Tridecapyth comma: Difference between revisions

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{{Infobox Interval

~~| JI glyph =~~

| Ratio = 3489660928/3486784401

~~| Monzo = 28 -20 0 0 0 1~~

~~| Cents = 1.42764~~

| Name = tridecapyth comma

| Color name =

| Color name = s33o3, trisatho 3rd, Trisatho comma

| ~~FJS name =~~

| Comma = yes

~~| Sound~~ =

}}

The '''tridecapyth comma''' ~~is the [[13~~-~~limit]]~~ [[~~unnoticeable comma~~]] ~~'''~~3489660928/3486784401''', which measures roughly 1.43 [[cent~~]]s~~. It is the interval which~~, when tempered out, not only equates~~ [[13/8]] ~~with~~ a stack of twenty [[3/2]] ~~perfect fifths~~ octave-reduced, ~~but also~~ a stack of ~~two~~ [[Pythagorean ~~comma~~]]~~s with~~ the [[1053/1024]] ~~tridecimal quartertone~~.

The '''tridecapyth comma''' ({{Monzo|legend=1| 28 -20 0 0 0 1 }}, [[ratio]]: 3489660928/3486784401), also described as the ''tridecaschisma'' (from "trideca-" (for 13) + "schisma"), is an [[unnoticeable comma]] in [[13-limit|13-limit just intonation]] which measures roughly 1.43 {{cent}}. It is the interval by which [[13/8]] exceeds a stack of twenty [[3/2|perfect fifths (3/2)]] octave reduced, and by which [[16/13]] falls short of a stack of four [[256/243|Pythagorean limmas (256/243)]]. It is perhaps more easily conceptualized as reaching [[13/4]] through ([[9/8]])10. In terms of commas, it is the amount by which [[1053/1024|tridecimal quartertone (1053/1024)]] is greater than a stack of two [[Pythagorean comma]]s.

[[~~Category~~:13-~~limit~~]]

== Temperaments ==

[[Category:~~Unnoticeable comma~~]]

=== Tridecapyth ===

{{ See also | No-fives subgroup temperaments#Temperaments with a 2.3.13 gene }}

Tempering out this comma in the 2.3.13 subgroup leads tridecapyth, which can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f [[val]] that takes the second-best, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, due to its small note count, [[41edo]] technically uses this mapping too, so that the val sum 41 + 53 = [[94edo]] also uses this mapping, suggesting it's of interest to flatter tunings of [[garibaldi]] with fifths tending close to pure; this corresponds to the extension of garibaldi called [[cassandra]]. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].

[[Subgroup]]: 2.3.13

: mapping generators: ~2, ~3/2

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.025

: [[error map]]: {{val| 0.000 +0.070 -0.019 }}

{{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788 }}

[[Badness]] (Sintel): 0.211

== Etymology ==

This comma was named by [[Aura]] in 2021.

== See also ==

* [[Tridecimal schisma]] (disambiguation page)

[[Category:Commas named for the intervals they stack]]

[[Category:Commas named after polymaths]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| JI glyph =
 | Ratio = 3489660928/3486784401
-| Monzo = 28 -20 0 0 0 1
-| Cents = 1.42764
 | Name = tridecapyth comma
-| Color name =
+| Color name = s<sup>3</sup>3o3, trisatho 3rd,<br>Trisatho comma
-| FJS name =
+| Comma = yes
-| Sound =
 }}
-The '''tridecapyth comma''' is the [[13-limit]] [[unnoticeable comma]] '''3489660928/3486784401''', which measures roughly 1.43 [[cent]]s.  It is the interval which, when tempered out, not only equates [[13/8]] with a stack of twenty [[3/2]] perfect fifths octave-reduced, but also a stack of two [[Pythagorean comma]]s with the [[1053/1024]] tridecimal quartertone.
+The '''tridecapyth comma''' ({{Monzo|legend=1| 28 -20 0 0 0 1 }}, [[ratio]]: 3489660928/3486784401), also described as the ''tridecaschisma'' (from "trideca-" (for 13) + "schisma"), is an [[unnoticeable comma]] in [[13-limit|13-limit just intonation]] which measures roughly 1.43 {{cent}}. It is the interval by which [[13/8]] exceeds a stack of twenty [[3/2|perfect fifths (3/2)]] octave reduced, and by which [[16/13]] falls short of a stack of four [[256/243|Pythagorean limmas (256/243)]]. It is perhaps more easily conceptualized as reaching [[13/4]] through ([[9/8]])<sup>10</sup>. In terms of commas, it is the amount by which [[1053/1024|tridecimal quartertone (1053/1024)]] is greater than a stack of two [[Pythagorean comma]]s.
-[[Category:13-limit]]
+== Temperaments ==
-[[Category:Unnoticeable comma]]
+=== Tridecapyth ===
+{{ See also | No-fives subgroup temperaments#Temperaments with a 2.3.13 gene }}
+Tempering out this comma in the 2.3.13 subgroup leads tridecapyth, which can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f [[val]] that takes the second-best, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, due to its small note count, [[41edo]] technically uses this mapping too, so that the val sum 41 + 53 = [[94edo]] also uses this mapping, suggesting it's of interest to flatter tunings of [[garibaldi]] with fifths tending close to pure; this corresponds to the extension of garibaldi called [[cassandra]]. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].
+[[Subgroup]]: 2.3.13
+{{mapping|legend=1| 1 1 -8 | 0 1 20 }}
+: mapping generators: ~2, ~3/2
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.025
+: [[error map]]: {{val| 0.000 +0.070 -0.019 }}
+{{Optimal ET sequence|legend=1| 12, 29f, 41, 53, 94, 147, 494, 641, 788 }}
+[[Badness]] (Sintel): 0.211
+== Etymology ==
+This comma was named by [[Aura]] in 2021.
+== See also ==
+* [[Tridecimal schisma]] (disambiguation page)
+[[Category:Commas named for the intervals they stack]]
+[[Category:Commas named after polymaths]]