41-comma: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Ratio = 36 893 488 147 419 103 232 / 36 472 996 377 170 786 403

| Monzo = 65 -41

~~| Cents = 19.84496~~

| Name = 41-comma, Pythagorean countercomma, countercomp comma

| Name = 41-comma, ~~ counterpyth~~ comma

| Color name = 41wM, fowowama

| Color name =

| Comma = yes

| ~~FJS name =~~

~~| Sound~~ =

}}

The '''41-comma''', '''Pythagorean countercomma''', or '''countercomp comma''' ([[monzo]]: {{monzo| 65 -41 }}), is a [[3-limit]] interval of 19.845 [[cent]]s. It is the amount by which a stack of 41 [[3/2|perfect fifths (3/2)]] falls short of 24 [[octave]]s, in other words 224/(3/2)41. It can also be described as the difference between a perfect fifth and a stack of six [[2187/2048|apotomes]], or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth.

~~{{monzo|65 -41}}, the~~ '''41-comma''' or '''~~counterpyth~~ comma''', is an interval of 19.845 [[cent]]s. It is the amount by which 41 fifths ([[3/2]]~~) fall~~ short of 24 [[octave]]s, in other words 224/(3/2)41.

== Temperaments ==

Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41~~. It leads to a number of regular temperaments in the [[counterpyth family]]~~. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]].

Tempering out this comma leads to the [[countercomp]] temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]]. See [[countercomp family]] for a number of rank-2 temperaments where it is tempered out.

~~=== Counterpyth (41&205) ===~~

~~Subgroup: 2.3.5~~

[[~~Comma list~~]]~~: {{monzo|65 -41}}~~

~~[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]~~

~~Mapping generators: ~531441/524288, ~5~~

~~[[POTE generator]]: ~5/4 = 386.668~~

~~{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}~~

~~[[Badness]]: 0.934310~~

~~=== 7~~-~~limit 41&123 ===~~

~~Subgroup:~~ 2.~~3.5.7~~

~~[[Comma list]]: 1029/1024, 537824/531441~~

~~[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]~~

~~[[POTE generator]]: ~5/4 = 385.731~~

~~{{Val list|legend=1| 41, 123, 164, 205d, 246d, 451dd }}~~

~~[[Badness]]: 0.161056~~

~~=== 7-limit 41&205 ===~~

~~Subgroup: 2.3.5.7~~

~~[[Comma list]]: 5120/5103, 2500000/2470629~~

~~[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]~~

~~[[POTE generator]]: ~5/4 = 385.667~~

~~{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}~~

~~[[Badness]]: 0.142344~~

~~=== 7-limit 41&369 ===~~

~~Subgroup: 2.3.5.7~~

~~[[Comma list]]: 2401/2400, 52613349376/52301766015~~

~~[[Mapping]]: [{{val| 41 65 95 115 }}, {{val| 0 0 2 1 }}]~~

~~[[POTE generator]]: ~8/7 = 231.045~~

~~{{Val list|legend=1| 41, 328, 369, 779 }}~~

[[~~Badness~~]]~~: 0~~.~~134559~~

== Terminology ==

The names were given by [[Flora Canou]] in 2022. ''Pythagorean countercomma'' was derived by analogy to ''Pythagorean comma'' as part of a series of Pythagorean interval names. ''Countercomp comma'' was derived from the temperament name, ''countercomp'', which was changed from ''counterpyth'' as in earlier materials, where this comma was also called ''counterpyth comma''. It was renamed after the convention was established that the temperament of the [[Pythagorean comma]] should be [[compton]] and never ''Pythagorean'', for fear of confusion with [[Pythagorean tuning]].

== See also ==

* [[Small comma]]

* [[Schismic–countercommatic equivalence continuum]]

[[Category:~~3-limit~~]]

[[Category:Countercomp]]

[[Category:~~Small comma~~]]

[[Category:Commas named systematically]]

[[Category:~~Counterpyth~~]]

[[Category:Commas named for their regular temperament properties]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Icon =
 | Ratio = 36 893 488 147 419 103 232 / <br>36 472 996 377 170 786 403
 | Monzo = 65 -41
-| Cents = 19.84496
+| Name = 41-comma, Pythagorean countercomma, countercomp comma
-| Name = 41-comma, <br>counterpyth comma
+| Color name = 41wM, fowowama
-| Color name =
+| Comma = yes
-| FJS name =
-| Sound =
 }}
+The '''41-comma''', '''Pythagorean countercomma''', or '''countercomp comma''' ([[monzo]]: {{monzo| 65 -41 }}), is a [[3-limit]] interval of 19.845 [[cent]]s. It is the amount by which a stack of 41 [[3/2|perfect fifths (3/2)]] falls short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup>. It can also be described as the difference between a perfect fifth and a stack of six [[2187/2048|apotomes]], or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth.
-{{monzo|65 -41}}, the '''41-comma''' or '''counterpyth comma''', is an interval of 19.845 [[cent]]s. It is the amount by which 41 fifths ([[3/2]]) fall short of 24 [[octave]]s, in other words 2<sup>24</sup>/(3/2)<sup>41</sup>.
 == Temperaments ==
-Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. It leads to a number of regular temperaments in the [[counterpyth family]]. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]].
+Tempering out this comma leads to the [[countercomp]] temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions ''N'' up to 1230, the comma is tempered out if and only if 41 divides ''N''. Examples are [[41edo]], [[164edo]], [[205edo]], [[246edo]], [[328edo]] and [[369edo]]. See [[countercomp family]] for a number of rank-2 temperaments where it is tempered out.
-=== Counterpyth (41&amp;205) ===
-Subgroup: 2.3.5
-[[Comma list]]: {{monzo|65 -41}}
-[[Mapping]]: [{{val| 41 65 0 }}, {{val| 0 0 1 }}]
-Mapping generators: ~531441/524288, ~5
-[[POTE generator]]: ~5/4 = 386.668
-{{Val list|legend=1| 41, 123, 164, 205, 369, 574, 779, 2132bc }}
-[[Badness]]: 0.934310
-=== 7-limit 41&amp;123 ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 1029/1024, 537824/531441
-[[Mapping]]: [{{val| 41 65 0 115 }}, {{val| 0 0 1 0 }}]
-[[POTE generator]]: ~5/4 = 385.731
-{{Val list|legend=1| 41, 123, 164, 205d, 246d, 451dd }}
-[[Badness]]: 0.161056
-=== 7-limit 41&amp;205 ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 5120/5103, 2500000/2470629
-[[Mapping]]: [{{val| 41 65 0 20 }}, {{val| 0 0 1 1 }}]
-[[POTE generator]]: ~5/4 = 385.667
-{{Val list|legend=1| 41, 123d, 164d, 205, 246, 451d, 697dd }}
-[[Badness]]: 0.142344
-=== 7-limit 41&amp;369 ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 2401/2400, 52613349376/52301766015
-[[Mapping]]: [{{val| 41 65 95 115 }}, {{val| 0 0 2 1 }}]
-[[POTE generator]]: ~8/7 = 231.045
-{{Val list|legend=1| 41, 328, 369, 779 }}
-[[Badness]]: 0.134559
+== Terminology ==
+The names were given by [[Flora Canou]] in 2022. ''Pythagorean countercomma'' was derived by analogy to ''Pythagorean comma'' as part of a series of Pythagorean interval names. ''Countercomp comma'' was derived from the temperament name, ''countercomp'', which was changed from ''counterpyth'' as in earlier materials, where this comma was also called ''counterpyth comma''. It was renamed after the convention was established that the temperament of the [[Pythagorean comma]] should be [[compton]] and never ''Pythagorean'', for fear of confusion with [[Pythagorean tuning]].
 == See also ==
 * [[Small comma]]
+* [[Schismic–countercommatic equivalence continuum]]
-[[Category:3-limit]]
+[[Category:Countercomp]]
-[[Category:Small comma]]
+[[Category:Commas named systematically]]
-[[Category:Counterpyth]]
+[[Category:Commas named for their regular temperament properties]]