Lesser tendoneutralisma: Difference between revisions
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| Ratio = 70368744177664/69894255367443 | | Ratio = 70368744177664/69894255367443 | ||
| Name = lesser tendoneutralisma | | Name = lesser tendoneutralisma | ||
| Color name = Sasa-quadtrithu comma | |||
| Comma = yes | | Comma = yes | ||
}} | }} | ||
The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. | The '''lesser tendoneutralisma''' is a [[small comma]] of the 2.3.13 subgroup which is the amount by which a stack of twelve [[16/13]]'s minus three [[octave]]s exceeds [[3/2]]; that is, it is equal to ([[16/13]])<sup>12</sup> / ([[3/2]]) / ([[2/1]])<sup>3</sup> and so equivalently also to ([[16/13]])<sup>11</sup> / ([[39/4]]). Because the ~11{{cent}} of error is distributed over twelve [[16/13]]'s in the pure-3's tuning, it is a very accurate way of connecting a chain of 13's to prime 3. A slightly less accurate but still good way of doing this is using the [[greater tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you can't combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. (A much simpler but (relatively) much higher error way of mapping 3 for those that prefer sharp fifths is by tempering ([[16/13]])<sup>2</sup>/([[3/2]]) = [[512/507]].) | ||
== Temperaments == | == Temperaments == | ||
=== Lesser Tendoneutralic === | === Lesser Tendoneutralic === | ||
{{ See also | Restles }} | |||
Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence). | Tempering the lesser tendoneutralisma in 2.3.13 leads to the highly notable 10 & 77 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[77edo]] can be used to approximate the pure-3's tuning, although [[67edo]] is an interesting choice for combining this temperament with meantone and perhaps more notably [[87edo]] is a very good choice for combining this temperament with [[parapyth]] and [[13-limit]] harmony generally (although it doesn't appear in the optimal ET sequence). | ||
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{{Optimal ET sequence|legend=1| 10, 47, 57, 67, 77, 164, 241, 405, 646f }} | {{Optimal ET sequence|legend=1| 10, 47, 57, 67, 77, 164, 241, 405, 646f }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 3.930 | ||
== See also == | == See also == | ||
* [[Greater tendoneutralisma]] | * [[Greater tendoneutralisma]] | ||
* [[512/507]] | |||
* [[Tridecapyth comma]] | * [[Tridecapyth comma]] | ||
[[Category:Commas with unknown etymology]] | |||