248edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|248}}
{{ED intro}}


== Theory ==
== Theory ==
248et tempers out [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]].  
248edo shares the mapping of [[harmonic]]s [[5/1|5]] and [[7/1|7]] with [[31edo]]. It has a decent 13-limit interpretation despite not being [[consistent]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[3136/3125]] and [[420175/419904]] in the 7-limit; [[441/440]], [[8019/8000]] in the 11-limit; [[729/728]], [[847/845]], [[1001/1000]], [[1575/1573]] and [[2200/2197]] in the 13-limit. It also notably tempers out the [[quartisma]]. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from [[131edt]]. Similarly, using the lower-error 248[[Wart notation|h]] val, the mappings of all its [[2.5.7_subgroup|no-3]] harmonics up to [[23-limit|28]] are multiples of 2 and derived from [[124edo]].


It [[support]]s [[Schismatic family #Bischismic|bischismic temperament]], providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for [[Varunismic temperaments #Essence|essence temperament]]. It is notable for its combination of precise intonation with an abundance of essentially tempered chords. 248 has divisors 2, 4, 8, 31, 62, and 124.
It [[support]]s the [[bischismic]] temperament, providing the [[optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the [[essence]] temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|248|}}
{{Harmonics in equal|248|}}
=== Subsets and supersets ===
Since 248 factors into {{factorization|248}}, 248edo has subset edos {{EDOs| 2, 4, 8, 31, 62, and 124 }}.


== 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]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| 287 -181 }}
| {{monzo| 287 -181 }}
| [{{val| 248 393 }}]
| {{mapping| 248 393 }}
| +0.108
| +0.108
| 0.108
| 0.108
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| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 12 32 -27 }}
| 32805/32768, {{monzo| 12 32 -27 }}
| [{{val| 248 393 576 }}]
| {{mapping| 248 393 576 }}
| -0.041
| -0.041
| 0.228
| 0.228
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| 2.3.5.7
| 2.3.5.7
| 3136/3125, 32805/32768, 420175/419904
| 3136/3125, 32805/32768, 420175/419904
| [{{val| 248 393 576 696 }}]
| {{mapping| 248 393 576 696 }}
| +0.066
| +0.066
| 0.270
| 0.270
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| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 3136/3125, 8019/8000, 41503/41472
| 441/440, 3136/3125, 8019/8000, 41503/41472
| [{{val| 248 393 576 696 858 }}]
| {{mapping| 248 393 576 696 858 }}
| +0.036
| +0.036
| 0.249
| 0.249
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 441/440, 729/728, 847/845, 1001/1000, 3136/3125
| 441/440, 729/728, 847/845, 1001/1000, 3136/3125
| [{{val| 248 393 576 696 858 918 }}]
| {{mapping| 248 393 576 696 858 918 }}
| +0.079
| +0.079
| 0.275
| 0.275
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 498.39
| 498.39
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 2
| 2
| 77\248<br>(47\248)
| 77\248<br />(47\248)
| 372.58<br>(227.42)
| 372.58<br />(227.42)
| 26/21<br>(154/135)
| 26/21<br />(154/135)
| [[Essence]]
| [[Essence]]
|-
|-
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|-
|-
| 8
| 8
| 117\248<br>(7\248)
| 117\248<br />(7\248)
| 566.13<br>(33.87)
| 566.13<br />(33.87)
| 104/75<br>(49/48)
| 104/75<br />(49/48)
| [[Octowerck]]
| [[Octowerck]]
|-
|-
| 31
| 31
| 103\248<br>(1\248)
| 103\248<br />(1\248)
| 498.39<br>(4.84)
| 498.39<br />(4.84)
| 4/3<br>(385/384)
| 4/3<br />(385/384)
| [[Birds]]
| [[Birds]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Bischismic]]
[[Category:Bischismic]]
[[Category:Essence]]
[[Category:Essence]]
[[Category:Quartismic]]

Latest revision as of 02:29, 17 April 2025

← 247edo 248edo 249edo →
Prime factorization 23 × 31
Step size 4.83871 ¢ 
Fifth 145\248 (701.613 ¢)
Semitones (A1:m2) 23:19 (111.3 ¢ : 91.94 ¢)
Consistency limit 11
Distinct consistency limit 11

248 equal divisions of the octave (abbreviated 248edo or 248ed2), also called 248-tone equal temperament (248tet) or 248 equal temperament (248et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 248 equal parts of about 4.84 ¢ each. Each step represents a frequency ratio of 21/248, or the 248th root of 2.

Theory

248edo shares the mapping of harmonics 5 and 7 with 31edo. It has a decent 13-limit interpretation despite not being consistent. The equal temperament tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440, 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It also notably tempers out the quartisma. 248edo, additionally, has the interesting property of its mapping for all prime harmonics 3 to 23 being a multiple of 3, and therefore derived from 131edt. Similarly, using the lower-error 248h val, the mappings of all its no-3 harmonics up to 28 are multiples of 2 and derived from 124edo.

It supports the bischismic temperament, providing the optimal patent val for 11-limit bischismic, and excellent tunings in the 7- and 13-limits. It also provides the optimal patent val for the essence temperament. It is notable for its combination of precise intonation with an abundance of essentially tempered chords.

Prime harmonics

Approximation of prime harmonics in 248edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.34 +0.78 -1.08 +0.29 +1.41 +1.50 -2.35 +0.76 +1.07 +1.74
Relative (%) +0.0 -7.1 +16.2 -22.4 +6.1 +29.1 +30.9 -48.6 +15.7 +22.1 +35.9
Steps
(reduced) 		248
(0) 		393
(145) 		576
(80) 		696
(200) 		858
(114) 		918
(174) 		1014
(22) 		1053
(61) 		1122
(130) 		1205
(213) 		1229
(237)

Subsets and supersets

Since 248 factors into 23 × 31, 248edo has subset edos 2, 4, 8, 31, 62, and 124.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [287 -181⟩ [⟨248 393]] +0.108 0.108 2.23
2.3.5 32805/32768, [12 32 -27⟩ [⟨248 393 576]] -0.041 0.228 4.70
2.3.5.7 3136/3125, 32805/32768, 420175/419904 [⟨248 393 576 696]] +0.066 0.270 5.58
2.3.5.7.11 441/440, 3136/3125, 8019/8000, 41503/41472 [⟨248 393 576 696 858]] +0.036 0.249 5.15
2.3.5.7.11.13 441/440, 729/728, 847/845, 1001/1000, 3136/3125 [⟨248 393 576 696 858 918]] +0.079 0.275 5.69

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 5\248 24.19 686/675 Sengagen
1 103\248 498.39 4/3 Helmholtz
2 77\248
(47\248) 		372.58
(227.42) 		26/21
(154/135) 		Essence
2 103\248 498.39 4/3 Bischismic
8 117\248
(7\248) 		566.13
(33.87) 		104/75
(49/48) 		Octowerck
31 103\248
(1\248) 		498.39
(4.84) 		4/3
(385/384) 		Birds

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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