248edo: Difference between revisions

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-The ''248 equal division'' divides the octave into 248 equal parts of 4.389 cents each. It tempers out 32805/32768 in the 5-limit; 3136/3125 and 420175/419904 in the 7-limit; 441/440 and 8019/8000 in the 11-limit; 729/728, 847/845, 1001/1000, 1575/1573 and 2200/2197 in the 13-limit. It supports [[Schismatic_family#Bischismic|bischismic temperament]], providing the [[Optimal_patent_val|optimal patent val]] for 11-limit bischismic, and excellent tunings in the 7 and 13-limits. It also provides the optimal patent val for [[Quince_clan#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.
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
+edo 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 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 ===
+{{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 ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 287 -181 }}
+| {{mapping| 248 393 }}
+| +0.108
+| 0.108
+| 2.23
+|-
+| 2.3.5
+| 32805/32768, {{monzo| 12 32 -27 }}
+| {{mapping| 248 393 576 }}
+| -0.041
+| 0.228
+| 4.70
+|-
+| 2.3.5.7
+| 3136/3125, 32805/32768, 420175/419904
+| {{mapping| 248 393 576 696 }}
+| +0.066
+| 0.270
+| 5.58
+|-
+| 2.3.5.7.11
+| 441/440, 3136/3125, 8019/8000, 41503/41472
+| {{mapping| 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
+| {{mapping| 248 393 576 696 858 918 }}
+| +0.079
+| 0.275
+| 5.69
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 5\248
+| 24.19
+| 686/675
+| [[Sengagen]]
+|-
+| 1
+| 103\248
+| 498.39
+| 4/3
+| [[Helmholtz (temperament)|Helmholtz]]
+|-
+| 2
+| 77\248<br />(47\248)
+| 372.58<br />(227.42)
+| 26/21<br />(154/135)
+| [[Essence]]
+|-
+| 2
+| 103\248
+| 498.39
+| 4/3
+| [[Bischismic]]
+|-
+| 8
+| 117\248<br />(7\248)
+| 566.13<br />(33.87)
+| 104/75<br />(49/48)
+| [[Octowerck]]
+|-
+| 31
+| 103\248<br />(1\248)
+| 498.39<br />(4.84)
+| 4/3<br />(385/384)
+| [[Birds]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+[[Category:Bischismic]]
+[[Category:Essence]]