Tour of regular temperaments: Difference between revisions

The diaschismic family tempers out the [[diaschisma]], {{Monzo|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. The major 2nd ~9/8 is divided in half, with each half equated to ~16/15. 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.

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, [[25/24]] (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into a neutral-sized 3rd of ~350¢ that is taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.

The augmented family tempers out the diesis of {{Monzo|7 0 -3}} = [[128/125]], the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" ([[3L 3s]]) in common 12-based music theory, as well as what is commonly called "[http://www.tcherepnin.com/alex/basic_elem1.htm#9step Tcherepnin's scale]" ([[3L 6s]]).

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]], [[22edo]], [[37edo]], and [[59edo]]. An important 7-limit extension also tempers out 64/63.

The tricot family tempers out the [[Tricot|tricot comma]], {{Monzo|39 -29 3}}. The generator is ~59049/40960 = [-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]].

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]]. 5/4 is equated to 1 fifth minus 1 period.

The tetracot family is a much higher accuracy affair than the dicot family. Instead of taking two neutral thirds to reach 3/2, it takes four minor (10/9) whole tones. Four of these exceed 3/2 by {{Monzo|5 -9 4}} (20000/19683), the minimal diesis or [[tetracot comma]]. 5/4 is equated to 9 generators minus an octave. [[7edo]] can also be considered a tetracot tuning, as can [[20edo]], [[27edo]], [[34edo]], and [[41edo]].

This tempers out the ripple comma, 6561/6250 = {{Monzo| -1 8 -5 }}, which equates a stack of four [[10/9]]'s with [[8/5]]. As one might expect, [[12edo]] is about as accurate as it can be.  

The magic family tempers out {{Monzo|-10 -1 5}} (3125/3072), known as the [[magic comma]] or small diesis, which is the difference between five 5/4's (3125/2048) and a 3/1. The generator is itself an approximate 5/4. The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.

The kleismic family of temperaments tempers out the [[kleisma]] {{Monzo|-6 -5 6}} = 15625/15552, which is the difference between six 6/5's and 3/1. It takes a slightly sharpened minor third as a generator, optimally tuned about 1.4 cents sharp.  5/4 is equated to 5 generators minus 1 octave. The kleismic family includes [[15edo]], [[19edo]], [[34edo]], [[49edo]], [[53edo]], [[72edo]], [[87edo]] and [[140edo]] among its possible tunings.

This tempers out the wesley comma, {{Monzo|-13 -2 7}} = 78125/73728. The generator is ~125/96 = ~412¢. Seven generators equals a double-compound 4th of ~16/3. 5/4 is equated to 1 octave minus 2 generators. An obvious 7-limit interpretation of the generator is 9/7, leading to the Lasepru temperament. An obvious 3-limit interpretation of the generator is 81/64, implying [[29-edo]].  

The sensipent (sensi) family tempers out the [[sensipent comma]], {{Monzo|2 9 -7}} (78732/78125), also known as the medium semicomma. Its generator is ~162/125 = ~443¢. Seven generators equals a double-compound 5th of ~6/1. 5/4 is equated to 9 generators minus 3 octaves. Tunings include [[8edo]], [[19edo]], [[46edo]], and [[65edo]]. An obvious 7-limit interpretation of the generator is 9/7, leading to the Sasepzo temperament.  

The würschmidt (or wuerschmidt) family tempers out the [[Würschmidt comma]], 393216/390625 = {{Monzo|17 1 -8}}. Würschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a double-compound perfect 5th); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOSs as [[Magic_family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as würschmidt tunings, as can [[65edo]], which is quite accurate.

The gammic family tempers out the gammic comma, {{Monzo|-29 -11 20}};. Nine generators of about 35¢ equals ~6/5, eleven equal ~5/4 and twenty equal ~3/2. 34-edo is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

This 2.3.11 clan tempers out 243/242 = {{Monzo|-1 5 0 0 -2}}. Its generator is ~11/9. Two generators equals ~3/2. 11/8 is equated to 5 generators minus an octave. This clan includes as a strong extension the [[Dicot]] temperament, which is in the Dicot family.

This 2.3.11 clan tempers out the Laquadlo comma {{Monzo|-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.

===[[Mistismic temperaments|Mistismic or sazoquadgu temperaments]]===