Tour of regular temperaments: Difference between revisions

Line 49:

: 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|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|10EDO]], and [[17edo|17EDO]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.

; [[~~Augmented_family~~|Augmented or Trigu family]] (P8/3, P5)

; [[Augmented family|Augmented or Trigu family]] (P8/3, P5)

: 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|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]]).

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: This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of four [[16/15]]'s with [[5/4]], and five of them with [[4/3]].

; [[~~16ed5/2~~|Quintaleap or Trisa-quingu family]] (P8, P4/5)

; [[Quintaleap family|Quintaleap or Trisa-quingu family]] (P8, P4/5)

: This tempers out the ''quintaleap'' comma, {{monzo|37 -16 -5}}. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives [[5/2]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

: This tempers out the ''quintaleap'' comma, {{monzo| 37 -16 -5 }}. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives [[5/2]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

; [[~~28ed5~~|Quindromeda or Quinsa-quingu family]] (P8, P4/5)

; [[Quindromeda family|Quindromeda or Quinsa-quingu family]] (P8, P4/5)

: This tempers out the ''quindromeda'' comma, {{monzo|56 -28 -5}}. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, [[5/1]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

: This tempers out the ''quindromeda'' comma, {{monzo| 56 -28 -5 }}. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, [[5/1]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.

; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)

@@ Line 49: / Line 49: @@
 : 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|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|10EDO]], and [[17edo|17EDO]]. An obvious 2.3.11 nterpretation of the generator is ~11/9, which leads to Rastmic aka Neutral aka Lulu.
-; [[Augmented_family|Augmented or Trigu  family]] (P8/3, P5)
+; [[Augmented family|Augmented or Trigu  family]] (P8/3, P5)
 : 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|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]]).
@@ Line 79: / Line 79: @@
 : This tempers out the passion comma, 262144/253125 = {{monzo| 18 -4 -5 }}, which equates a stack of four [[16/15]]'s with [[5/4]], and five of them with [[4/3]].
-; [[16ed5/2|Quintaleap or Trisa-quingu family]] (P8, P4/5)
+; [[Quintaleap family|Quintaleap or Trisa-quingu family]] (P8, P4/5)
-: This tempers out the ''quintaleap'' comma, {{monzo|37 -16 -5}}. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives [[5/2]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.
+: This tempers out the ''quintaleap'' comma, {{monzo| 37 -16 -5 }}. The generator is ~135/128, five of them gives ~4/3, and sixteen of them gives [[5/2]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.
-; [[28ed5|Quindromeda or Quinsa-quingu family]] (P8, P4/5)
+; [[Quindromeda family|Quindromeda or Quinsa-quingu family]] (P8, P4/5)
-: This tempers out the ''quindromeda'' comma, {{monzo|56 -28 -5}}. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, [[5/1]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.
+: This tempers out the ''quindromeda'' comma, {{monzo| 56 -28 -5 }}. The generator is ~4428675/4194304, five of them gives ~4/3, and 28 of them gives the fifth harmonic, [[5/1]]. An obvious 17-limit interpretation of the generator is ~18/17, which makes saquinso.
 ; [[Amity family|Amity or Saquinyo family]] (P8, P11/5)