6656/6561: Difference between revisions

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The [[~~tetris~~ comma]] is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]]. ~~(It is named as a contraction of "tetratone" and "tridecimal".)~~

The '''6656/6561''', the '''tetris comma''', is a [[small comma|small]] [[13-limit]] [[comma]]. It is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]].

== Temperaments ==

When tempered, it implies a sharp fifth or a doubly as sharp [[9/8|tone]], and it features as the important comma that reduces rank 3 [[parapythic]] to [[#No-5's leapday|no-5's leapday]], which is notable as having much lower [[badness]], as discussed there on this page.

When [[tempering out|tempered out]], it implies a sharp fifth or a doubly as sharp [[9/8|tone]], and it features as the important comma that reduces rank-3 [[parapythic]] to [[#No-5's leapday|no-5's leapday]], which is notable as having much lower [[badness]], as discussed there on this page.

=== Tetris ===

If tempered only on the 2.3.13 subgroup, you get tetris. [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is ~~17 + 29 =~~ [[46edo]], which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related ~~no-5's 13-limit parapythic temperament documented below as~~ [[~~#No-5's Leapday|no-5's leapday~~]]; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.

If tempered out only on the 2.3.13 subgroup, you get tetris. [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related [[leapfrog]] temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.

[[Subgroup]]: 2.3.13

: mapping generators: ~2, ~3, ~13

[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~3/2 = 704.822

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 704.822

{{Optimal ET sequence|legend=1| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}

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Badness (Dirichlet): 0.522

==~~= No-5's Leapday =~~==

== Etymology ==

~~In regular [[13-limit]]~~ [[leapday]], the mapping for prime 5 is very complex at -25 gens, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. Furthermore, adding prime 5 to rank 3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: ~~a notable [[patent val~~]] ~~tuning not appearing~~ in ~~the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used~~ as a ~~[[consistent circle]]~~ of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "~~disqualified~~" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient.

This comma was named by [[User:Godtone|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".

~~[[Subgroup]]: 2.3.7.11.13~~

~~[[Comma list]]: [[896/891]], [[352/351]], [[6656/6561]]~~

~~{{mapping|legend=1| 1 0 -21 -14 -9 | 0 1 15 11 8 }}~~

~~: mapping generators: ~2, ~3, ~7, ~11, ~13~~

~~[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~3/2 = 704.633~~

~~{{Optimal ET sequence|legend=1| 12de, 17, 46, 63 }}~~

~~Badness (Dirichlet): 0~~.~~436~~

@@ Line 3: / Line 3: @@
 | Comma = yes
 }}
-The [[tetris comma]] is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]]. (It is named as a contraction of "tetratone" and "tridecimal".)
+The '''6656/6561''', the '''tetris comma''', is a [[small comma|small]] [[13-limit]] [[comma]]. It is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]].
 == Temperaments ==
-When tempered, it implies a sharp fifth or a doubly as sharp [[9/8|tone]], and it features as the important comma that reduces rank 3 [[parapythic]] to [[#No-5's leapday|no-5's leapday]], which is notable as having much lower [[badness]], as discussed there on this page.
+When [[tempering out|tempered out]], it implies a sharp fifth or a doubly as sharp [[9/8|tone]], and it features as the important comma that reduces rank-3 [[parapythic]] to [[#No-5's leapday|no-5's leapday]], which is notable as having much lower [[badness]], as discussed there on this page.
 === Tetris ===
-If tempered only on the 2.3.13 subgroup, you get tetris. [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is 17 + 29 = [[46edo]], which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related no-5's 13-limit parapythic temperament documented below as [[#No-5's Leapday|no-5's leapday]]; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.
+If tempered out only on the 2.3.13 subgroup, you get tetris. [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related [[leapfrog]] temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.
 [[Subgroup]]: 2.3.13
-{{mapping|legend=1| 1 0 -9 | 0 1 8 }}
+{{Mapping|legend=1| 1 0 -9 | 0 1 8 }}
 : mapping generators: ~2, ~3, ~13
-[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~3/2 = 704.822
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 704.822
 {{Optimal ET sequence|legend=1| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}
@@ Line 22: / Line 23: @@
 Badness (Dirichlet): 0.522
-=== No-5's Leapday ===
+== Etymology ==
-In regular [[13-limit]] [[leapday]], the mapping for prime 5 is very complex at -25 gens, which is also in the opposite direction of how all other primes in the [[13-limit]] are reached. Furthermore, adding prime 5 to rank 3 [[parapythic]] is arguably against the original vision of it as a 2.3.7.11.13-subgroup temperament, so avoiding prime 5 may be preferred for this reason also. This results in no-5's leapday, which as aforementioned is much lower in badness, but it also allows more tunings to be used: a notable [[patent val]] tuning not appearing in the [[optimal ET sequence]] is [[80edo]], which is approximately the just-13's tuning (as [[10edo]] is used as a [[consistent circle]] of [[~]][[16/13]]'s therein), with 13/8 still tuned slightly flat so qualifying a reasonable tuning for the 2.3.13 subgroup (as evidenced by appearing in the sequence for tetris). In other words, the only reason 80edo was "disqualified" from leapday is that the mapping for prime 5 constrains the tuning range which is naturally more flexible as a no-5's 13-limit temperament, which this case is also a sign of leapday being very efficient.
+This comma was named by [[User:Godtone|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".
-[[Subgroup]]: 2.3.7.11.13
-[[Comma list]]: [[896/891]], [[352/351]], [[6656/6561]]
-{{mapping|legend=1| 1 0 -21 -14 -9 | 0 1 15 11 8 }}
-: mapping generators: ~2, ~3, ~7, ~11, ~13
-[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~3/2 = 704.633
-{{Optimal ET sequence|legend=1| 12de, 17, 46, 63 }}
-Badness (Dirichlet): 0.436