248edo: Difference between revisions

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== 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 ===

=== 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" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| 287 -181 }}

| [{{~~val~~| 248 393 }}]

| {{mapping| 248 393 }}

| +0.108

| 0.108

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| 2.3.5

| 32805/32768, {{monzo| 12 32 -27 }}

| [{{~~val~~| 248 393 576 }}]

| {{mapping| 248 393 576 }}

| -0.041

| 0.228

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| 2.3.5.7

| 3136/3125, 32805/32768, 420175/419904

| [{{~~val~~| 248 393 576 696 }}]

| {{mapping| 248 393 576 696 }}

| +0.066

| 0.270

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| 2.3.5.7.11

| 441/440, 3136/3125, 8019/8000, 41503/41472

| [{{~~val~~| 248 393 576 696 858 }}]

| {{mapping| 248 393 576 696 858 }}

| +0.036

| 0.249

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| 2.3.5.7.11.13

| 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

| 0.275

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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

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| [[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|###]] ~~

[[Category:Bischismic]]

[[Category:Essence]]

~~[[Category:Quartismic]]~~

@@ Line 3: / Line 3: @@
 == Theory ==
-et 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]].
+= 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 ===
 {{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" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! 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]] (¢)
@@ Line 23: / Line 26: @@
 | 2.3
 | {{monzo| 287 -181 }}
-| [{{val| 248 393 }}]
+| {{mapping| 248 393 }}
 | +0.108
 | 0.108
@@ Line 30: / Line 33: @@
 | 2.3.5
 | 32805/32768, {{monzo| 12 32 -27 }}
-| [{{val| 248 393 576 }}]
+| {{mapping| 248 393 576 }}
 | -0.041
 | 0.228
@@ Line 37: / Line 40: @@
 | 2.3.5.7
 | 3136/3125, 32805/32768, 420175/419904
-| [{{val| 248 393 576 696 }}]
+| {{mapping| 248 393 576 696 }}
 | +0.066
 | 0.270
@@ Line 44: / Line 47: @@
 | 2.3.5.7.11
 | 441/440, 3136/3125, 8019/8000, 41503/41472
-| [{{val| 248 393 576 696 858 }}]
+| {{mapping| 248 393 576 696 858 }}
 | +0.036
 | 0.249
@@ Line 51: / Line 54: @@
 | 2.3.5.7.11.13
 | 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
 | 0.275
@@ Line 60: / Line 63: @@
 {| class="wikitable center-all left-5"
 |+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
 |-
@@ Line 102: / Line 105: @@
 | [[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:Essence]]
-[[Category:Quartismic]]