13edt: Difference between revisions

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

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

'''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.

'''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.

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.

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 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 [[no-2]] [[3/1-equave-7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].

== Theory ==

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! [[4L 5s (3/1-equivalent)|Enneatonic]] degree

! Corresponding 3.5.7 subgroup intervals

! [[Lambda ups and downs notation|Lambda]] ~~(sLsLsLsLs,~~ E = 1/1)

! [[Lambda ups and downs notation|Lambda]] (sLsLsLsLs, {{nowrap|E {{=}} 1/1}})

|-

| 0

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=== Rank-2 temperaments ===

{| class="wikitable center-all right-3 left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per tritave

|-

! Generator (reduced)

! Periods per tritave

! Cents (reduced)

! Generator (reduced)

! Associated ratio

! Cents (reduced)

! Associated ratio

! Temperament

|-

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| [[Canopus]]

|-

|1

| 1

|5\13

| 5\13

|731.63

| 731.63

|75/49

| 75/49

|

|-

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* [[23ed7|23ED7]]: relative ED7

~~[[Category:Edt]]~~

[[Category:Tritave]]

[[Category:Macrotonal]]

[[Category:Nonoctave]]

[[Category:~~Bohlen-Pierce~~]]

[[Category:Bohlen–Pierce]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[The_Riemann_Zeta_Function_and_Tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
+[[File:13edt.png|thumb|alt=13edt.png|A plot of the [[the Riemann zeta function and tuning#Removing primes|no-twos Z-function]], in terms of which 13edt is the fourth no-twos zeta peak [[EDT]].]]
-'''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.
+'''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.
-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.
+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 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 [[no-2]] [[3/1-equave-7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
+In the [[no-2]] [[3/1]]-[[equave]]-[[7-limit]], [[13edt]] maintains the smallest relative error of any EDT until [[258edt]] and [[271edt]], and the smallest absolute error until [[56edt]].
 == Theory ==
@@ Line 26: / Line 26: @@
 ! [[4L 5s (3/1-equivalent)|Enneatonic]]<br />degree
 ! Corresponding<br />3.5.7 subgroup<br />intervals
-! [[Lambda ups and downs notation|Lambda]] (sLsLsLsLs,<br />E = 1/1)
+! [[Lambda ups and downs notation|Lambda]]<br />(sLsLsLsLs, {{nowrap|E {{=}} 1/1}})
 |-
 | 0
@@ Line 151: / Line 151: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all right-3 left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per tritave
+|-
-! Generator<br>(reduced)
+! Periods<br />per tritave
-! Cents<br>(reduced)
+! Generator<br />(reduced)
-! Associated<br>ratio
+! Cents<br />(reduced)
+! Associated<br />ratio
 ! Temperament
 |-
@@ Line 182: / Line 183: @@
 | [[Canopus]]
 |-
-|1
+| 1
-|5\13
+| 5\13
-|731.63
+| 731.63
-|75/49
+| 75/49
 |
 |-
@@ Line 202: / Line 203: @@
 * [[23ed7|23ED7]]: relative ED7
-[[Category:Edt]]
 [[Category:Tritave]]
 [[Category:Macrotonal]]
 [[Category:Nonoctave]]
-[[Category:Bohlen-Pierce]]
+[[Category:Bohlen–Pierce]]