1012edo: Difference between revisions

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== Theory ==

1012edo is a strong 13-limit system, ~~distinctly~~ [[consistent]] through the 15-odd-limit. It is a [[~~The Riemann zeta function and tuning #Zeta EDO lists|~~zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit ~~commas is~~ [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

In the ~~2.3~~.7~~.11.101~~, it tempers out [[~~7777~~/~~7776~~]] ~~and~~ is a ~~tuning~~ for ~~the~~ [[~~neutron star~~]] temperament.

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}.

Since 1012 factors into {{nowrap| 22 × 11 × 23 }}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

~~=== Trivia ===~~

~~In addition to containing 22edo and 23edo, it contains a~~ [[~~22L 1s]] scale produced by generator of 45\1012 associated with [[33/32~~]], ~~and is associated with~~ the ~~45 & 1012 temperament~~, ~~making it [[concoctic]]. A comma basis~~ for the ~~13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1 0 -3 }}~~.

== Regular temperament properties ==

=== Rank-2 temperaments ===

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

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

|-

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

| 1

| 361\1012

| 428.~~066~~

| 428.06

| 2800/2187

| [[Osiris]]

|-

| 2

| ~~491~~\1012

| 145\1012

| 498.~~023~~

| 171.94

| 243/220

| [[Semiosiris]]

|-

| 2

| 420\1012

| 498.02

| 7/5

| [[Quarvish]]

Line 38:

Line 44:

| 44

| 420\1012 (6\1012)

| 498.~~023~~ (7.~~115~~)

| 498.02 (7.11)

| 4/3 (18375/18304)

| [[Ruthenium]]

|}

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|1012}}
+{{ED intro}}
 == Theory ==
-edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.
+edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.
-In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.
+In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.
 === Prime harmonics ===
 {{Harmonics in equal|1012}}
-=== Divisors ===
+=== Subsets and supersets ===
-has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}.
+Since 1012 factors into {{nowrap| 2<sup>2</sup> × 11 × 23 }}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.
-=== Trivia ===
-In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 & 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.
 == Regular temperament properties ==
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
 | 1
 | 361\1012
-| 428.066
+| 428.06
 | 2800/2187
 | [[Osiris]]
 |-
 | 2
-| 491\1012
+| 145\1012
-| 498.023
+| 171.94
+| 243/220
+| [[Semiosiris]]
+|-
+| 2
+| 420\1012
+| 498.02
 | 7/5
 | [[Quarvish]]
@@ Line 38: / Line 44: @@
 | 44
 | 420\1012<br>(6\1012)
-| 498.023<br>(7.115)
+| 498.02<br>(7.11)
 | 4/3<br>(18375/18304)
 | [[Ruthenium]]
 |}
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]