Tour of regular temperaments: Difference between revisions

: The porcupine family tempers out {{nowrap|{{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 Quadsa-triyoti family]] (P8, P11/3)

: The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo| 39 -29 3 }}. The generator is {{nowrap|~59049/40960 {{=}} {{monzo| -13 10 -1 }} {{=}} 633¢}}, or its octave inverse {{nowrap|~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 {{nowrap|81/56 {{=}} 639¢}}, a much simpler ratio which leads to the [[Tour of Regular Temperaments#Latriruti clan (P8, P11/3)|Latriruti clan]]. An obvious 13-limit interpretation is {{nowrap|13/9 {{=}} 637¢}}, an even simpler ratio implying the [[Tour of Regular Temperaments #Satrithoti clan (P8, P11/3)|Satrithoti clan]].

: The dimipent (or diminished) family tempers out the major diesis or diminished comma, {{Monzo| 3 4 -4 }} or [[648/625]], the amount by which four 6/5 minor thirds exceed an octave, and so identifies the minor third with the quarter-octave. Hence it has the same 300-cent 6/5-approximations as [[12edo|12EDO]]. 5/4 is equated to 1 fifth minus 1 period.