2016edo: Difference between revisions

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-rank-2 temps (until they're named; otherwise there would be infinitely many); misc cleanup
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| Step size = 0.59524¢
| Step size = 0.59524¢
| Fifth = 1179\2016 (701.79¢) (→[[224edo|131\224]])
| Fifth = 1179\2016 (701.79¢) (→[[224edo|131\224]])
|Semitones=342:153}}
| Semitones = 342:153}}
The '''2016 equal divisions of the octave''' ('''2016edo'''), or the '''2016-tone equal temperament''' ('''2016tet'''), '''2016 equal temperament''' ('''2016et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 2016 [[equal]] parts of about 595  milli[[cent]]s each.  
The '''2016 equal divisions of the octave''' ('''2016edo'''), or the '''2016-tone equal temperament''' ('''2016tet'''), '''2016 equal temperament''' ('''2016et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 2016 [[equal]] parts of about 595  milli[[cent]]s each.  
== Theory ==
== Theory ==
{{Harmonics in equal|2016}}
{{Harmonics in equal|2016}}
Line 26: Line 27:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" | '[[Subgroup]]
! rowspan="2" |[[Comma list]]
! rowspan="2" | [[Comma list]]
(zeroes skipped for clarity)
! rowspan="2" | [[Mapping]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" |Optimal
! colspan="2" | Tuning error
8ve stretch (¢)
! colspan="2" |Tuning error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3.5
| 2.3.5
|{{Monzo|-83, 26, 18}}, {{Monzo|30, 47, -45}}
| {{Monzo| -83 26 18 }}, {{Monzo| 30 47 -45 }}
|[{{Val|2016 3195 4681}}]
| [{{Val| 2016 3195 4681 }}]
|0.036
| 0.036
|0.050
| 0.050
|8.4
| 8.4
|-
|-
|2.3.5.7
| 2.3.5.7
|250047/250000, {{monzo|7, 18, -2, -11}}, {{monzo|43, -1, -13, -4}}
| 250047/250000, {{monzo| 7 18 -2 -11 }}, {{monzo| 43 -1 -13 -4 }}
|[{{Val|2016 3195 4681 5660}}]
| [{{Val| 2016 3195 4681 5660 }}]
|0.007
| 0.007
|0.066
| 0.066
|11.1
| 11.1
|-
|-
|2.3.5.7
| 2.3.5.7
|5250987/5242880, 40353607/40310784, {{monzo|14, 11, -22, 7}}
| 5250987/5242880, 40353607/40310784, {{monzo| 14 11 -22 7 }}
|[{{Val|2016 3195 4681 5659}}] (2016d)
| [{{Val| 2016 3195 4681 5659 }}] (2016d)
|0.060
| 0.060
|0.060
| 0.060
|10.1
| 10.1
|-
|2.5.11.13
|{{monzo|5 -6 9 6}}, {{monzo|-38 12 4 -1}}, {{monzo|0 -22 3 11}}
|[{{Val|2016 4681 6974 7460}}]
|0.013
|0.015
|2.5
|-
|2.5.11.13.19.41.47
|7943/7942, 322465/322373, 415292/415207, 511225/511024,
 
5078491/5078125, 22151168/22150865
|{{Val|2016 4681 6974 7460 8564 10801 11198}}
|0.002
|0.019
|3.2
|}
 
== Rank two temperaments by generator ==
{| class="wikitable center-all left-5"
!Periods
per octave
!Generator
(reduced)
!Cents
(reduced)
!Temperaments
|-
|-
|21
| 2.5.11.13
|23\2016
| {{monzo| 5 -6 9 6 }}, {{monzo| -38 12 4 -1 }}, {{monzo| 0 -22 3 11 }}
|13.690
| [{{Val| 2016 4681 6974 7460 }}]
|441 & 1407
| 0.013
| 0.015
| 2.5
|-
|-
|24
| 2.5.11.13.19.41.47
|29\2016
| 7943/7942, 322465/322373, 415292/415207, 511225/511024, <br>5078491/5078125, 22151168/22150865
|17.262
| {{Val| 2016 4681 6974 7460 8564 10801 11198 }}
|72 & 624
| 0.002
| 0.019
| 3.2
|}
|}

Revision as of 15:32, 18 April 2022

← 2015edo 2016edo 2017edo →
Prime factorization 25 × 32 × 7
Step size 0.595238 ¢ 
Fifth 1179\2016 (701.786 ¢) (→ 131\224)
Semitones (A1:m2) 189:153 (112.5 ¢ : 91.07 ¢)
Consistency limit 5
Distinct consistency limit 5

The 2016 equal divisions of the octave (2016edo), or the 2016-tone equal temperament (2016tet), 2016 equal temperament (2016et) when viewed from a regular temperament perspective, divides the octave into 2016 equal parts of about 595 millicents each.

Theory

Approximation of odd harmonics in 2016edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.169 -0.004 +0.222 +0.257 -0.127 -0.051 -0.173 -0.194 +0.106 +0.052 +0.297
Relative (%) -28.4 -0.7 +37.2 +43.1 -21.4 -8.6 -29.1 -32.5 +17.8 +8.8 +49.9
Steps
(reduced) 		3195
(1179) 		4681
(649) 		5660
(1628) 		6391
(343) 		6974
(926) 		7460
(1412) 		7876
(1828) 		8240
(176) 		8564
(500) 		8855
(791) 		9120
(1056)

2016 is a significantly composite number, with its divisors being 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008. It's abundancy index is 2.25.

Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.

2016 shares the mapping for 3 with 224edo, albeit with a 28 relative cent error.

2016edo has two reasonable mappings for 7. The 2016d val, ⟨2016 3195 4681 5659], tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and [14, 11, -22, 7⟩. As such, its circle of the interval 7/6 is the same as in 9edo.

The patent val, ⟨2016 3195 4681 5658] tempers out 250047/250000, along with [7, 18, -2, -11⟩ and [43, -1, -13, -4⟩. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to 63/50.

In the 11-limit, 2016edo tempers out the [0 0 -22 0 3 11⟩ comma, which equates a stack of 11 25/13s with 3 hendecataves.

Fractional octave temperaments

The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of 72 & 624 with period 24 and 441 & 1407, period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18. Using the 441 & 1407 temperament, 2016edo also tempers out the akjaysma.

If we assume that 2016edo is a dual-seventh system, where 5659th and 5660th steps represent 7- and 7+, two distinct dimensions amounting to 49/1, then this allows for mixing of these two temperaments. 236 & 600 is the temperament which best represents that.

Regular temperament properties

'Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [-83 26 18⟩, [30 47 -45⟩ [⟨2016 3195 4681]] 0.036 0.050 8.4
2.3.5.7 250047/250000, [7 18 -2 -11⟩, [43 -1 -13 -4⟩ [⟨2016 3195 4681 5660]] 0.007 0.066 11.1
2.3.5.7 5250987/5242880, 40353607/40310784, [14 11 -22 7⟩ [⟨2016 3195 4681 5659]] (2016d) 0.060 0.060 10.1
2.5.11.13 [5 -6 9 6⟩, [-38 12 4 -1⟩, [0 -22 3 11⟩ [⟨2016 4681 6974 7460]] 0.013 0.015 2.5
2.5.11.13.19.41.47 7943/7942, 322465/322373, 415292/415207, 511225/511024,
5078491/5078125, 22151168/22150865 		⟨2016 4681 6974 7460 8564 10801 11198] 0.002 0.019 3.2
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