Kite's thoughts on pergens: Difference between revisions

For example, srutal could be (P8/2, M2/2), but P5 is preferred because it is unsplit. Or it could be (P8/2, P12), but P5 is preferred because it is smaller. Or it could be (P8/2, P4), but P5 is always preferred over P4. Note that P5/2 is <u>not</u> preferred over P4/2. For example, decimal is (P8/2, P4/2), not (P8/2, P5/2).

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! colspan="2" | pergen

! colspan="2" | [[Color notation|color name]]

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(P8/2, P4/2) is called half-everything because the fifth is also split in half, and since every 3-limit interval can be expressed as the sum/difference of octaves and fifths, every single 3-limit interval is also split in half.

Rank-3 pergens are trickier to find. For example, Breedsmic is 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. [http://x31eq.com/cgi-bin/rt.cgi?ets=130_171_270&limit=7 x31.com] gives us this matrix:

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! | period

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! | gen1

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! | gen2

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! | period

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! | gen1

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! | gen2

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Use an [http://wims.unice.fr/wims/wims.cgi?session=GF84B8C7BF.1&lang=en&cmd=reply&module=tool%2Flinear%2Fmatmult.en&matA=1+1+1%0D%0A0+2+1%0D%0A0+0+2&matB=&show=A%5E-1 online tool] to invert it. Here "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

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! | 2/1

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! | 3/1

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! | 5/1

| | /4

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Thus the period = (4,0,0) / 4 = 2/1= P8, gen1 = (-2,2,0) / 4 = (-1,1,0) / 2 = P5/2, and gen2 = (-1,-1,2) / 4 = (25:6)/4.

Fortunately, there is a better way. Discard the 3rd column instead, and keep the 4th one:

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! | period

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! | gen1

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! | gen2

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This inverts to this matrix:

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! | 2/1

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! | 3/1

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! | 7/1

| | /2

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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 multigen. A double half-5th is a 5th = (-1,1,0), and this gives us (-4,0,2)/2 = (-2,0,1) = 7/4. The fraction disappears, the multigen becomes the gen, and we can add/subtract the period and the gen1 directly. Subtracting an octave and inverting makes gen2 = 8/7. Adding an octave and subtracting 4 half-5ths makes 64/63. The pergen is (P8, P5/2, /1) with /1 = 64/63, rank-3 half-5th. This is far better than (P8, P5/2, (96:25)/4).

If the octave is split, the table has a '''perchain''' ("peer-chain", chain of periods) that shows the octave: C -- F#v=Gb^ -- C. Genchains have a theoretically infinite length, but perchains have a finite length. The full rank-2 lattice has genchains running horizontally and perchains running vertically.

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unsplit

schismic, mavila, archy, etc.

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half-8ve

&gt; 700¢

^1 = 81/80

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&lt; 700¢)

^1 = 64/63

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^1 = 27/26

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half-4th

^1 = 64/63

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^1 = 81/80

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half-5th

^1 = 33/32

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half-

everything

vvA1,

vv\\M2

C^^ = C#

C^^// = D

P5/2 = ^m3 = vM3

= ^/4 = v\P5

C - Eb^=Ev - G,

C - F^/=Gv\ - C

^1 = 33/32

/1 = 64/63

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\\m2,

vv\\A1

C// = Db

C^^// = C#

P4/2 = /M2 = \m3

P5/2 = ^/m3 = v\M3

C - D/=Eb\ - F,

C - Eb^/=Ev\ - G

^1 = 81/80

/1 = 64/63

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\\A1,

^^\\m2

C// = C#

C^^\\ = B

P5/2 = /m3 = \M3

P4/2 =v/M2 = ^\m3

C - Eb/=E\ - G,

C - Dv/=Eb^\ - F

^1 = 81/80

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third-8ve

^1 = 81/80

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third-4th

^1 = 81/80

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third-5th

^1 = 64/63

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third-11th

^1 = 729/704

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^1 = 33/32

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third-8ve, half-4th

P4/2 = ^<span style="vertical-align: super;">3</span>m2 = v<span style="vertical-align: super;">3</span>M3

C - Db^<span style="vertical-align: super;">3</span>=Ev<span style="vertical-align: super;">3</span> - F

^1 = 33/32

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\\m2

C// = Db

P4/2 = /M2 = \m3

C - D/=Eb\ - F

^1 = 81/80, /1 = 64/63

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third-8ve, half-5th

\\A1

C// = C#

P5/2 = /m3 = \M3

C - Eb/=E\ - G

^1 = 49/48, /1 = 343/324

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half-8ve, third-4th

\<span style="vertical-align: super;">3</span>A1

C/<span style="vertical-align: super;">3</span> = C#

P4/3 = \M2 = //m2

C - D\ - Eb/ - F

^1 = 1029/1024, /1 = 49/48

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half-8ve, third-5th

P5/3 = vvA2 = ^<span style="vertical-align: super;">4</span>dd3

C - D#vv - Fb^^ - G

^1 = 81/80

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\\\m2

C/// = Db

P5/3 = /M2 = \\m3

C - D/ - F\ - G

^1 = (10,-6,1,-1), /1 = 64/63

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half-8ve, third-11th

P11/3 = ^^4 = v<span style="vertical-align: super;">4</span>5

C - F^^ - Cvv - F

^1 = 33/32

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third-

everything

\<span style="vertical-align: super;">3</span>A1

C/3 = C#

P4/3 = \M2 = //m2

P5/3 = v/M2

C - D\ - Eb/ - F

C - Dv/ - F^\ - G

^1 = 64/63

/1 = 81/80

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\<span style="vertical-align: super;">3</span>m2

C/<span style="vertical-align: super;">3</span> = Db

P5/3 = /M2 = \\m3

P4/3 = v\M2

C - D/ - F\ - G

C - Dv\ - Eb^/ - F

^1 = 81/80

/1 = 64/63

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\<span style="vertical-align: super;">3</span>m2

C/3 = Db

P5/3 = /M2 = \\m3

P8/3 = v/M3

C - D/ - F\ - G

C - Ev/ - Ab^\ - C

^1 = 81/80

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The disadvantage to the lexicographical ordering above is that more complex pergens are listed before simpler ones, e.g. half-8ve third-5th before quarter-5th. However, the former can arise from two simple commas, so arguably it isn't particularly complex.

Heptatonic 5th-based notation is only possible if the 5th ranges from 600¢ to 720¢. For every bare enharmonic, the following table shows in what parts of this range this interval should be upped or downed. The tipping point edo is simply the 3-exponent of the bare enharmonic.

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! colspan="2" | bare enharmonic

! | if the 5th is just

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==Naming very large intervals==

The following table shows the secondary splits for all pergens up to the third-splits. A split interval is only included if it falls in the range from d5 to A5 on the genchain of 5ths, others are too remote. For convenience, naturally occurring splits are listed too, under "all pergens". (P8/3, P4/2) has all the secondary splits that (P8/3, P5) and (P8, P4/2) have, plus additional ones.

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! colspan="2" | pergen

! | secondary splits of a 12th or less

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! colspan="2" | half-splits

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! colspan="2" | third-splits

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The following table shows how to notate various 5-limit rank-2 temperaments. The sweet spot isn't precisely defined, thus all cents are approximate. The up symbol's ratio is always the mapping comma, or its inverse.

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! | <u>5-limit temperament</u>

! | cents

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The schismic comma is a negative (i.e. descending) dim 2nd because it takes you down the scale, but up in pitch. The mavila temperament could perhaps be notated without ups and downs, because 5/4 is still a 3rd, and the 4:5:6 triad still looks like a triad.

Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one. Enharmonics are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank. Examples:

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! | tuning

! | enharmonics

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