Kite's thoughts on pergens: Difference between revisions

Line 1:

<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>2018-01-08 00:01:44 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-08 01:59:28 UTC</tt>.

: The original revision id was <tt>~~624535539~~</tt>.

: The original revision id was <tt>624537149</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.

Line 32:

||= (P8/2, P5) ||= half-octave ||= (11, -4, -2) ||= srutal ||= small deep green ||= sggT ||

||= " ||= " ||= 81/80, 50/49 ||= injera ||= deep reddish and green ||= rryy&gT ||

||= (P8, P5/2} ||= half-fifth ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||

||= (P8, P5/2) ||= half-fifth ||= 25/24 ||= dicot ||= deep yellow ||= yyT ||

||= " ||= " ||= (-1,5,0,0,-2) ||= mohajira ||= deep amber ||= aaT ||

||= (P8, P4/2) ||= half-fourth ||= 49/48 ||= semaphore ||= deep blue ||= bbT ||

Line 237:

everything ||= ||= ||= ||= ||= ||

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

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

||= (P8/4, P5) ||= ^4d2 ||= C^4 ``=`` B# ||= P8/4 = vm3 = ^3A2 ||= C Ebv Gbvv=F#^^ A^ C ||= diminished,

^1 = 81/80 ||

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

||= (P8, P4/4) ||= ^4dd2 ||= C^4 ``=`` B## ||= P4/4 = ^m2 = v3AA1 ||= C Db^ Ebb^^=D#vv Ev F ||= ||

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

||= (P8, P5/4) ||= v4A1 ||= C^4 ``=`` C# ||= P5/4 = vM2 = ^3m2 ||= C Dv Evv=Eb^^ F^ G ||= tetracot ||

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

||= (P8, P11/4) ||= v4dd3 ||= C^4 ``=`` Eb3 ||= P11/4 = ^M3 = v3dd5 ||= C E^ G#^^ Dbv F ||= ||

||= {P8, P12/4} ||= v4m2 ||= C^4 ``=`` Db ||= P12/4 = v4 = ^3M3 ||= C Fv Bbvv=A^^ D^ G ||= ||

||= (P8, P12/4) ||= v4m2 ||= C^4 ``=`` Db ||= P12/4 = v4 = ^3M3 ||= C Fv Bbvv=A^^ D^ G ||= ||

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

||= (P8/4, P4/2) ||= ||= ||= ||= ||= ||

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

||= (P8/2, M2/4) ||= ||= ||= ||= ||= ||

||= {P8/2, P5/4} ||= ||= ||= ||= ||= ||

||= (P8/2, P4/4) ||= ||= ||= ||= ||= ||

||= {P8/4, P4/3} ||= ||= ||= ||= ||= ||

||= (P8/2, P5/4) ||= ||= ||= ||= ||= ||

||= {P8/4, P5/3} ||= ||= ||= ||= ||= ||

||= (P8/4, P4/3) ||= ||= ||= ||= ||= ||

||= {P8/4, P11/3} ||= ||= ||= ||= ||= ||

||= (P8/4, P5/3) ||= ||= ||= ||= ||= ||

||= {P8/3, P4/4} ||= ||= ||= ||= ||= ||

||= (P8/4, P11/3) ||= ||= ||= ||= ||= ||

||= {P8/3, P5/4} ||= ||= ||= ||= ||= ||

||= (P8/3, P4/4) ||= ||= ||= ||= ||= ||

||= {P8/3, P11/4} ||= ||= ||= ||= ||= ||

||= (P8/3, P5/4) ||= ||= ||= ||= ||= ||

||= {P8/3, P12/4} ||= ||= ||= ||= ||= ||

||= (P8/3, P11/4) ||= ||= ||= ||= ||= ||

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

||= (P8/3, P12/4) ||= ||= ||= ||= ||= ||

||= (P8/4, P4/4) ||= ||= ||= ||= ||= ||

Removing the ups and downs from an enharmonic interval makes a "bare" enharmonic, a conventional 3-limit 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¢. 12-edo's fifth is the "tipping point": 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. Therefore __**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. In the table, the other pergens' enharmonic intervals are upped or downed as if the 5th were just.

Line 353:

Line 354:

==Alternate enharmonics==

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12*[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded way up to [4,2] = M3. The enharmonic becomes [33,19] - 12*[4,2] = [-15,-5] = -5*[3,1] = -5 * v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2.

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12*[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12*[4,2] = [-15,-5] = -5*[3,1] = -5 * v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2.

P1 -- ^4m3 -- v4M6 -- C

C -- Eb^4 -- Av4 -- C

Line 383:

Line 384:

==Alternate keyspans and stepspans==

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. ~~These~~ edos would also work, 12-edo is merely the most convenient choice, ~~mostly~~ because of its familiarity.

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.

== ==

==Combining pergens==

Line 395:

Line 396:

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

==Pergens and EDOs==

Line 401:

Pergens have much in common with edos. Pergens (rank-2 ones) assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but far fewer than a hundred have been explored. There are an infinite number of pergens, but far fewer than a hundred will suffice most composers.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported.

Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. However (P8/3, P5) is supported. In practice, if the generator's keyspan is very small, a partially supported pergen. For example, 22edo and 2\22 generator.

How many pergens are fully supported by a given edo? Surprisingly, an infinite number! For example, 12edo supports (P8, P4/5), (P8, P11/17), (P8, WWP4/29), (P8, W3P4/41), etc.

How many edos support a given pergen? Again, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and if k is M's N-edo keyspan~~, k by n~~. To be fully supported, N/m and k/n must be coprime.

How many edos support a given pergen? Again, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is M's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.

Given an edo, a period p\N and a generator g\N, what is the pergen? The octave fraction n is N/p. Let the edo's 5th be f\N. To find the multigen M, we must find a monzo (a,b) such that a*N + b*f is a multiple of g. If n = 1, |b| = 1.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. 13-edo and 18-edo are incompatible with heptatonic notation, and 13b-edo and 18b-edo are used instead.

Line 416:

Line 418:

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

||= (P8/2, P5) ||= half-octave ||= 12, 14, 16, 18b, 20*, 22, 24*, 26, 28*, 30* ||

||= (P8, P4/2) ||= half-fourth ||= 14, 15*, 18b*, 19, 20*, 23,24, 25*, 28*, 29, 30* ||

||= (P8, P4/2) ||= half-fourth ||= 13b, 14, 15*, 18b*, 19, 20*, 23, 24, 25*, 28*, 29, 30* ||

||= (P8, P5/2) ||= half-fifth ||= 14*, 17, 18b, 20*, 21*, 24, 27, 28*, 30*, 31 ||

||= (P8/2, P4/2) ||= half-everything ||= 14, 18b, 20*, 24, 28*, 30* ||

Line 423:

Line 425:

||= (P8, P4/3) ||= third-fourth ||= 13b, 14*, 15, 21*, 22, 28*, 29, 30* ||

||= (P8, P5/3) ||= third-fifth ||= 15*, 16, 20*, 21, 25*, 26, 30*, 31 ||

||= (P8, P11/3) ||= third-11th ||= ||

||= (P8, P11/3) ||= third-11th ||= 15, 17, 21, 23, 30* ||

||= (P8/3, P4/2) ||= third-8ve, half-4th ||= 15, 18b*, 24, 30 ||

||= (P8/3, P4/2) ||= third-8ve, half-4th ||= 15, 18b*, 24, 30* ||

||= (P8/3, P5/2) ||= third-8ve, half-5th ||= 18b, 24, 30 ||

||= (P8/3, P5/2) ||= third-8ve, half-5th ||= 18b, 21, 24, 27, 30 ||

||= (P8/2, P4/3) ||= half-8ve, third-4th ||= 14, 22, 28*, 30* ||

||= (P8/2, P4/3) ||= half-8ve, third-4th ||= 14, 22, 28*, 30 ||

||= (P8/2, P5/3) ||= half-8ve, third-5th ||= 16, 20*, 26, 30* ||

||= (P8/2, P11/3) ||= half-8ve, third-11th ||= ||

||= (P8/2, P11/3) ||= half-8ve, third-11th ||= 19, 30 ||

||= (P8/3, P4/3) ||= third-everything ||= 15, 21, 30* ||

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

||= {P8/4, P5} ||= ||= 12, 16, 20, 24*, 28 ||

||= (P8/4, P5) ||= quarter-octave ||= 12, 16, 20, 24*, 28 ||

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

||= (P8, P4/4) ||= quarter-fourth ||= 18b*, 19, 20*, 28, 29, 30* ||

||= {P8, P5/4} ||= ||= ||

||= (P8, P5/4) ||= quarter-fifth ||= 14*, 20, 21*, 27, 28* ||

||= {P8, P11/4} ||= ||= ||

||= (P8, P11/4) ||= quarter-eleventh ||= 14, 17, 20, 28*, 31 ||

||= {P8, P12/4} ||= ||= ||

||= (P8, P12/4) ||= quarter-twelfth ||= 13b, 15*, 18b, 20*, 23, 25*, 28, 30* ||

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

||= (P8/4, P4/2) ||= quarter-octave, half-fourth ||= 20, 24, 28 ||

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

||= (P8/2, M2/4) ||= half-octave, quarter-tone ||= 20, 22, 24, 26, 28 ||

||= {P8/2, P5/4} ||= ||= ||

||= (P8/2, P4/4) ||= half-octave, quarter-fourth ||= 18b, 20*, 28, 30* ||

||= {P8/4, P4/3} ||= ||= ||

||= (P8/2, P5/4) ||= half-octave, quarter-fifth ||= 14, 20, 28* ||

||= {P8/4, P5/3} ||= ||= ||

||= (P8/4, P4/3) ||= quarter-octave, third-fourth ||= 28 ||

||= {P8/4, P11/3} ||= ||= ||

||= (P8/4, P5/3) ||= quarter-octave, third-fifth ||= 16, 20 ||

||= {P8/3, P4/4} ||= ||= ||

||= (P8/4, P11/3) ||= quarter-octave, third-eleventh ||= ||

||= {P8/3, P5/4} ||= ||= ||

||= (P8/3, P4/4) ||= third-octave, quarter-fourth ||= 18b*, 30 ||

||= {P8/3, P11/4} ||= ||= ||

||= (P8/3, P5/4) ||= third-octave, quarter-fifth ||= 21, 27 ||

||= {P8/3, P12/4} ||= ||= ||

||= (P8/3, P11/4) ||= third-octave, quarter-eleventh ||= ||

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

||= (P8/3, P12/4) ||= third-octave, quarter-twelfth ||= 15, 18b, 30* ||

||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||

Line 646:

Line 649:

</tr>

<tr>

<td style="text-align: center;">(P8, P5/2}

<td style="text-align: center;">(P8, P5/2)

</td>

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

Line 1,553:

Line 1,556:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P5}

<td style="text-align: center;">(P8/4, P5)

</td>

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

Line 1,568:

Line 1,571:

</tr>

<tr>

<td style="text-align: center;">{P8, P4/4}

<td style="text-align: center;">(P8, P4/4)

</td>

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

Line 1,582:

Line 1,585:

</tr>

<tr>

<td style="text-align: center;">{P8, P5/4}

<td style="text-align: center;">(P8, P5/4)

</td>

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

Line 1,596:

Line 1,599:

</tr>

<tr>

<td style="text-align: center;">{P8, P11/4}

<td style="text-align: center;">(P8, P11/4)

</td>

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

Line 1,610:

Line 1,613:

</tr>

<tr>

<td style="text-align: center;">{P8, P12/4}

<td style="text-align: center;">(P8, P12/4)

</td>

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

Line 1,624:

Line 1,627:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P4/2}

<td style="text-align: center;">(P8/4, P4/2)

</td>

</td>

</td>

</td>

</td>

</td>

</tr>

<tr>

<td style="text-align: center;">(P8/2, M2/4)

</td>

Line 1,638:

Line 1,655:

</tr>

<tr>

<td style="text-align: center;">{P8/2, P4/4}

<td style="text-align: center;">(P8/2, P4/4)

</td>

Line 1,652:

Line 1,669:

</tr>

<tr>

<td style="text-align: center;">{P8/2, P5/4}

<td style="text-align: center;">(P8/2, P5/4)

</td>

Line 1,666:

Line 1,683:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P4/3}

<td style="text-align: center;">(P8/4, P4/3)

</td>

Line 1,680:

Line 1,697:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P5/3}

<td style="text-align: center;">(P8/4, P5/3)

</td>

Line 1,694:

Line 1,711:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P11/3}

<td style="text-align: center;">(P8/4, P11/3)

</td>

Line 1,708:

Line 1,725:

</tr>

<tr>

<td style="text-align: center;">{P8/3, P4/4}

<td style="text-align: center;">(P8/3, P4/4)

</td>

Line 1,722:

Line 1,739:

</tr>

<tr>

<td style="text-align: center;">{P8/3, P5/4}

<td style="text-align: center;">(P8/3, P5/4)

</td>

Line 1,736:

Line 1,753:

</tr>

<tr>

<td style="text-align: center;">{P8/3, P11/4}

<td style="text-align: center;">(P8/3, P11/4)

</td>

Line 1,750:

Line 1,767:

</tr>

<tr>

<td style="text-align: center;">{P8/3, P12/4}

<td style="text-align: center;">(P8/3, P12/4)

</td>

Line 1,764:

Line 1,781:

</tr>

<tr>

<td style="text-align: center;">{P8/4, P4/4}

<td style="text-align: center;">(P8/4, P4/4)

</td>

Line 2,099:

Line 2,116:

<h2 id="toc9"><a name="Further Discussion-Alternate enharmonics"></a>Alternate enharmonics</h2>

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12*[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded way up to [4,2] = M3. The enharmonic becomes [33,19] - 12*[4,2] = [-15,-5] = -5*[3,1] = -5 * v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2.

Sometimes the enharmonic found by rounding off the gedra can be greatly improved by rounding off differently. For example, (P8/3, P4/4) unreduces to (P8/3, WWM6/12), a false double. The bare alternate generator is WWM6/12 = [33,19]/12 = [3,2] = m3. The bare enharmonic is [33,19] - 12*[3,2] = [-3,-5] = a quintuple-diminished 6th! This would make for a very confusing notation. However, [33,19]/12 can be rounded very inaccurately all the way up to [4,2] = M3. The enharmonic becomes [33,19] - 12*[4,2] = [-15,-5] = -5*[3,1] = -5 * v12A2, which is an improvement but still awkward. The period is ^4m3 and the generator is v3M2.

P1 -- ^4m3 -- v4M6 -- C

C -- Eb^4 -- Av4 -- C

Line 2,129:

Line 2,146:

<h2 id="toc10"><a name="Further Discussion-Alternate keyspans and stepspans"></a>Alternate keyspans and stepspans</h2>

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. ~~These~~ edos would also work, 12-edo is merely the most convenient choice, ~~mostly~~ because of its familiarity.

One might wonder, when using gedras, why 12-edo keyspans? Why heptatonic stepspans? Heptatonic is best because conventional notation is heptatonic, and we want to minimize the heptatonic stepspan of the enharmonic. In order for the matrix to be invertible, the edo must be connected to the 4\7 kite on the scale tree. The only choices are 5-edo, 12-edo, 19-edo, 26-edo, 33-edo, 40-edo and 47-edo, and on the other side, 2-edo, 9-edo, 16-edo and 23-edo. Any of these edos would also work. 12-edo is merely the most convenient choice, because of its familiarity. Dividing the gedra directly only gives you an estimate of the best period or generator. As noted in the previous section, to improve the enharmonic, this initial estimate must often be revised. So the choice of estimating edo isn't very important.

<h2 id="toc11"> </h2>

<h2 id="toc12"><a name="Further Discussion-Combining pergens"></a>Combining pergens</h2>

Line 2,138:

Line 2,155:

<ul><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li></ul>

However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.

~~ ~~

<h2 id="toc13"><a name="Further Discussion-Pergens and EDOs"></a>Pergens and EDOs</h2>

Line 2,144:

Line 2,160: