13edt: Difference between revisions

Line 1:

{{ED intro}} It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.

[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[Riemann zeta function#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]

[[File:pts-3-5-7-e3-twtop.jpg|thumb|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]

'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-~~tempered version of the [[Bohlen–Pierce]] scale,~~ and ~~therefore has received by far the most attention among equal divisions of the tritave.~~

== Theory ==

{{See also| Relationship between Bohlen-Pierce and octave-ful temperaments }}

It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[~~Sensamagic_clan#Bohpier|~~bohpier ~~temperament~~]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.

13edt provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as the rank-2 [[bohpier]] temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.

13edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].

In the [[3.5.7 subgroup]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

In the 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

~~== Theory ==~~

~~{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}~~

~~{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}~~

* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]

=== Odd harmonics ===

{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 13edt (continued)}}

== Intervals ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
+{{ED intro}} It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.
 [[File:13edt.png|thumb|alt=13edt.png|A plot of the [[Riemann zeta function#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
 [[File:pts-3-5-7-e3-twtop.jpg|thumb|Projective tuning space of the [[3.5.7 subgroup]], showing 13edt near the center.]]
-'''13 equal divisions of the tritave''' ('''13edt''') is the [[nonoctave]] [[tuning system]] derived by dividing the [[tritave]] (3/1) into 13 equal steps of 146.3 [[cent]]s each, or the thirteenth root of 3. It is best known as the equal-tempered version of the [[Bohlen–Pierce]] scale, and therefore has received by far the most attention among equal divisions of the tritave.
+== Theory ==
+{{See also| Relationship between Bohlen-Pierce and octave-ful temperaments }}
-It provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as [[Sensamagic_clan#Bohpier|bohpier temperament]]. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.
+edt provides an excellent approximation to the [[3.5.7 subgroup]], especially for its size, being comparable to [[34edo]]'s accuracy in the 5-limit. In this subgroup, it tempers out [[245/243]] and [[3125/3087]], the same commas as the rank-2 [[bohpier]] temperament. It is less impressive in higher prime limits, but makes for excellent no-twos 7-limit harmony. For higher limits, the multiples of 13 ([[26edt]], [[39edt]], and [[52edt]]) come to the fore.
 edt can be described as approximately 8.202[[edo]]. This implies that each step of 13edt can be approximated by 5 steps of [[41edo]].
-In the [[3.5.7 subgroup]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
+In the 3.5.7 subgroup, 13edt maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
-== Theory ==
-{{Harmonics in equal|13|3|1|prec=2|intervals=odd}}
-{{Harmonics in equal|13|3|1|prec=2|intervals=odd|start=12}}
-* [[Relationship between Bohlen-Pierce and octave-ful temperaments]]
+=== Odd harmonics ===
+{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=11}}
+{{Harmonics in equal|13|3|1|prec=2|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 13edt (continued)}}
 == Intervals ==

13edt: Difference between revisions

FloraC (talk | contribs)