248edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Plumtree (talk | contribs)
autopatrolled
1,976 edits
m Infobox ET now computes most parameters automatically
Expand on theory and misc. cleanup
Line 3: Line 3:


== 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]].  
248 = 8 × 31, and 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]].  


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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 23: Line 26:
| 2.3
| 2.3
| {{monzo| 287 -181 }}
| {{monzo| 287 -181 }}
| [{{val| 248 393 }}]
| {{mapping| 248 393 }}
| +0.108
| +0.108
| 0.108
| 0.108
Line 30: Line 33:
| 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
Line 37: Line 40:
| 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
Line 44: Line 47:
| 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
Line 51: Line 54:
| 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
Line 60: Line 63:
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 102: Line 105:
| [[Birds]]
| [[Birds]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 15:17, 22 March 2024

← 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

Template:EDO intro

Theory

248 = 8 × 31, and 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.

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 it is distinct

Retrieved from "https://en.xen.wiki/index.php?title=248edo&oldid=139931"