Tridecapyth comma: Difference between revisions

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The '''tridecapyth comma''' ({{Monzo|legend=1| 28 -20 0 0 0 1 }}, [[ratio]]: 3489660928/3486784401) is an [[unnoticeable comma~~|unnoticeable~~]] [[13-limit~~]] [[comma~~]] which measures roughly 1.43 [[cent~~]]s~~. 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, and the ~~amount by which~~ [[~~262144~~/~~255879~~]] ~~exceeds~~ [[~~177147~~/~~173056~~]].

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.

== Temperaments ==

=== 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]]

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-The '''tridecapyth comma''' ({{Monzo|legend=1| 28 -20 0 0 0 1 }}, [[ratio]]: 3489660928/3486784401) is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[comma]] which measures roughly 1.43 [[cent]]s. 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, and the amount by which [[262144/255879]] exceeds [[177147/173056]].
+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.
+== Temperaments ==
+=== 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]]