Xenharmonic Wiki

3136/3125: Difference between revisions

← Older editNewer edit →
VisualWikitext
Improve descriptions
Improve temperament data and commenting out the latter two due to name conflicts
Line 16: Line 16:


=== Orion ===
=== Orion ===
As [[28/25]] is close to [[19/17]] and as the latter is a precise approximation of half of [[5/4]], it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]] and the [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]] which together imply tempering [[3136/3125]] and [[2128/2125]], resulting in a rank 3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea#The_Milky_Way|The Milky Way realm]].
As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[5/4]], it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]] and the [[square superparticular|semiparticular]] ([[5/4]])/([[19/17]])<sup>2</sup> = [[1445/1444]], which together imply tempering out [[3136/3125]] and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].


==== 2.5.7.17.19 ====
==== 2.5.7.17.19 ====
Comma list: 3136/3125, 476/475, 1445/1444 = S17/S19, 2128/2125
[[Subgroup]]: 2.5.7.17.19


[[Mapping]]:<br>
[[Comma list]]: 476/475, 1445/1444
[{{val| 1 2 2 4 4 }}<br>
{{val| 0 2 5 0 1 }}<br>
{{val| 0 0 0 1 1 }}]


[[CTE]] generators: 2/1, ~28/25 = 193.642, ~17/16 = 104.434
[[Mapping]]: [{{val| 1 0 -3 0 -1 }}, {{val| 0 2 5 0 1 }}, {{val| 0 0 0 1 1 }}]


[[Val]]s: {{Val list| 12, 25, 31, 37, 43, 50, 56, 68, 93}}
Mapping generators: ~2, ~56/25, ~17


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434
{{Val list|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}
[[Badness]]: 0.0150
<!-- names await consolidation
==== 2.3.5.7.17.19 ====
==== 2.3.5.7.17.19 ====
As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16 = S18 = S20 and S17 = S19, it can be considered a natural and precise extension, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion.
As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16 = S18 = S20 and S17 = S19, it can be considered a natural and precise extension, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion.


Comma list: 3136/3125, 1445/1444 = S17/S19, 1216/1215 = S18/S20
Subgroup: 2.3.5.7.17.19


[[Mapping]]:<br>
Comma list: 476/475, 1216/1215, 1445/1444
[{{val| 1 1 2 2 1 1 }}<br>
{{val| 0 1 0 0 5 5 }}<br>
{{val| 0 0 2 5 1 2 }}]


[[CTE]] generators: 2/1, ~3/2 = 702.132, ~28/25 = 193.647
Mapping: [{{val| 1 0 0 -3 -5 -6 }}, {{val| 0 1 0 0 5 5 }}, {{val| 0 0 2 5 1 2 }}]


[[Val]]s: {{Val list| 12, 87, 99, 111, 118, 130}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.132, ~28/25 = 193.647
 
Optimal GPV sequence: {{Val list| 12, …, 87, 99, 118, 210gh, 217, 229, 328h, 446 }}
 
Badness: 0.456


==== Semiorion ====
==== Semiorion ====
Line 49: Line 54:
Subgroup: 2.3.5.7.17.19
Subgroup: 2.3.5.7.17.19


Comma list: 3136/3125, 289/288 = S17, 361/360 = S19
Comma list: 289/288, 361/360, 476/475
 
Mapping: [{{val| 2 0 0 -6 5 3 }}, {{val| 0 1 0 0 1 1 }}, {{val| 0 0 2 5 0 1 }}]


[[Mapping]]:<br>
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.509, ~28/25 = 193.669
[{{val| 2 2 4 4 7 7 }}<br>
{{val| 0 1 0 0 1 1 }}<br>
{{val| 0 0 2 5 0 1 }}]


[[CTE]] generators: ~17/12 = 600.0, ~3/2 = 702.509, ~28/25 = 193.669
Optimal GPV sequence: {{Val list| 12, …, 50, 68, 106d, 118, 248g, 316g }}


[[Val]]s: {{Val list|12, 50, 56, 62, 68, 80, 118, 130}}
Badness: 0.569
-->


[[Category:Hemimean]]
[[Category:Hemimean]]
Retrieved from "https://en.xen.wiki/w/3136/3125"