:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:09 --&gt; = Db&lt;br /&gt;
+C&lt;!-- ws:start:WikiTextRawRule:012:``///`` --&gt;///&lt;!-- ws:end:WikiTextRawRule:012 --&gt; = Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - F#v=Gb^ - C&lt;br /&gt;
@@ Line 1,317: / Line 1,320: @@
          &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:WikiTextRawRule:010:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:010 --&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;!-- ws:start:WikiTextRawRule:013:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:013 --&gt; B#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Ebv Gbvv A^ C&lt;br /&gt;
@@ Line 1,334: / Line 1,337: @@
          &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:WikiTextRawRule:011:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:011 --&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;!-- ws:start:WikiTextRawRule:014:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:014 --&gt; B##&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Db^ Ebb^^ Ev F&lt;br /&gt;
@@ Line 1,350: / Line 1,353: @@
          &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:WikiTextRawRule:012:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:012 --&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;!-- ws:start:WikiTextRawRule:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&gt; C#&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Dv Evv F^ G&lt;br /&gt;
@@ Line 1,366: / Line 1,369: @@
          &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:WikiTextRawRule:013:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:013 --&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;!-- ws:start:WikiTextRawRule:016:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:016 --&gt; Eb&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C E^ G#^^ Dbv F&lt;br /&gt;
@@ Line 1,382: / Line 1,385: @@
          &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:WikiTextRawRule:014:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:014 --&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;!-- ws:start:WikiTextRawRule:017:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:017 --&gt; Db&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C Fv Bbvv D^ G&lt;br /&gt;
@@ Line 1,494: / Line 1,497: @@
          &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:WikiTextRawRule:015:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:015 --&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;!-- ws:start:WikiTextRawRule:018:``=`` --&gt;=&lt;!-- ws:end:WikiTextRawRule:018 --&gt; B#&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C - Db^ - Ebb^ -&lt;br /&gt;
@@ Line 1,525: / Line 1,528: @@
 [&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, {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 above 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 &amp;quot;bare&amp;quot; conventional interval, which vanishes in certain edos. For example, {P8/2, P5}'s enharmonic interval is ^^d2, the bare enharmonic 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 above 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;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. The implied edo is just the 3-factor of the bare enharmonic interval.&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 bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The implied edo is simply the 3-factor of the bare enharmonic.&lt;br /&gt;
@@ Line 1,734: / Line 1,737: @@
 [&lt;em&gt;Question: what if there are highs and lows?&lt;/em&gt;]&lt;br /&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;
+Not all enharmonics work with all pergens. The possible enharmonics can be deduced from the pergen as follows:&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;
+For {P8/M, P5}, the bare enharmonic is the difference between M bare periods and an octave. If x is the 3-factor of the period, the enharmonic interval's 3-factor is Mx, and the implied edo is |Mx|. For example, for {P8/2, P5}, the implied edo is |2x|, i.e., an even number. Possible bare enharmonics are M2, d2, and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8/4, P5}, the implied edo is a multiple of 4, and only d2 is possible.&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;
+For {P8, multi-gen/N}, since the octave is unsplit, the only possible multi-gens are P4, P5, P11, P12, or some other voicing of the 5th. Thus the multi-gen's 3-factor is 1. The bare enharmonic is the difference between N bare generators and the multi-gen. If y is the 3-factor of the generator, the bare enharmonic's 3-factor is Ny ± 1, and the implied edo is |Ny ± 1|. For example, for {P8, P4/2}, the implied edo is 2y ± 1, thus it must be an odd number, which rules out 2, 12 and 26, and thus M2, d2 and d&lt;span style="vertical-align: super;"&gt;3&lt;/span&gt;2. For {P8, P12/5}, the implied edo is 5y ± 1.&lt;br /&gt;
 &lt;br /&gt;
-G (enharmonic) = N * G (gen) - G(multi-gen)&lt;br /&gt;
+For {P8/M, multi-gen/N}, there are two conditions on the enharmonic. If T is the 3-factor of the multi-gen, the conditions are edo = Mx and edo = Ny ± T. For {P8/2, P4/2}, the two conditions are mutually exclusive: the edo must be both even and odd.&lt;br /&gt;
+Therefore there must be two accidental pairs, each and two implied edos, and &lt;br /&gt;
 &lt;br /&gt;
-P5/2 = m3, G(enh) = 2 (-3) - 1 = -7 = d8, inverts to A1&lt;br /&gt;
+For {P8/M, P5}, the implied edo = Mx&lt;br /&gt;
-P5/2 = M3, enh = 2(4) - 1 = A1&lt;br /&gt;
+For {P8, multi-gen/N}, the implied edo = Ny ± 1&lt;br /&gt;
+For {P8/M, multi-gen/N}, the implied edo = Mx = Ny ± T, where T is the 3-factor of the multi-gen,&lt;br /&gt;
+or, the 1st implied edo = Mx and the 2nd implied edo = Ny ± 1&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;