Quintile family: Difference between revisions

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The pental family tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}.
The pental family tempers out the pental comma, 847288609443/838860800000 = {{Monzo|-28 25 -5}}.


== Pental ==
== Pental ==
Subgroup: 2.3.5
 
[[Comma]]: 847288609443/838860800000
[[Comma]]: 847288609443/838860800000


Line 12: Line 14:
[[Badness]]: 0.240050
[[Badness]]: 0.240050


=== 7-limit ===
=== 7-limit ===
The pental temperament can be described as 5&60 temperament (''pentacloud'', named by [[User:Xenllium|Xenllium]]), tempering out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.
The pental temperament can be described as 5&60 temperament (also known as ''pentacloud''), tempering out the [[cloudy comma]], 16807/16384 and the [[245/243|sensamagic comma]], 245/243 in the 7-limit.
 
Subgroup: 2.3.5.7


[[Comma list]]: 245/243, 16807/16384
[[Comma list]]: 245/243, 16807/16384
Line 25: Line 29:
[[Badness]]: 0.120942
[[Badness]]: 0.120942


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 3087/3025
Comma list: 245/243, 385/384, 3087/3025


Line 36: Line 42:
Badness: 0.093248
Badness: 0.093248


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 245/243, 3087/3025
Comma list: 105/104, 144/143, 245/243, 3087/3025


Line 47: Line 55:
Badness: 0.067549
Badness: 0.067549


== Hemipental ==
== Hemipental ==
The ''hemipental'' temperament (125&130, named by [[User:Xenllium|Xenllium]]) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[Hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 (laquadzo-agu).
The ''hemipental'' temperament (125&130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 ([[Hewuermera temperaments|hewuermera]], satribiru-agu) and 5250987/5242880 (laquadzo-agu).
 
Subgroup: 2.3.5.7


[[Comma list]]: 19683/19600, 589824/588245
[[Comma list]]: 19683/19600, 589824/588245
Line 60: Line 70:
[[Badness]]: 0.104163
[[Badness]]: 0.104163


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 8019/8000, 180224/180075
Comma list: 540/539, 8019/8000, 180224/180075


Line 71: Line 83:
Badness: 0.047624
Badness: 0.047624


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000
Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000


Line 82: Line 96:
Badness: 0.041043
Badness: 0.041043


== Decal ==
== Decal ==
The decal temperament (130&190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the [[15/14ths equal temperament|linus comma]], {{Monzo|11 -10 -10 10}}.
The decal temperament (130&190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the [[15/14ths equal temperament|linus comma]], {{Monzo|11 -10 -10 10}}.
Subgroup: 2.3.5.7


[[Comma list]]: 235298/234375, 321489/320000
[[Comma list]]: 235298/234375, 321489/320000
Line 95: Line 111:
[[Badness]]: 0.104859
[[Badness]]: 0.104859


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 8019/8000, 234375/234256
Comma list: 441/440, 8019/8000, 234375/234256


Line 106: Line 124:
Badness: 0.040633
Badness: 0.040633


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 441/440, 729/728, 1001/1000, 4225/4224
Comma list: 441/440, 729/728, 1001/1000, 4225/4224


Line 119: Line 139:
[[Category:Theory]]
[[Category:Theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Pental]]

Revision as of 04:14, 5 June 2021

The pental family tempers out the pental comma, 847288609443/838860800000 = [-28 25 -5⟩.

Pental

Subgroup: 2.3.5

Comma: 847288609443/838860800000

Mapping: [⟨5 0 -28], ⟨0 1 5]]

POTE generator: ~3/2 = 701.210

Optimal ET sequence: 5, 60, 65, 190, 255, 575, 830b, 1405b

Badness: 0.240050

7-limit

The pental temperament can be described as 5&60 temperament (also known as pentacloud), tempering out the cloudy comma, 16807/16384 and the sensamagic comma, 245/243 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 245/243, 16807/16384

Mapping: [⟨5 0 -28 14], ⟨0 1 5 0]]

POTE generator: ~3/2 = 700.548

Template:Val list

Badness: 0.120942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3087/3025

Mapping: [⟨5 0 -28 14 49], ⟨0 1 5 0 -4]]

POTE generator: ~3/2 = 701.377

Vals: Template:Val list

Badness: 0.093248

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 245/243, 3087/3025

Mapping: [⟨5 0 -28 14 49 -29], ⟨0 1 5 0 -4 6]]

POTE generator: ~3/2 = 700.996

Vals: Template:Val list

Badness: 0.067549

Hemipental

The hemipental temperament (125&130) tempers out the cataharry comma, 19683/19600 in the 7-limit, as well as 589824/588245 (hewuermera, satribiru-agu) and 5250987/5242880 (laquadzo-agu).

Subgroup: 2.3.5.7

Comma list: 19683/19600, 589824/588245

Mapping: [⟨5 8 12 14], ⟨0 -2 -10 1]]

POTE generator: ~8/7 = 230.654

Template:Val list

Badness: 0.104163

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000, 180224/180075

Mapping: [⟨5 8 12 14 18], ⟨0 -2 -10 1 -18]]

POTE generator: ~8/7 = 230.634

Vals: Template:Val list

Badness: 0.047624

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1575/1573, 4096/4095, 8019/8000

Mapping: [⟨5 8 12 14 18 18], ⟨0 -2 -10 1 -18 13]]

POTE generator: ~8/7 = 230.668

Vals: Template:Val list

Badness: 0.041043

Decal

The decal temperament (130&190) tempers out the varunisma, 321489/320000 in the 7-limit, as well as 235298/234375 (triwellisma, tribizo-asepgu), 2460375/2458624 (sesquiquartisma, laquadru-atriyo), and the linus comma, [11 -10 -10 10⟩.

Subgroup: 2.3.5.7

Comma list: 235298/234375, 321489/320000

Mapping: [⟨10 0 -56 -67], ⟨0 1 5 6]]

POTE generator: ~3/2 = 701.303

Template:Val list

Badness: 0.104859

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 234375/234256

Mapping: [⟨10 0 -56 -67 -108], ⟨0 1 5 6 9]]

POTE generator: ~3/2 = 701.240

Vals: Template:Val list

Badness: 0.040633

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 4225/4224

Mapping: [⟨10 0 -56 -67 -108 37], ⟨0 1 5 6 9 0]]

POTE generator: ~3/2 = 701.252

Vals: Template:Val list

Badness: 0.023948

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