2048edo: Difference between revisions

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== Theory ==
== Theory ==
{{Harmonics in equal|2048}}
2048edo has an excellent [[3/2|perfect fifth]], which derives from [[1024edo]]. It tempers out the breedsma ([[2401/2400]]) in the 7-limit, and supports the rank-3 breed temperament.
2048edo has an excellent perfect fifth, which derives from 1024edo. It tempers out the breedsma (2401/2400) in the 7-limit, and supports the rank 3 breed temperament.


2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13s limit patent val is unambiguous and has an error of only 0.019 cents/octave.
2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13's limit patent val is unambiguous and has an error of only 0.019 cents/octave.


2048edo is the 11th power of two EDO.
In the 5-limit, it supports [[monzismic]] temperament. In the 11-limit, 2048edo tempers out the [[quartisma]].


In the 5-limit, it supports [[monzismic]] temperament.
=== Prime harmonics ===
{{Harmonics in equal|2048}}


In the 11-limit, 2048edo tempers out the [[quartisma]].
=== Subsets and supersets ===
2048edo is the 11th-power-of-two edo.


== 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|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal
! rowspan="2" | Optimal<br>8ve Stretch (¢)
8ve stretch (¢)
! colspan="2" | Tuning Error
! 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|54, -37, 2}},{{monzo|-111, -12, 56}}
| {{monzo| 54 -37 2 }}, {{monzo| -111 -12 56 }}
|[{{val|2048 3246 4755}}]
| {{mapping| 2048 3246 4755 }}
|0.0264
| 0.0264
|0.0364
| 0.0364
|6.22
| 6.22
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, {{monzo|-18, -13, 13, 3}}, {{monzo|49, -38, 0, 4}}
| 2401/2400, {{monzo| -18 -13 13 3 }}, {{monzo| 49 -38 0 4 }}
|[{{val|2048 3246 4755 5749}}]
| {{mapping| 2048 3246 4755 5749 }}
|0.0439
| 0.0439
|0.0438
| 0.0438
|7.48
| 7.48
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179
| 2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179
|[{{val|2048 3246 4755 5749 7085}}]
| {{mapping| 2048 3246 4755 5749 7085 }}
|0.0323
| 0.0323
|0.0456
| 0.0456
|7.78
| 7.78
|}
|}
[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->

Revision as of 13:32, 15 October 2023

← 2047edo 2048edo 2049edo →
Prime factorization 211
Step size 0.585938 ¢ 
Fifth 1198\2048 (701.953 ¢) (→ 599\1024)
Semitones (A1:m2) 194:154 (113.7 ¢ : 90.23 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

2048edo has an excellent perfect fifth, which derives from 1024edo. It tempers out the breedsma (2401/2400) in the 7-limit, and supports the rank-3 breed temperament.

2048edo is usable as high as 43-limit, with the 2048f val regular temperament having an error of 0.023 cents (0.04 edosteps) per octave, and the no-13's limit patent val is unambiguous and has an error of only 0.019 cents/octave.

In the 5-limit, it supports monzismic temperament. In the 11-limit, 2048edo tempers out the quartisma.

Prime harmonics

Approximation of prime harmonics in 2048edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.002 -0.181 -0.271 +0.049 +0.293 -0.073 +0.143 -0.149 -0.085 -0.114
Relative (%) +0.0 -0.3 -30.9 -46.3 +8.4 +49.9 -12.4 +24.4 -25.5 -14.5 -19.4
Steps
(reduced) 		2048
(0) 		3246
(1198) 		4755
(659) 		5749
(1653) 		7085
(941) 		7579
(1435) 		8371
(179) 		8700
(508) 		9264
(1072) 		9949
(1757) 		10146
(1954)

Subsets and supersets

2048edo is the 11th-power-of-two edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3.5 [54 -37 2⟩, [-111 -12 56⟩ [⟨2048 3246 4755]] 0.0264 0.0364 6.22
2.3.5.7 2401/2400, [-18 -13 13 3⟩, [49 -38 0 4⟩ [⟨2048 3246 4755 5749]] 0.0439 0.0438 7.48
2.3.5.7.11 2401/2400, 1890625/1889568, 1953125/1951488, 117440512/117406179 [⟨2048 3246 4755 5749 7085]] 0.0323 0.0456 7.78
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