330edo: Difference between revisions

(10 intermediate revisions by 6 users not shown)

Line 1:

==Theory==

== Theory ==

330edo has a flat tendency, with its [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.

330edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.

The equal temperament tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].

It tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[~~meter comma|~~meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].

It provides a nice tuning for [[keenanismic family|keenanismic]], the rank-4 temperament that tempers out 385/384 (even better than its [[optimal patent val]] [[284edo]]), and actually a close-to-optimal tuning for 11-limit [[Canou family #semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 14641/14580.

Line 14:

Line 13:

=== Subsets and supersets ===

Since 330 factors into ~~2 × 3 × 5 × 11~~, it has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.

Since 330 factors into {{factorisation|330}}, 330edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.

==Regular temperament properties==

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

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

|-

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

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

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

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

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

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

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

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

|-

! colspan="2" | Tuning error

|2.3

|-

|{{monzo|-523 330}}

! [[TE error|Absolute]] (¢)

|{{~~val~~|330 523}}

! [[TE simple badness|Relative]] (%)

| 0.0432

|-

| 2.3

| {{monzo| -523 330 }}

| {{mapping| 330 523 }}

| +0.0432

| 0.0432

| 1.19

|-

|2.3.5

| 2.3.5

|32805/32768, {{monzo|-2 -50 35}}

| 32805/32768, {{monzo| -2 -50 35 }}

|{{~~val~~|330 523 766}}

| {{mapping| 330 523 766 }}

| 0.1521

| +0.1521

| 0.1581

| 4.35

|-

|2.3.5.7

| 2.3.5.7

|32805/32768, 250047/250000, ~~703125~~/~~702464~~

| 32805/32768, 250047/250000, 823543/819200

|{{~~val~~|330 523 766 926}}

| {{mapping| 330 523 766 926 }}

| 0.2524

| +0.2524

| 0.2212

| 6.08

Line 48:

Line 51:

=== 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 8ve

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

|1

| 1

|137\330

| 137\330

|498.182

| 498.182

|4/3

| 4/3

|[[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 3

| 137\330 (27\330)

| 498.182 (98.182)

| 4/3 (200/189)

| [[Term (temperament)|Term]]

|-

| 6

| 137\330 (27\330)

| 498.182 (98.182)

| 4/3 (200/189)

| [[Semiterm]] (330eff)

|-

|3

| 22

|137\330 (27\330)

| 137\330 (2\330)

|498.182 (98.~~182~~)

| 498.182 (7.273)

|4/3 (~~200/189~~)

| 4/3 ({{monzo| 16 -13 2 }})

|[[~~Term~~]]

| [[Major arcana]] (5-limit)

|}

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

[[Category:Canou]]

[[Category:Semicanousmic]]

[[Category:Keenanismic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|330}}
+{{ED intro}}
-==Theory==
+== Theory ==
+edo has a flat tendency, with its [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.
-edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.
+The equal temperament tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].
-It tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter comma|meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].
 It provides a nice tuning for [[keenanismic family|keenanismic]], the rank-4 temperament that tempers out 385/384 (even better than its [[optimal patent val]] [[284edo]]), and actually a close-to-optimal tuning for 11-limit [[Canou family #semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 14641/14580.
@@ Line 14: / Line 13: @@
 === Subsets and supersets ===
-Since 330 factors into 2 × 3 × 5 × 11, it has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.
+Since 330 factors into {{factorisation|330}}, 330edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.
-==Regular temperament properties==
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+|-
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Comma list]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | [[Mapping]]
-! colspan="2" |Tuning Error
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-|-
+! colspan="2" | Tuning error
-|2.3
+|-
-|{{monzo|-523 330}}
+! [[TE error|Absolute]] (¢)
-|{{val|330 523}}
+! [[TE simple badness|Relative]] (%)
-| 0.0432
+|-
+| 2.3
+| {{monzo| -523 330 }}
+| {{mapping| 330 523 }}
+| +0.0432
 | 0.0432
 | 1.19
 |-
-|2.3.5
+| 2.3.5
-|32805/32768, {{monzo|-2 -50 35}}
+| 32805/32768, {{monzo| -2 -50 35 }}
-|{{val|330 523 766}}
+| {{mapping| 330 523 766 }}
-| 0.1521
+| +0.1521
 | 0.1581
 | 4.35
 |-
-|2.3.5.7
+| 2.3.5.7
-|32805/32768, 250047/250000, 703125/702464
+| 32805/32768, 250047/250000, 823543/819200
-|{{val|330 523 766 926}}
+| {{mapping| 330 523 766 926 }}
-| 0.2524
+| +0.2524
 | 0.2212
 | 6.08
@@ Line 48: / Line 51: @@
 === 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 8ve
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|137\330
+| 137\330
-|498.182
+| 498.182
-|4/3
+| 4/3
-|[[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
+|-
+| 3
+| 137\330<br />(27\330)
+| 498.182<br />(98.182)
+| 4/3<br />(200/189)
+| [[Term (temperament)|Term]]
+|-
+| 6
+| 137\330<br />(27\330)
+| 498.182<br />(98.182)
+| 4/3<br />(200/189)
+| [[Semiterm]] (330eff)
 |-
-|3
+| 22
-|137\330<br>(27\330)
+| 137\330<br />(2\330)
-|498.182<br>(98.182)
+| 498.182<br />(7.273)
-|4/3<br>(200/189)
+| 4/3<br />({{monzo| 16 -13 2 }})
-|[[Term]]
+| [[Major arcana]] (5-limit)
 |}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 [[Category:Canou]]
 [[Category:Semicanousmic]]
 [[Category:Keenanismic]]