Tridecapyth comma: Difference between revisions

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The '''tridecapyth comma''' ({{Monzo|legend=1| 28 -20 0 0 0 1 }}, [[ratio]]: 3489660928/3486784401), also described as the ''tridecaschisma'' (after the 2.3.13 temperament), 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.  
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 ==
== Temperaments ==
=== Tridecapyth ===
[[Tempering out]] this comma in the 2.3.13 subgroup leads to the [[tridecapyth]] temperament.  
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 tuning 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 ==
== Etymology ==

Latest revision as of 12:53, 30 June 2026

Interval information
Ratio 3489660928/3486784401
Factorization 228 × 3-20 × 13
Monzo [28 -20 0 0 0 1⟩
Size in cents 1.427644¢
Name tridecapyth comma
Color name s33o3, trisatho 3rd,
Trisatho comma
FJS name [math]\displaystyle{ \text{ddd3}^{13} }[/math]
Special properties reduced
Tenney norm (log2 nd) 63.3997
Weil norm (log2 max(n, d)) 63.4009
Wilson norm (sopfr(nd)) 129
Comma size unnoticeable
Open this interval in xen-calc

The tridecapyth comma (monzo: [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 just intonation which measures roughly 1.43  ¢. It is the interval by which 13/8 exceeds a stack of twenty perfect fifths (3/2) octave reduced, and by which 16/13 falls short of a stack of four 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 tridecimal quartertone (1053/1024) is greater than a stack of two Pythagorean commas.

Temperaments

Tempering out this comma in the 2.3.13 subgroup leads to the tridecapyth temperament.

Etymology

This comma was named by Aura in 2021.

See also

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