Kite's thoughts on pergens: Difference between revisions

Line 1,221:

It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. It might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozo), and its double-pair notation can also combine both~~. Half-4th has E = vvm2 and G = ^M2 = vm3~~. Third-4th has E' = \3A1 and G' = \M2 = ~~//m2~~. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - ~~\M2~~ = ^/1.

Sixth-4th with single-pair notation has an awkward ^6d64 enharmonic. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Third-4th has E = v3A1 and G'= vM2 = ^^m2. Half-4th has E' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent enharmonics are

P1 — ^/1=v/m2 — //m2=~~\M2~~ — ^M2=~~vm3~~ — /m3=~~\\M3~~ — ^\M3=v\4 — P4

P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4

C — C^/=~~Dbv/~~ — Db//=D\ — D^=~~Ebv~~ — Eb/~~=E\\ —~~ E^/=Fv\ — F

C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F

When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, E must be an A1, with any number of downs except 3. v5A1 creates m ^m v~ ^~ vM M. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.

Line 1,232:

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, ccM6/12), a false double. The bare alternate generator is ccM6/12 = [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. ^1 = 25¢ + 0.75·c, about an eighth-tone.

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

P1 -- ^4m3 -- v4M6 -- P8

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

P1 -- v3M2 -- v6M3=^6m2 -- ^3m3 -- P4

@@ Line 1,221: / Line 1,221: @@
 It's not yet known if every pergen can avoid large enharmonics (those of a 3rd or more) with double-pair notation. One situation in which very large enharmonics occur is the "half-step glitch". This is when the stepspan of the multigen is half (or a third, a quarter, etc.) of the multigen's splitting fraction. For example, in sixth-4th, six generators must cover three scale steps, and each one must cover a half-step. Each generator is either a unison or a 2nd, which causes the enharmonic's stepspan to equal the multigen's stepspan.
-Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 enharmonic. It might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore  commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Half-4th has E = vvm2 and G = ^M2 = vm3. Third-4th has E' = \<span style="vertical-align: super;">3</span>A1 and G' = \M2 = //m2. G - G' = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G - G' = ^M2 - \M2 = ^/1.
+Sixth-4th with single-pair notation has an awkward ^<span style="vertical-align: super;">6</span>d<span style="vertical-align: super;">6</span>4 enharmonic. This pergen might result from combining third-4th and half-4th (e.g. tempering out both the Porcupine and Semaphore  commas, aka Triyo & Zozo), and its double-pair notation can also combine both. Third-4th has E = v<span style="vertical-align: super;">3</span>A1 and G'= vM2 = ^^m2. Half-4th has E' = \\m2 and G' = /M2 = \m3. G' - G = P4/2 - P4/3 = P4/6. Thus the sixth-4th generator is G' - G = /M2 - vM2 = ^/1. Equivalent enharmonics are
-<span style="display: block; text-align: center;">P1 — ^/1=v/m2 — //m2=\M2 — ^M2=vm3 — /m3=\\M3 — ^\M3=v\4 — P4
+<span style="display: block; text-align: center;">P1 — ^/1=^\m2 — ^^m2=vM2 — /M2=\m3 — ^m3=vvM3 — v/M3=v\4 — P4
-<span style="display: block; text-align: center;">C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — F
+<span style="display: block; text-align: center;">C — ^/C=^\Db — ^^Db=vD — /D=\Eb — ^Eb=vvE — v/E=v\F — F
 When ups and downs are used to notate edos, a third symbol is used, a '''mid''' , written ~. The mid is only for relative notation (intervals), never for absolute notation (notes). For imperfect intervals, the mid interval is the interval exactly midway between major and minor. In addition, the mid-4th is midway between perfect and augmented, i.e. half-augmented, and the mid-5th is half-diminished. For example, 17edo's 2nds and 3rds run minor-mid-major. This avoids having to constantly choose between the equivalent terms upminor and downmajor. 72edo's 2nds and 3rds run minor, upminor, downmid, mid, upmid, downmajor, major. This makes the terms more concise. Mids can be used in pergen notation too. For single-pair, E must be an A1, with any number of downs except 3. v<sup>5</sup>A1 creates m ^m v~ ^~ vM M. Using mids with double-pair notation is trickier. Mids never appear in the perchain. If one accidental pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, if at all.
@@ Line 1,232: / Line 1,232: @@
 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, ccM6/12), a false double. The bare alternate generator is ccM6/12 = [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·v<span style="vertical-align: super;">12</span>A2, which is an improvement but still awkward. The period is ^<span style="vertical-align: super;">4</span>m3 and the generator is v<span style="vertical-align: super;">3</span>M2. ^1 = 25¢ + 0.75·c, about an eighth-tone.
-<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- C
+<span style="display: block; text-align: center;">P1 -- ^<span style="vertical-align: super;">4</span>m3 -- v<span style="vertical-align: super;">4</span>M6 -- P8
 <span style="display: block; text-align: center;">C -- Eb^<span style="vertical-align: super;">4</span> -- Av<span style="vertical-align: super;">4</span> -- C
 <span style="display: block; text-align: center;">P1 -- v<span style="vertical-align: super;">3</span>M2 -- v<span style="vertical-align: super;">6</span>M3=^<span style="vertical-align: super;">6</span>m2 -- ^<span style="vertical-align: super;">3</span>m3 -- P4