No-twos subgroup temperaments: Difference between revisions

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= No-twos no-threes subgroup temperaments =

No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, ~~it makes~~ the ~~5th harmonic into~~ the equave. Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.

No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.

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Discovered and named by {{u|CompactStar}} in 2026, Pentadacus is an efficient 5.7.11 subgroup temperament generated by a [[meantone]]-esque small whole tone interval that represents [[55/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]]. It is a no-twos version of [[~~undecimal~~ didacus]], and is named as a play on "didacus", with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).

Discovered and named by {{u|CompactStar}} in 2026, Pentadacus is an efficient 5.7.11 subgroup temperament generated by a [[meantone]]-esque small whole tone interval that represents [[55/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]], basically making it a compressed [[14ed5]] generator chain. It is a no-twos version of 2.5.7.11 [[didacus]], and is named as a play on "didacus", with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).

[[Subgroup]]: 5.7.11

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[[Badness]] (Sintel): 0.216

=== Pentapassion ===

Named by [[User:2^67-1|Cole]] in 2026, Pentapassion is a [[weak extension]] of Pentadacus to the 5.7.11.13 subgroup that splits the generator in half, making the new generator a semitone. [[13/5]] (the pentave-reduced 13th harmonic) is found at 17 of these new generators up. It is named in reference to the octave-equivalent [[Passion]] temperament, which has a similar generator, and in both temperaments [[7/5]] is mapped to 6 generators up.

[[Subgroup]]: 5.7.11.13

[[Comma list]]: [[847/845]], 831875/823543

: mapping generators: ~5, ~343/325

[[Optimal tuning]]s:

* [[WE]]: ~5 = 2785.855{{c}}, ~343/325 = 97.393{{c}}

: [[error map]]: {{val| -0.459 +1.384 -1.968 +1.000 }}

* [[CWE]]: ~5 = 2786.314{{c}}, ~343/325 = 97.386{{c}}

: error map: {{val| 0.000 +1.803 -1.601 +1.347 }}

[[Optimal ET sequence]]: TBA

[[Badness]] (Sintel): 0.277

== Antipyth ==

@@ Line 965: / Line 965: @@
 = No-twos no-threes subgroup temperaments =
-No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, it makes the 5th harmonic into the equave. Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.
+{{See also|Ed5}}
+No-twos temperaments exclude the second harmonic, thus meaning the next lowest prime, 3, is the best remaining option for an equave. If you then also remove the 3rd harmonic, the best remaining option for the equave would be [[5/1]] (also known as the pentane). Of course you can keep removing even more primes making 7th, 11th, etc. harmonics as the equave, but it will probably be impractical.
@@ Line 971: / Line 973: @@
 {{Main| Pentadacus }}
-Discovered and named by {{u|CompactStar}} in 2026, Pentadacus is an efficient 5.7.11 subgroup temperament generated by a [[meantone]]-esque small whole tone interval that represents [[55/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]]. It is a no-twos version of [[undecimal didacus]], and is named as a play on "didacus", with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).
+Discovered and named by {{u|CompactStar}} in 2026, Pentadacus is an efficient 5.7.11 subgroup temperament generated by a [[meantone]]-esque small whole tone interval that represents [[55/49]]. Stacking 3 of these tones gives [[7/5]] and 7 of them give [[11/5]], basically making it a compressed [[14ed5]] generator chain. It is a no-twos version of 2.5.7.11 [[didacus]], and is named as a play on "didacus", with the "di-" part being rebracketed as the Greek prefix for 2 and changed to "penta-" (5).
 [[Subgroup]]: 5.7.11
@@ Line 989: / Line 991: @@
 [[Badness]] (Sintel): 0.216
+=== Pentapassion ===
+Named by [[User:2^67-1|Cole]] in 2026, Pentapassion is a [[weak extension]] of Pentadacus to the 5.7.11.13 subgroup that splits the generator in half, making the new generator a semitone. [[13/5]] (the pentave-reduced 13th harmonic) is found at 17 of these new generators up. It is named in reference to the octave-equivalent [[Passion]] temperament, which has a similar generator, and in both temperaments [[7/5]] is mapped to 6 generators up.
+[[Subgroup]]: 5.7.11.13
+[[Comma list]]: [[847/845]], 831875/823543
+{{Mapping|legend=3| 1 1 1 1 | 0 6 14 17 }}
+: mapping generators: ~5, ~343/325
+[[Optimal tuning]]s:
+* [[WE]]: ~5 = 2785.855{{c}}, ~343/325 = 97.393{{c}}
+: [[error map]]: {{val| -0.459 +1.384 -1.968 +1.000 }}
+* [[CWE]]: ~5 = 2786.314{{c}}, ~343/325 = 97.386{{c}}
+: error map: {{val| 0.000 +1.803 -1.601 +1.347 }}
+[[Optimal ET sequence]]: TBA
+[[Badness]] (Sintel): 0.277
 == Antipyth ==