301edo: Difference between revisions

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~~'''301edo''' is the [[EDO|equal division of the octave]] into 301 parts of 3.98671 [[cent]]s each.~~

301edo is a strong 7-limit system, and distinctly consistent through the [[17-odd-limit]]. It tempers out [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], 5632/5625, [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. ~~Because~~ it tempers out both 32805/32768 and 2401/2400, it ~~supports~~ the [[sesquiquartififths]] temperament.

== Theory ==

301edo is a strong 7-limit system, and distinctly [[consistent]] through the [[17-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], [[5632/5625]], [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it [[support]]s the [[sesquiquartififths]] temperament.

301 ~~is a composite number~~, ~~since 301 = 7 × 43~~. This is related to the proposal of the deaf French mathematician and acoustician ~~[[Wikipedia: Joseph Sauveur~~|Joseph Sauveur]] to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 301 factors into {{factorisation|301}}, 301edo has [[7edo]] and [[43edo]] as its subsets. This is related to the proposal of the deaf French mathematician and acoustician {{w|Joseph Sauveur}} to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.

301edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]].

~~=== Prime harmonics ===~~

~~{{Primes in edo|edo=301|columns=11|prec=3}}~~

== 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| -477 301 }}

| [{{~~val~~| 301 477 }}]

| {{mapping| 301 477 }}

| +0.0927

| 0.0927

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| 2.3.5

| 32805/32768, {{monzo| 3 45 -32 }}

| [{{~~val~~| 301 477 699 }}]

| {{mapping| 301 477 699 }}

| +0.0048

| 0.1456

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| 2.3.5.7

| 2401/2400, 32805/32768, 1959552/1953125

| [{{~~val~~| 301 477 699 845 }}]

| {{mapping| 301 477 699 845 }}

| +0.0085

| 0.1262

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 5632/5625, 8019/8000

| [{{~~val~~| 301 477 699 845 1041 }}]

| {{mapping| 301 477 699 845 1041 }}

| +0.0734

| 0.1720

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| 2.3.5.7.11.13

| 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024

| [{{~~val~~| 301 477 699 845 1041 1114 }}]

| {{mapping| 301 477 699 845 1041 1114 }}

| +0.0310

| 0.1834

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| 2.3.5.7.11.13.17

| 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088

| [{{~~val~~| 301 477 699 845 1041 1114 1230 }}]

| {{mapping| 301 477 699 845 1041 1114 1230 }}

| +0.0721

| 0.1973

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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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| 25\301

| 99.67

| ~~200~~/~~189~~

| 18/17

| [[Quintaschis]]

|-

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| 498.34

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 7

| 125\301 (4\301)

| 125\301 (4\301)

| 498.34 (15.95)

| 498.34 (15.95)

| 4/3 (245/243)

| 4/3 (245/243)

| [[Septant]]

|-

| 43

| 125\301 (1\301)

| 125\301 (1\301)

| 498.34 (3.99)

| 498.34 (3.99)

| 4/3 (540/539)

| 4/3 (540/539)

| [[Meridic]]

|}

<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:Equal divisions of the octave]]~~

[[Category:Meridic]]

@@ Line 1: / Line 1: @@
-'''301edo''' is the [[EDO|equal division of the octave]] into 301 parts of 3.98671 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
-edo is a strong 7-limit system, and distinctly consistent through the [[17-odd-limit]]. It tempers out [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], 5632/5625, [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Because it tempers out both 32805/32768 and 2401/2400, it supports the [[sesquiquartififths]] temperament.
+== Theory ==
+edo is a strong 7-limit system, and distinctly [[consistent]] through the [[17-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], [[5632/5625]], [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it [[support]]s the [[sesquiquartififths]] temperament.
-is a composite number, since 301 = 7 × 43. This is related to the proposal of the deaf French mathematician and acoustician [[Wikipedia: Joseph Sauveur|Joseph Sauveur]] to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.
+=== Prime harmonics ===
+{{Harmonics in equal|301}}
+=== Subsets and supersets ===
+Since 301 factors into {{factorisation|301}}, 301edo has [[7edo]] and [[43edo]] as its subsets. This is related to the proposal of the deaf French mathematician and acoustician {{w|Joseph Sauveur}} to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.
 edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]].
-=== Prime harmonics ===
-{{Primes in edo|edo=301|columns=11|prec=3}}
 == 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 23: / Line 27: @@
 | 2.3
 | {{monzo| -477 301 }}
-| [{{val| 301 477 }}]
+| {{mapping| 301 477 }}
 | +0.0927
 | 0.0927
@@ Line 30: / Line 34: @@
 | 2.3.5
 | 32805/32768, {{monzo| 3 45 -32 }}
-| [{{val| 301 477 699 }}]
+| {{mapping| 301 477 699 }}
 | +0.0048
 | 0.1456
@@ Line 37: / Line 41: @@
 | 2.3.5.7
 | 2401/2400, 32805/32768, 1959552/1953125
-| [{{val| 301 477 699 845 }}]
+| {{mapping| 301 477 699 845 }}
 | +0.0085
 | 0.1262
@@ Line 44: / Line 48: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 5632/5625, 8019/8000
-| [{{val| 301 477 699 845 1041 }}]
+| {{mapping| 301 477 699 845 1041 }}
 | +0.0734
 | 0.1720
@@ Line 51: / Line 55: @@
 | 2.3.5.7.11.13
 | 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024
-| [{{val| 301 477 699 845 1041 1114 }}]
+| {{mapping| 301 477 699 845 1041 1114 }}
 | +0.0310
 | 0.1834
@@ Line 58: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088
-| [{{val| 301 477 699 845 1041 1114 1230 }}]
+| {{mapping| 301 477 699 845 1041 1114 1230 }}
 | +0.0721
 | 0.1973
@@ Line 66: / Line 70: @@
 === 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 76: / Line 81: @@
 | 25\301
 | 99.67
-| 200/189
+| 18/17
 | [[Quintaschis]]
 |-
@@ Line 101: / Line 106: @@
 | 498.34
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 7
-| 125\301<br>(4\301)
+| 125\301<br />(4\301)
-| 498.34<br>(15.95)
+| 498.34<br />(15.95)
-| 4/3<br>(245/243)
+| 4/3<br />(245/243)
 | [[Septant]]
 |-
 | 43
-| 125\301<br>(1\301)
+| 125\301<br />(1\301)
-| 498.34<br>(3.99)
+| 498.34<br />(3.99)
-| 4/3<br>(540/539)
+| 4/3<br />(540/539)
 | [[Meridic]]
 |}
+<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:Equal divisions of the octave]]
 [[Category:Meridic]]