Xenharmonic Wiki

Tour of regular temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
→Temperaments defined by a comma involving 11: added extra info about the lattice/pergen. moved the quintuple-prime lattices to the rank-4 section.
Line 577: Line 577:
: The mirkwai family of rank three temperaments tempers out the mirkwai comma, {{Monzo|0 3 4 -5}} = 16875/16807. Four ~7/5 generators equal the pergen's compound upmajor 7th of  ~27/7.
: The mirkwai family of rank three temperaments tempers out the mirkwai comma, {{Monzo|0 3 4 -5}} = 16875/16807. Four ~7/5 generators equal the pergen's compound upmajor 7th of  ~27/7.


=== Temperaments defined by a comma involving 11 ===
=== Temperaments defined by an 11-limit comma ===
; [[Ptolemismic clan|Ptolemismic or Luyoyo clan]]  
; [[Ptolemismic clan|Ptolemismic or Luyoyo clan]] (P8, P5, ^1)
: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} = 100/99.
: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} = 100/99. 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI.  In the pergen, ^1 = ~81/80.


; [[Biyatismic clan|Biyatismic or Lologu clan]]
; [[Biyatismic clan|Biyatismic or Lologu clan]] (P8, P5, ^1)
: These temper out the biyatisma, {{monzo| -3 -1 -1 0 2 }} = 121/120.
: These temper out the biyatisma, {{monzo| -3 -1 -1 0 2 }} = 121/120. 5/4 is equated to 121/96, which is two 11/8's minus a 3/2 fifth. Since 121/96 is an ila (11-limit no-fives no-sevens) interval, every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.


; [[Valinorsmic clan|Valinorsmic or Lorugugu clan]]  
; [[Valinorsmic clan|Valinorsmic or Lorugugu clan]]  
: These temper out the valinorsma, {{monzo| 4 0 -2 -1 1 }} = 176/175.
: These temper out the valinorsma, {{monzo| 4 0 -2 -1 1 }} = 176/175. To be a rank-3 temperament, either an additional comma must vanish or the prime subgroup must omit prime 3. Thus no assumptions can be made about the pergen.


; [[Rastmic rank three clan|Rastmic or Lulu rank-3 clan]]
; [[Rastmic rank three clan|Rastmic or Lulu rank-3 clan]]
: These temper out the rastma, {{monzo| 1 5 0 0 -2 }} = 243/242. As an ila (11-limit no-fives no-sevens) rank-2 temperament, the pergen is (P8, P5/2).
: These temper out the rastma, {{monzo| 1 5 0 0 -2 }} = 243/242. In the corresponding [[Tour of regular temperaments#Clans defined by a 2.3.11 .28ila.29 comma|2.3.11 rank-2 temperament]], the pergen is (P8, P5/2).


; [[Keenanismic temperaments|Keenanismic or Lozoyo temperaments]]
; [[Pentacircle clan|Pentacircle or Saluzo clan]] (P8, P5, ^1)
: These temper out the keenanisma, {{monzo| -7 -1 1 1 1 }} = 385/384.  
: These temper out the pentacircle comma, {{monzo| 7 -4 0 1 -1 }} = 896/891. The interval between 11/8 and 7/4 is equated to 81/64. Since that is a 3-limit interval, every 2.3.11 interval is equated to a 2.3.7 interval and vice versa, and both the pergen and the lattice are identical to that of either 2.3.7 JI or 2.3.11 JI. In the pergen, ^1 is either ~64/63 or ~33/32 or ~729/704.


; [[Werckismic temperaments|Werckismic or Luzozogu temperaments]]
; [[Semicanousmic clan|Semicanousmic or Quadlo-agu clan]] (P8, P5, ^1)
: These temper out the werckisma, {{monzo| -3 2 -1 2 -1 }} = 441/440.
: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580. 5/4 is equated to an ila (11-limit no-fives no-sevens) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.


; [[Swetismic temperaments|Swetismic or Lururuyo temperaments]]
; [[Semiporwellismic clan|Semiporwellismic or Salulugu clan]] (P8, P5, ^1)
: These temper out the swetisma, {{monzo| 2 3 1 -2 -1 }} = 540/539.
: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335. 5/4 is equated to an ila (11-limit no-fives no-sevens) interval, thus every 2.3.5.11 interval is equated to an ila interval, and both the pergen and the lattice are identical to that of ila JI.  In the pergen, ^1 is either ~33/32 or ~729/704.


; [[Pentacircle clan|Pentacircle or Saluzo clan]]  
; [[Alpharabian rank three clan|Alpharabian or Laquadlo rank-3 clan]]
: These temper out the pentacircle comma, {{monzo| 7 -4 0 1 -1 }} = 896/891.
: These temper out the alpharabian comma, {{monzo| -17 2 0 0 4 }} = 131769/131072. In the corresponding [[Tour of regular temperaments#Clans defined by a 2.3.11 .28ila.29 comma|2.3.11 rank-2 temperament]], the pergen is (P8/2, M2/4).


; [[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]]
== Rank-4 temperaments ==
: These temper out the lehmerisma, {{monzo| -4 -3 2 -1 2 }} = 3025/3024.
{{main|Rank-4 temperament}}


; [[Kalismic temperaments|Kalismic or Bilorugu temperaments]]
Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.
: These temper out the kalisma, {{monzo| -3 4 -2 -2 2 }} = 9801/9800.


; [[Semicanousmic clan|Semicanousmic or Quadlo-agu clan]]  
; [[Keenanismic temperaments|Keenanismic or Lozoyo temperaments]] (P8, P5, ^1, /1)
: These temper out the semicanousma, {{monzo| -2 -6 -1 0 4 }} = 14641/14580.
: These temper out the keenanisma, {{monzo| -7 -1 1 1 1 }} = 385/384. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 isn't), ~33/32 or ~729/704.


; [[Semiporwellismic clan|Semiporwellismic or Salulugu clan]]  
;[[Werckismic temperaments|Werckismic or Luzozogu temperaments]] (P8, P5, ^1, /1)
: These temper out the semiporwellisma, {{monzo| 14 -3 -1 0 -2 }} = 16384/16335.
: These temper out the werckisma, {{monzo| -3 2 -1 2 -1 }} = 441/440. 11/8 is equated to [-6 2 -1 2> and 5/4 is equated to [-5 2 0 2 -1>, thus the lattice can be thought of as either 7-limit JI or 11-limit no-fives JI. In the pergen, ^1 is ~64/63, and /1 is either ~81/80, ~33/32 or ~729/704.


; [[Alpharabian rank three clan|Alpharabian or Laquadlo rank-3 clan]]  
;[[Swetismic temperaments|Swetismic or Lururuyo temperaments]] (P8, P5, ^1, /1)
: These temper out the alpharabian comma, {{monzo| -17 2 0 0 4 }} = 131769/131072.
: These temper out the swetisma, {{monzo| 2 3 1 -2 -1 }} = 540/539. 11/8 is equated to [-1 3 1 -2> = 135/98 and 5/4 is equated to [-4 -3 0 2 1>, thus the lattice can be thought of as either 7-limit JI or 11-limit no-fives JI. In the pergen, ^1 is ~64/63, and /1 can be either ~81/80, ~33/32 or ~729/704.


== Rank-4 temperaments ==
; [[Lehmerismic temperaments|Lehmerismic or Loloruyoyo temperaments]] (P8, P5, ^1, /1)
{{main|Rank-4 temperament}}
: These temper out the lehmerisma, {{monzo| -4 -3 2 -1 2 }} = 3025/3024. Since 7/4 is equated to a yala (11-limit no-sevens) interval, both the pergen and the lattice are identical to that of yala JI. In the pergen, ^1 = ~81/80 and /1 = either ~33/32 or ~729/704.


Even less explored than rank three temperaments are rank four temperaments. In fact, unless one counts 7-limit JI they don't seem to have been explored at all. However, they could be used; for example [[Hobbits|hobbit scales]] can be constructed for them.
;[[Kalismic temperaments|Kalismic or Bilorugu temperaments]] (P8/2, P5, ^1, /1)
: These temper out the kalisma, {{monzo| -3 4 -2 -2 2 }} = 9801/9800. The octave is split into two ~99/70 periods. In the pergen, ^1 could be either ~81/80 or ~64/63, and /1 could be either ~64/63 (if ^1 isn't), ~33/32 or ~729/704.


== Subgroup temperaments ==
== Subgroup temperaments ==
Retrieved from "https://en.xen.wiki/w/Tour_of_regular_temperaments"