Kite's thoughts on pergens: Difference between revisions

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If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.

Sometimes the mapping comma needs to be inverted. In ~~diminished, which sets 6/5 = P8/4, ^1 = 80/81. in~~ every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. See also blackwood-like pergens below.

Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also blackwood-like pergens below.

Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.

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

C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — 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). It only applies to imperfect intervals. The mid interval is the interval exactly midway between major and minor. 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 has not yet been fully investigated. There may be pergens for which a 2nd mid symbol is useful. If one pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, and only if the genchain's E is an A1 that isn't triple-downed.

==Alternate enharmonics==

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gedra

count

mid

edomapping

upspan

liftspan

==Miscellaneous Notes==

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d = -r

Gedras can also be expanded by adding an entry for ups/downs, ~~and perhaps one~~ for ~~lifts/drops too~~. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].

'''Height of a pergen'''

@@ Line 1,138: / Line 1,138: @@
 If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.
-Sometimes the mapping comma needs to be inverted. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. in every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. See also blackwood-like pergens below.
+Sometimes the mapping comma needs to be inverted. In every temperament except those in the meantone family, the 81/80 comma is not tempered out, but it is still tempered, just like every ratio. Occasionally 81/80 is tempered so far that it becomes a descending interval. In diminished, which sets 6/5 = P8/4, ^1 = 80/81. See also blackwood-like pergens below.
 Cents can be assigned to each accidental symbol, even if no specific commas are specified. Let c = the cents of the tuning's 5th from 700¢, the 12edo 5th. Thus P5 = 700¢ + c. From this we can calculate the cents of any 3-limit interval. The sharp always equals A1 = 100¢ + 7c. Since the enharmonic = 0¢, we can derive the cents of the up symbol. If the enharmonic is vvA1, then vvA1 = 0¢, and ^1 = (A1)/2 = (100¢ + 7c)/2 = 50¢ + 3.5c. If the 5th is 696¢, c = -4 and the up symbol equals 36¢. /1 can be similarly derived from its enharmonic.
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 <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;">C — C^/=Dbv/ — Db//=D\ — D^=Ebv — Eb/=E\\ — E^/=Fv\ — 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). It only applies to imperfect intervals. The mid interval is the interval exactly midway between major and minor. 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 has not yet been fully investigated. There may be pergens for which a 2nd mid symbol is useful. If one pair appears only in the perchain and the other pair only in the genchain, then the mids appear only in the genchain, and only if the genchain's E is an A1 that isn't triple-downed.
 ==Alternate enharmonics==
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 gedra
+count
+mid
 edomapping
+upspan
+liftspan
 ==Miscellaneous Notes==
@@ Line 4,245: / Line 4,255: @@
 d = -r
-Gedras can also be expanded by adding an entry for ups/downs, and perhaps one for lifts/drops too. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].
+Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1].
 <u>'''Height of a pergen'''</u>