Tour of regular temperaments: Difference between revisions

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; [[Meantone family|Meantone or Gu family]] (P8, P5)

: The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.

: The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[55edo|55]] and [[81edo|81]] EDOs. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.

; [[Schismatic family|Schismatic or Layo family]] (P8, P5)

: The schismatic family tempers out the schisma of {{Monzo|-15 8 1}} = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]].

: The schismatic family tempers out the schisma of {{Monzo|-15 8 1}} = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], and [[118edo|118]] EDOs.

; [[Pelogic family|Pelogic or Layobi family]] (P8, P5)

: This tempers out the pelogic comma, {{Monzo|-7 3 1}} = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].

: This tempers out the pelogic comma, {{Monzo|-7 3 1}} = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo|9]], [[16edo|16]], [[23edo|23]], and [[25edo|25]] EDOs.

; [[Father family|Father or Gubi family]] (P8, P5)

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; [[Negri|Negri or Laquadyo family]] (P8, P4/4)

: This tempers out the [[negri comma]], {{Monzo|-14 3 4}};. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.

: This tempers out the [[negri comma]], {{Monzo|-14 3 4}}. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.

; [[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)

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; [[Pental family|Pental or Trila-quingu family]] (P8/5, P5)

: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}. The period is 59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. 5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}. The period is ~59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio. 5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.

; [[Ripple family|Ripple or Quingu family]] (P8, P4/5)

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; [[Passion family|Passion or Saquingu family]] (P8, P4/5)

: 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)

: 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)

: 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)

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; [[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)

: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}};. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.

: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}}. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.

; [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6)

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; [[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)

: 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. 34EDO is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

: 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. 34EDO is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.

=== Clans defined by a 2.3.7 (za) comma ===

@@ Line 26: / Line 26: @@
 ; [[Meantone family|Meantone or Gu family]] (P8, P5)
-: The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo]], [[19edo]], [[31edo]], [[43edo]], [[50edo]], [[55edo]] and [[81edo]]. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.
+: The meantone family tempers out [[81/80]], also called the syntonic comma. This comma manifests as the difference between a stack of four 3/2's (81/16, or (3/2)^4) and 5/1 harmonic (5/1, or 80/16). It is so named because it splits the major third into two equal sized tones, signifying that 9/8 and 10/9 are equated, with each tone being sized as a mean of the two tones. It has a flattened fifth or sharpened fourth as generator. Some meantone tunings are [[12edo|12]], [[19edo|19]], [[31edo|31]], [[43edo|43]], [[50edo|50]], [[55edo|55]] and [[81edo|81]] EDOs. Aside from tuning meantone as a subset of some equal division of the octave, some common rank-2 tunings include having a generator of 3/2 flattened by 1/3, 2/7, 1/4, 1/5 or 1/6 of the syntonic comma.
 ; [[Schismatic family|Schismatic or Layo family]] (P8, P5)
-: The schismatic family tempers out the schisma of {{Monzo|-15 8 1}} = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo]], [[29edo]], [[41edo]], [[53edo]], and [[118edo]].
+: The schismatic family tempers out the schisma of {{Monzo|-15 8 1}} = [[32805/32768]], which is the amount by which the Pythagorean comma exceeds the syntonic comma. The 5-limit version of the temperament is a [[microtemperament]] which flattens the fifth by a fraction of a schisma, but other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity; whereas meantone equates four 3/2's with 5/1, schismatic equates eight 4/3's with 10/1, so that the Pythagorean diminished fourth of 8192/6561 is equated with 5/4. Tunings include [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], and [[118edo|118]] EDOs.
 ; [[Pelogic family|Pelogic or Layobi family]] (P8, P5)
-: This tempers out the pelogic comma, {{Monzo|-7 3 1}} = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].
+: This tempers out the pelogic comma, {{Monzo|-7 3 1}} = [[135/128]], also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates [[2L 5s]] "anti-diatonic" scales. 5/4 is equated to 3 fourths minus 1 octave. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo|9]], [[16edo|16]], [[23edo|23]], and [[25edo|25]] EDOs.
 ; [[Father family|Father or Gubi family]] (P8, P5)
@@ Line 62: / Line 62: @@
 ; [[Negri|Negri or Laquadyo family]] (P8, P4/4)
-: This tempers out the [[negri comma]], {{Monzo|-14 3 4}};. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.
+: This tempers out the [[negri comma]], {{Monzo|-14 3 4}}. Its generator is ~16/15, four of which make ~4/3. 5/4 is equated to 3 generators.
 ; [[Tetracot family|Tetracot or Saquadyo family]] (P8, P5/4)
@@ Line 71: / Line 71: @@
 ; [[Pental family|Pental or Trila-quingu family]] (P8/5, P5)
-: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}. The period is 59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio.  5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.
+: This tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}. The period is ~59049/51200, and 5 periods make an octave. The generator is a 5th, or equivalently, 3/5 of an 8ve minus a 5th. This alternate generator is only about 18¢, thus the scales have a very lopsided L/s ratio.  5/4 is equated to 2/5 of an octave minus 5 of these 18¢ generators. An obvious 7-limit interpretation of the generator is 8/7, which leads to Laquinzo.
 ; [[Ripple family|Ripple or Quingu family]] (P8, P4/5)
@@ Line 78: / Line 78: @@
 ; [[Passion family|Passion or Saquingu family]] (P8, P4/5)
 : 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)
+: 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)
+: 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 92: / Line 98: @@
 ; [[Trisedodge family|Trisedodge or Saquintrigu family]] (P8/5, P4/3)
-: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}};. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.
+: This tempers out the trisedodge comma, 30958682112/30517578125 = {{Monzo|19 10 -15}}. The period is ~144/125 = 240¢. The generator is ~6/5. Six periods minus three generators equals ~4/3. 5/4 is equated to 2 generators minus 1 period. An obvious 7-limit interpretation of the period is 8/7.
 ; [[Ampersand|Ampersand or Lala-tribiyo family]] (P8, P5/6)
@@ Line 140: / Line 146: @@
 ; [[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)
-: 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. 34EDO is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.
+: 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. 34EDO is an obvious tuning. The head of the family is 5-limit gammic, whose generator chain is [[Carlos Gamma]]. Another member is Neptune temperament.
 === Clans defined by a 2.3.7 (za) comma ===