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]].

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 [[~~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" | [[Mapping]]

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

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

! colspan="2" | Tuning error

|-

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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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=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

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| 498.39

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 2

| 77\248 (47\248)

| 77\248 (47\248)

| 372.58 (227.42)

| 372.58 (227.42)

| 26/21 (154/135)

| 26/21 (154/135)

| [[Essence]]

|-

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|-

| 8

| 117\248 (7\248)

| 117\248 (7\248)

| 566.13 (33.87)

| 566.13 (33.87)

| 104/75 (49/48)

| 104/75 (49/48)

| [[Octowerck]]

|-

| 31

| 103\248 (1\248)

| 103\248 (1\248)

| 498.39 (4.84)

| 498.39 (4.84)

| 4/3 (385/384)

| 4/3 (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:Equal divisions of the octave]]~~

[[Category:Bischismic]]

[[Category:Essence]]

~~[[Category:Quartismic]]~~

@@ Line 1: / Line 1: @@
-{{EDO intro|248}}
+{{Infobox ET}}
+{{ED intro}}
 == 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]].
+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 [[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" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 22: / Line 27: @@
 | 2.3
 | {{monzo| 287 -181 }}
-| [{{val| 248 393 }}]
+| {{mapping| 248 393 }}
 | +0.108
 | 0.108
@@ Line 29: / Line 34: @@
 | 2.3.5
 | 32805/32768, {{monzo| 12 32 -27 }}
-| [{{val| 248 393 576 }}]
+| {{mapping| 248 393 576 }}
 | -0.041
 | 0.228
@@ Line 36: / Line 41: @@
 | 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 43: / Line 48: @@
 | 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 50: / Line 55: @@
 | 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 58: / Line 63: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 75: / Line 81: @@
 | 498.39
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 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]]
 |-
@@ Line 90: / Line 96: @@
 |-
 | 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]]
 |-
 | 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]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave]]
 [[Category:Bischismic]]
 [[Category:Essence]]
-[[Category:Quartismic]]