Tour of regular temperaments: Difference between revisions

Line 53:

; [[Misty family|Misty or Sasa-trigu family]] (P8/3, P5)

: The misty family tempers out the [[misty comma]] of {{monzo| 26 -12 -3 }}, the difference between the Pythagorean comma and a stack of three schismas.

: The misty family tempers out the [[misty comma]] of {{monzo| 26 -12 -3 }}, the difference between the [[Pythagorean comma]] and a stack of three [[Schisma|schismas]]. The period is ~512/405 and the generator is ~3/2 (or alternatively ~135/128). 5/4 is equated to 8 periods minus 4 fifths, thus 5/4 is split into 4 equal parts, each 2 periods minus a fifth.

; [[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)

: The porcupine family tempers out {{Monzo|1 -5 3}} = [[250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include [[15edo|15]], [[22edo|22]], [[37edo|37]], and [[59edo|59]] EDOs. An important 7-limit extension also tempers out 64/63.

; [[Tricot family|Tricot or ~~Quadsatriyo~~ family]] (P8, P11/3)

; [[Tricot family|Tricot or Quadsa-triyo family]] (P8, P11/3)

: The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = {{monzo|-13 10 -1}} = 633¢, or its octave inverse ~81920/59049 = 567¢. Three of the latter generators equals a compound 4th of ~8/3. 5/4 is equated to 14 octaves minus 29 of these generators. An obvious 7-limit interpretation of the generator is 81/56 = 639¢, a much simpler ratio which leads to the [[Tour of Regular Temperaments#Latriru clan (P8, P11/3)|Latriru clan]]. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the [[Tour of Regular Temperaments #Satritho clan (P8, P11/3)|Satritho clan]].

@@ Line 53: / Line 53: @@
 ; [[Misty family|Misty or Sasa-trigu family]] (P8/3, P5)
-: The misty family tempers out the [[misty comma]] of {{monzo| 26 -12 -3 }}, the difference between the Pythagorean comma and a stack of three schismas.
+: The misty family tempers out the [[misty comma]] of {{monzo| 26 -12 -3 }}, the difference between the [[Pythagorean comma]] and a stack of three [[Schisma|schismas]]. The period is ~512/405 and the generator is ~3/2 (or alternatively ~135/128). 5/4 is equated to 8 periods minus 4 fifths, thus 5/4 is split into 4 equal parts, each 2 periods minus a fifth.
 ; [[Porcupine family|Porcupine or Triyo family]] (P8, P4/3)
 : The porcupine family tempers out {{Monzo|1 -5 3}} = [[250/243]], the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. 5/4 is equated to 1 octave minus 5 generators. Some porcupine temperaments include [[15edo|15]], [[22edo|22]], [[37edo|37]], and [[59edo|59]] EDOs. An important 7-limit extension also tempers out 64/63.
-; [[Tricot family|Tricot or Quadsatriyo family]] (P8, P11/3)
+; [[Tricot family|Tricot or Quadsa-triyo family]] (P8, P11/3)
 : The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = {{monzo|-13 10 -1}} = 633¢, or its octave inverse ~81920/59049 = 567¢. Three of the latter generators equals a compound 4th of ~8/3. 5/4 is equated to 14 octaves minus 29 of these generators. An obvious 7-limit interpretation of the generator is 81/56 = 639¢, a much simpler ratio which leads to the [[Tour of Regular Temperaments#Latriru clan (P8, P11/3)|Latriru clan]]. An obvious 13-limit interpretation is 13/9 = 637¢, an even simpler ratio implying the [[Tour of Regular Temperaments #Satritho clan (P8, P11/3)|Satritho clan]].