248edo: Difference between revisions

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== Theory ==

~~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]]. 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]].

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 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.

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== Regular temperament properties ==

{~~{comma basis begin}}~~

{| 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

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| 0.275

| 5.69

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| 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

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| 498.39

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 2

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| 4/3<br />(385/384)

| [[Birds]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|248}}
+{{ED intro}}
 == Theory ==
-= 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]]. 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]].
+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.
@@ Line 14: / Line 14: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 50: / Line 59: @@
 | 0.275
 | 5.69
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 65: / Line 81: @@
 | 498.39
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 2
@@ Line 90: / Line 106: @@
 | 4/3<br />(385/384)
 | [[Birds]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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]]