248edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| Line 58: | Line 50: | ||
| 0.275 | | 0.275 | ||
| 5.69 | | 5.69 | ||
{{comma basis end}} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 82: | Line 68: | ||
|- | |- | ||
| 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]] | ||
|- | |- | ||
| Line 94: | Line 80: | ||
|- | |- | ||
| 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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
[[Category:Bischismic]] | [[Category:Bischismic]] | ||
[[Category:Essence]] | [[Category:Essence]] | ||
Revision as of 04:29, 16 November 2024
| ← 247edo | 248edo | 249edo → |
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. 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.
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
| 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
Template:Comma basis begin |- | 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 Template:Comma basis end
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf