Tour of regular temperaments: Difference between revisions

The diaschismic family tempers out the [[diaschisma]], [11 -4 -2> or 2048/2025, such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period of an approximate 45/32 or 64/45, and its generator is an approximate 3/2. 5/4 is equated to 3 periods minus 2 fifths. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. An obvious 7-limit interpretation of the period is 7/5, which makes [[pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out. [[22edo]] is an excellent pajara tuning.

This clan tempers out [-13 10 0 -1> =  50.7¢. It equates 7/4 to an augmented 6th. Its best downward extension is [[Meantone family|Septimal Meantone]].

This clan tempers out the garischisma, [25 -14 0 -1> = 33554432/33480783. It equates 8/7 to two apotomes ([-11 7> = 2187/2048), and 7/4 to a double-diminished 8ve. This clan includes [[Vulture family|vulture]], [[Breedsmic temperaments|newt]], [[Schismatic family|garibaldi]], [[Landscape microtemperaments|sextile]], and satin.

This clan tempers out [15 -13 0 2> = 12.2¢, and includes as a strong extension the [[Hemififths]] temperament. 7/4 is equated to 13 generators minus 3 octaves. An obvious 11-limit interpretation of the ~351¢ generator is 11/9, leading to the Lulu temperament.

This clan tempers out the gamelisma, [-10 1 0 3> = 1029/1024. Three ~8/7 generators equals a 5th. A particularly noteworthy member of the gamelismic clan is miracle, but other members include valentine, unidec, mothra, rodan, and hemithirds. Miracle temperament divides the fifth into 6 equal steps, thus it's a weak extension. Its 21-note scale called "blackjack" and 31-note scale called "canasta" have some useful properties. It is the most efficient 11-limit temperament for many purposes, with a tuning close to 72-EDO.

This clan tempers out [-9 11 0 -3> = 15.0¢. Generator = ~112/81 = ~566¢. Three generators equals ~8/3. 7/4 is equated to 11 generators minus 5 octaves. An obvious 2.3.5.7 interpretation of the generator is 7/5, leading to the [[Liese]] temperament, which is a weak extension of Meantone.

This clan tempers out [-3 9 0 -4> = 42.3¢. its generator is ~9/7. Four generators equals ~8/3. 7/4 is equated to 4 octaves minus 9 generators. This clan includes as a strong extension the [[Squares]] temperament, which is a weak extension of Meantone.

This clan tempers out [16 -3 0 -4> = 18.8¢. Its generator is ~21/16. Four generators makes ~3/1. 7/4 is equated to 2 octaves minus 3 generators. This clan includes as a strong extension the [[Vulture family|Vulture]] temperament, which is in the Vulture family.

This clan tempers out [5 -12 0 5> = 20.7¢. Its generator is ~243/196 = ~380¢. Five generators makes ~3/1. 7/4 is equated to 12 generators minus 3 octaves. An obvious 5-limit interpretation of the generator is 5/4, leading to the [[Magic]] temperament, which is in the Magic family.

This clan tempers out [7 8 0 -7> = 33.8¢. Its generator is ~7/6. Seven generators equals ~3/1. 7/4 is equated to 8 generators minus 1 octave. This clan includes as a strong extension the [[Orwell]] temperament, which is in the Semicomma family.

This 2.3.11 clan tempers out [-17 2 0 0 4>. Its half-ocave period is ~363/256, and its generator is ~33/32. Four generators equals ~9/8. 3/2 is equated to a period plus 2 generators, and 11/8 is equated to a period minus a generator. This clan includes as a strong extension the Comic aka Saquadyobi temperament, which is in the Comic family.

This 2.3.13 clan tempers out 512/507 = [0 -7 0 0 0 3>. Its generator is ~18/13. Three generators equals ~8/3. 13/8 is equated to 7 generators minus three octaves. This clan is related to the Latriru clan.

This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The M3 generator is ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.