User:Aura/4191814edo: Difference between revisions

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~~== Theory ==~~

This edo has a [[consistency]] limit of 21, which is the most impressive out of all the 3-2 [[telicity|telic]] multiples of [[190537edo]]. It tempers out the [[archangelic comma]] in the [[3-limit]], and though this system's 5-limit and 7-limit are rather lackluster for an edo this size, the representation of the prime [[11/1|11]] is a bit better, and the representations of the [[13/1|13]], [[17/1|17]], and [[19/1|19]] are excellent, all which help to bridge the lackluster [[5/1|5]] and [[7/1|7]]. Thus, this system is worthy of a great deal of further exploration in the [[19-limit]].

This ~~EDO~~ has a consistency limit of 21, which is the most impressive out of all the 3-2 [[telicity|telic]] multiples of [[190537edo]]. It tempers out the [[~~Archangelic~~ comma]] in the 3-limit, and though this system's 5-limit and 7-limit are rather lackluster for an ~~EDO~~ this size, the representation of the 11~~-prime~~ is a bit better, and the representations of the 13~~-prime~~, 17~~-prime~~, and 19~~-prime~~ are excellent, all which help to bridge the lackluster 5~~-prime~~ and 7~~-prime~~. Thus, this system is worthy of a great deal of further exploration in the [[19-limit]].

In this system, the [[perfect fifth]] at 2452054\4191814 is divisible by the prime factors of 2, 11, 227 and 491. However, the [[perfect fourth]], at 1739760\4191814, has more prime divisors, namely the prime factors of 2^4, 3, 5, 11 and 659. The latter means that just as in [[159edo]], the perfect fourth is divisible by 33, and thus, this system can offer not only a more accurate version of [[Ozan Yarman]]'s original 79-tone system, but also [[metatemperament]]s to [[~~Yarman~~ I]] and [[~~Yarman~~ II]].

In this system, the [[perfect fifth]] at 2452054\4191814 is divisible by the prime factors of 2, 11, 227 and 491. However, the [[perfect fourth]], at 1739760\4191814, has more prime divisors, namely the prime factors of 2<sup>4</sup>, 3, 5, 11 and 659. The latter means that just as in [[159edo]], the perfect fourth is divisible by 33, and thus, this system can offer not only a more accurate version of [[Ozan Yarman]]'s original 79-tone system, but also [[metatemperament]]s to [[yarman I]] and [[yarman II]].

=== Prime harmonics ===

~~[[Category:Equal divisions~~ of ~~the octave|#######]] ~~

{{Harmonics in equal|4191814|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 4191814edo (continued)}}

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-{{Infobox ET}}
+{{Mathematical interest}}
-{{EDO intro|4191814}}
+{{Infobox ET|debug=1}}
+{{ED intro}}
-== Theory ==
+This edo has a [[consistency]] limit of 21, which is the most impressive out of all the 3-2 [[telicity|telic]] multiples of [[190537edo]]. It tempers out the [[archangelic comma]] in the [[3-limit]], and though this system's 5-limit and 7-limit are rather lackluster for an edo this size, the representation of the prime [[11/1|11]] is a bit better, and the representations of the [[13/1|13]], [[17/1|17]], and [[19/1|19]] are excellent, all which help to bridge the lackluster [[5/1|5]] and [[7/1|7]]. Thus, this system is worthy of a great deal of further exploration in the [[19-limit]].
-This EDO has a consistency limit of 21, which is the most impressive out of all the 3-2 [[telicity|telic]] multiples of [[190537edo]].  It tempers out the [[Archangelic comma]] in the 3-limit, and though this system's 5-limit and 7-limit are rather lackluster for an EDO this size, the representation of the 11-prime is a bit better, and the representations of the 13-prime, 17-prime, and 19-prime are excellent, all which help to bridge the lackluster 5-prime and 7-prime.  Thus, this system is worthy of a great deal of further exploration in the [[19-limit]].
-In this system, the [[perfect fifth]] at 2452054\4191814 is divisible by the prime factors of 2, 11, 227 and 491.  However, the [[perfect fourth]], at 1739760\4191814, has more prime divisors, namely the prime factors of 2^4, 3, 5, 11 and 659.  The latter means that just as in [[159edo]], the perfect fourth is divisible by 33, and thus, this system can offer not only a more accurate version of [[Ozan Yarman]]'s original 79-tone system, but also [[metatemperament]]s to [[Yarman I]] and [[Yarman II]].
+In this system, the [[perfect fifth]] at 2452054\4191814 is divisible by the prime factors of 2, 11, 227 and 491. However, the [[perfect fourth]], at 1739760\4191814, has more prime divisors, namely the prime factors of 2<sup>4</sup>, 3, 5, 11 and 659. The latter means that just as in [[159edo]], the perfect fourth is divisible by 33, and thus, this system can offer not only a more accurate version of [[Ozan Yarman]]'s original 79-tone system, but also [[metatemperament]]s to [[yarman I]] and [[yarman II]].
-{{Harmonics in equal|4191814|columns=12}}
+=== Prime harmonics ===
+{{Harmonics in equal|4191814|columns=9}}
-[[Category:Equal divisions of the octave|#######]] <!-- 7-digit number -->
+{{Harmonics in equal|4191814|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 4191814edo (continued)}}