Tour of regular temperaments: Difference between revisions

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; Lee or Latriru clan (P8, P11/3)

: This clan tempers out the Latriru comma, {{Monzo|-9 11 0 -3}} = 177147/175616. 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.

; [[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)

: This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5.

; Buzzardismic or Saquadru clan (P8, P12/4)

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; Saquinzo clan (P8, P12/5)

: This clan tempers out the Saquinzo comma, {{Monzo|5 -12 0 5}} = 537824/531441. 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.

~~; [[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)~~

: This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5.

; Lasepzo clan (P8, P11/7)

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; [[Rastmic_temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2)

: 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.

; [[Tribilo_family| Tribilo Clan]] (P8/3, P4/2)

: This 2.3.11 clan tempers out the [[Nexus_comma|Tribilo or Nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its third-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3.

; [[Alpharabian_temperaments|Alpharabian or Laquadlo clan]] (P8/2, M2/4)

: This 2.3.11 clan tempers out the [[alpharabian comma]] {{monzo| -17 2 0 0 4 }}. Its half-octave 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 or saquadyobi temperament, which is in the comic family.

~~; [[Tribilo_family| Tribilo Clan]] (P8/3, P4/2)~~

: This 2.3.11 clan tempers out the [[Nexus_comma|Tribilo or Nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its third-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3.

=== Clans defined by a 2.3.13 (tha) comma ===

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=== Clans defined by a 2.5.7 (yaza nowa) comma ===

These are defined by a yaza nowa or 7-limit-no-threes comma. See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-threes ratio such as 5/4, 8/5, 25/8, etc.

; [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)

: This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd = ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.

; Bapbo or Rurugu Nowa clan (P8, M3/2)

: This clan tempers out the bapbo comma, [[256/245]]. The genarator is ~8/7 = ~202¢ and two of them equals ~5/4.

~~; [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)~~

: This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd = ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.

~~; [[Llywelyn clan|Llywelyn or Sasepru-agu Nowa clan]] (P8, cM3/7)~~

~~: This clan tempers out the llywelyn comma, {{Monzo|22 0 -1 -7}} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.~~

~~; Slither or Satritriru-aquadyo Nowa clan (P8, ccm6/9)~~

: This clan tempers out the slither comma, {{Monzo|16 0 4 -9}} = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.

; [[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M3/2)

: This clan tempers out the hemimean comma, {{Monzo|6 0 -5 2}} = 3136/3125. The generator is ~28/25 = ~194¢. Two generators equals the nowa major 3rd = ~5/4, three of them equals ~7/5, and five of them equals ~7/4.

~~; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)~~

: This clan tempers out the quince, {{Monzo|-15 0 -2 7}} = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan.

; [[Mabilismic clan|Mabilismic or Latrizo-aquiniyo Nowa clan]] (P8, cM3/3)

: This clan tempers out the mabilisma, {{Monzo|-20 0 5 3}} = 1071875/1048576. The generator is ~175/128 = ~527¢. Three generators equals ~5/2 and five of them equals ~32/7.

; Vorwell or Sasatriru-aquadbigu Nowa clan (P8, m6/3)

: This clan tempers out the vorwell comma, {{Monzo|27 0 -8 -3}} = 134217728/133984375. The rutrigu generator is ~1024/875 = ~272¢. Three generators equals ~8/5 and eight of them equals ~7/2.

; Rainy or Quinzo-atriyo Nowa clan (P8, M3/5)

: This clan tempers out the [[rainy comma]], {{Monzo|-21 0 3 5}} = 2100875/2097152. The rurugu generator is ~256/245 = ~77¢. Three generators equals ~8/7 and five of them equals the nowa major 3rd = ~5/4.

; ~~Vorwell~~ or ~~Sasatriru~~-~~aquadbigu~~ Nowa clan (P8, m6/3)

; [[Llywelyn clan|Llywelyn or Sasepru-agu Nowa clan]] (P8, cM3/7)

: This clan tempers out the ~~vorwell~~ comma, {{Monzo|27 0 -8 -3}} = ~~134217728~~/~~133984375~~. The ~~rutrigu~~ generator is ~~~1024~~/~~875~~ = ~~~272¢~~. ~~Three~~ generators equals ~8/5 and ~~eight~~ of them equals ~7/2.

: This clan tempers out the llywelyn comma, {{Monzo|22 0 -1 -7}} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.

; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)

: This clan tempers out the quince, {{Monzo|-15 0 -2 7}} = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan.

; Slither or Satritriru-aquadyo Nowa clan (P8, ccm6/9)

: This clan tempers out the slither comma, {{Monzo|16 0 4 -9}} = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.

=== Clans defined by a 3.5.7 (yaza noca) comma ===

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; [[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M6/2)

: This 3.5.7 clan tempers out the sensamagic comma {{Monzo|0 -5 1 2}} = 245/243. The generator is ~9/7, and two generators equals the noca major 6th = ~5/3.

~~; Betelgeuse or Satritrizo-agugu Noca clan (P12, c3M6)~~

: This 3.5.7 clan tempers out the Betelgeuse comma {{Monzo|0 -13 -2 9}} = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th = ~45/1.

; [[Gariboh clan|Gariboh or Triru-aquinyo Noca clan]] (P12, M6/3)

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; Procyon or Sasepzo-atrigu Noca clan (P12, m7/7)

: This 3.5.7 clan tempers out the Procyon comma {{Monzo|0 -8 -3 7}} = 823543/820125. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the noca minor seventh = ~9/5.

; Betelgeuse or Satritrizo-agugu Noca clan (P12, c3M6/9)

: This 3.5.7 clan tempers out the Betelgeuse comma {{Monzo|0 -13 -2 9}} = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th = ~45/1.

; Izar or Saquadtrizo-asepgu Noca clan (P12, c5m7/12)

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=== Temperaments defined by an 11-limit comma ===

; [[Ptolemismic clan|Ptolemismic or Luyoyo clan]] (P8, P5, ^1)

: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} = 100/99. 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, ^1 = ~81/80.

; [[Biyatismic clan|Biyatismic or Lologu clan]] (P8, P5, ^1)

@@ Line 204: / Line 204: @@
 ; Lee or Latriru clan (P8, P11/3)
 : This clan tempers out the Latriru comma, {{Monzo|-9 11 0 -3}} = 177147/175616. 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.
+; [[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)
+: This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5.
 ; Buzzardismic or Saquadru clan (P8, P12/4)
@@ Line 219: / Line 222: @@
 ; Saquinzo clan (P8, P12/5)
 : This clan tempers out the Saquinzo comma, {{Monzo|5 -12 0 5}} = 537824/531441. 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.
-; [[Stearnsmic clan|Stearnsmic or Latribiru clan]] (P8/2, P4/3)
-: This clan temper out the stearnsma, {{Monzo|1 10 0 -6}} = 118098/117649. The period is ~486/343 = ~600¢. The generator is ~9/7 = ~434¢, or alternatively one period minus ~9/7, which equals ~54/49 = ~166¢. Three of these alternate generators equals ~4/3. 7/4 is equated to 5 ~9/7 generators minus an octave. Equating the ~54/49 generator to ~10/9 creates a weak extension of the [[porcupine]] temperament, as does equating the period to ~7/5.
 ; Lasepzo clan (P8, P11/7)
@@ Line 243: / Line 243: @@
 ; [[Rastmic_temperaments|Rastmic or Neutral or Lulu clan]] (P8, P5/2)
 : 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.
+; [[Tribilo_family| Tribilo Clan]] (P8/3, P4/2)
+: This 2.3.11 clan tempers out the [[Nexus_comma|Tribilo or Nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its third-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3.
 ; [[Alpharabian_temperaments|Alpharabian or Laquadlo clan]] (P8/2, M2/4)
 : This 2.3.11 clan tempers out the [[alpharabian comma]] {{monzo| -17 2 0 0 4 }}. Its half-octave 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 or saquadyobi temperament, which is in the comic family.
-; [[Tribilo_family| Tribilo Clan]] (P8/3, P4/2)
-: This 2.3.11 clan tempers out the [[Nexus_comma|Tribilo or Nexus comma]] {{monzo| -16 -3 0 0 6 }}. Its third-octave period is ~121/96 and its least-cents generator is ~12/11. A period plus a generator equals ~11/8. Six of these generators equals ~27/16. A period minus a generator equals ~1331/1152 or ~1536/1331. Two of these alternative generators equals ~4/3.
 === Clans defined by a 2.3.13 (tha) comma ===
@@ Line 261: / Line 261: @@
 === Clans defined by a 2.5.7 (yaza nowa) comma ===
 These are defined by a yaza nowa or 7-limit-no-threes comma. See also [[subgroup temperaments]]. The pergen's multigen (the 2nd term, omitting any fraction) is always a 5-limit-no-threes ratio such as 5/4, 8/5, 25/8, etc.
+; [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)
+: This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd = ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.
 ; Bapbo or Rurugu Nowa clan (P8, M3/2)
 : This clan tempers out the bapbo comma, [[256/245]]. The genarator is ~8/7 = ~202¢ and two of them equals ~5/4.
-; [[Jubilismic clan|Jubilismic or Biruyo Nowa clan]] (P8/2, M3)
-: This clan tempers out the jubilisma, [[50/49]], which is the difference between 10/7 and 7/5. The generator is the nowa major 3rd = ~5/4. The half-octave period is ~7/5 or ~10/7. 7/4 is equated to 1 period plus 1 generator.
-; [[Llywelyn clan|Llywelyn or Sasepru-agu Nowa clan]] (P8, cM3/7)
-: This clan tempers out the llywelyn comma, {{Monzo|22 0 -1 -7}} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.
-; Slither or Satritriru-aquadyo Nowa clan (P8, ccm6/9)
-: This clan tempers out the slither comma, {{Monzo|16 0 4 -9}} = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.
 ; [[Hemimean clan|Hemimean or Zozoquingu Nowa clan]] (P8, M3/2)
 : This clan tempers out the hemimean comma, {{Monzo|6 0 -5 2}} = 3136/3125. The generator is ~28/25 = ~194¢. Two generators equals the nowa major 3rd = ~5/4, three of them equals ~7/5, and five of them equals ~7/4.
-; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)
-: This clan tempers out the quince, {{Monzo|-15 0 -2 7}} = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan.
 ; [[Mabilismic clan|Mabilismic or Latrizo-aquiniyo Nowa clan]] (P8, cM3/3)
 : This clan tempers out the mabilisma, {{Monzo|-20 0 5 3}} = 1071875/1048576. The generator is ~175/128 = ~527¢. Three generators equals ~5/2 and five of them equals ~32/7.
+; Vorwell or Sasatriru-aquadbigu Nowa clan (P8, m6/3)
+: This clan tempers out the vorwell comma, {{Monzo|27 0 -8 -3}} = 134217728/133984375. The rutrigu generator is ~1024/875 = ~272¢. Three generators equals ~8/5 and eight of them equals ~7/2.
 ; Rainy or Quinzo-atriyo Nowa clan (P8, M3/5)
 : This clan tempers out the [[rainy comma]], {{Monzo|-21 0 3 5}} = 2100875/2097152. The rurugu generator is ~256/245 = ~77¢. Three generators equals ~8/7 and five of them equals the nowa major 3rd = ~5/4.
-; Vorwell or Sasatriru-aquadbigu Nowa clan (P8, m6/3)
+; [[Llywelyn clan|Llywelyn or Sasepru-agu Nowa clan]] (P8, cM3/7)
-: This clan tempers out the vorwell comma, {{Monzo|27 0 -8 -3}} = 134217728/133984375. The rutrigu generator is ~1024/875 = ~272¢. Three generators equals ~8/5 and eight of them equals ~7/2.
+: This clan tempers out the llywelyn comma, {{Monzo|22 0 -1 -7}} = 4194304/4117715. The generator is ~8/7 = ~227¢ and seven of them equals ~5/2.
+; [[Quince clan|Quince or Lasepzo-agugu Nowa clan]] (P8, m6/7)
+: This clan tempers out the quince, {{Monzo|-15 0 -2 7}} = 823543/819200. The trizo-agu generator is ~343/320 = ~116¢. Two generators equals ~8/7, five generators equals ~7/5, and seven generators equals the nowa minor 6th ~8/5. An obvious 5-limit interpretation of the generator is 16/15, leading to the [[miracle]] temperament, which is in the gamelismic clan.
+; Slither or Satritriru-aquadyo Nowa clan (P8, ccm6/9)
+: This clan tempers out the slither comma, {{Monzo|16 0 4 -9}} = 40960000/40353607. The generator is ~49/40 = ~357¢. Four generators equals ~16/7, five of them equals ~14/5, and nine of them equals a double-compound minor 6th of ~32/5.
 === Clans defined by a 3.5.7 (yaza noca) comma ===
@@ Line 297: / Line 297: @@
 ; [[Sensamagic clan|Sensamagic or Zozoyo Noca clan]] (P12, M6/2)
 : This 3.5.7 clan tempers out the sensamagic comma {{Monzo|0 -5 1 2}} = 245/243. The generator is ~9/7, and two generators equals the noca major 6th = ~5/3.
-; Betelgeuse or Satritrizo-agugu Noca clan (P12, c<sup>3</sup>M6)
-: This 3.5.7 clan tempers out the Betelgeuse comma {{Monzo|0 -13 -2 9}} = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th = ~45/1.
 ; [[Gariboh clan|Gariboh or Triru-aquinyo Noca clan]] (P12, M6/3)
@@ Line 309: / Line 306: @@
 ; Procyon or Sasepzo-atrigu Noca clan (P12, m7/7)
 : This 3.5.7 clan tempers out the Procyon comma {{Monzo|0 -8 -3 7}} = 823543/820125. The generator is ~49/45, three generators equals ~9/7, four equals ~7/5, and seven equals the noca minor seventh = ~9/5.
+; Betelgeuse or Satritrizo-agugu Noca clan (P12, c<sup>3</sup>M6/9)
+: This 3.5.7 clan tempers out the Betelgeuse comma {{Monzo|0 -13 -2 9}} = 40353607/39858075. The generator is ~3645/2401, two generators equals ~7/3, and nine generators equals the noca triple-compound major 6th = ~45/1.
 ; Izar or Saquadtrizo-asepgu Noca clan (P12, c<sup>5</sup>m7/12)
@@ Line 582: / Line 582: @@
 === Temperaments defined by an 11-limit comma ===
 ; [[Ptolemismic clan|Ptolemismic or Luyoyo clan]] (P8, P5, ^1)
-: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} = 100/99. 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI.  In the pergen, ^1 = ~81/80.
+: These temper out the ptolemisma, {{monzo| 2 -2 2 0 -1 }} = 100/99. 11/8 is equated to 25/18, which is an octave minus two 6/5's. Since 25/18 is a 5-limit interval, every 2.3.5.11 interval is equated to a 5-limit interval, and both the pergen and the lattice are identical to that of 5-limit JI. In the pergen, ^1 = ~81/80.
 ; [[Biyatismic clan|Biyatismic or Lologu clan]] (P8, P5, ^1)