Valinorsmic clan: Difference between revisions

Line 13:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.~~311~~{{c}}, ~5/4 = 389.~~540~~{{c}}, ~7/4 = 971.~~553~~{{c}}

* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~5/4 = 389.~~393~~{{c}}, ~7/4 = 971.~~520~~{{c}}

: [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}

: error map: {{val| 0.000 +3.080 +2.694 -1.011 }}

{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

[[Badness]] (Sintel): 0.~~079~~

[[Badness]] (Sintel): 0.0793

=== Overview to extensions ===

The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. ~~Julius, a.k.a. varda~~, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have ~~neat~~ extensions to the 13-limit via tempering out both [[351/350]] and [[352/351]].

The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]].

Discussed elsewhere are:

Line 49:

Line 51:

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.~~360~~{{c}}, ~3/2 = 702.~~695~~{{c}}, ~5/4 = 389.~~312~~{{c}}

* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}

* [[CWE]]: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~592~~{{c}}, ~5/4 = 389.~~376~~{{c}}

: [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}

: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }}

{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}

[[Badness]] (Sintel): 0.816

Line 64:

Line 68:

Optimal tunings:

* WE: ~2 = 1199.~~435~~{{c}}, ~3/2 = 702.~~506~~{{c}}, ~5/4 = 389.~~233~~{{c}}

* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}

* CWE: ~2 = 1200.~~000~~{{c}}, ~3/2 = 702.~~703~~{{c}}, ~5/4 = 389.~~430~~{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}

{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}

Badness (Sintel): 1.~~073~~

Badness (Sintel): 1.07

== Augenic ==

Line 80:

Line 84:

[[Optimal tuning]]s:

* [[WE]]: ~5/4 = 398.~~924~~{{c}}, ~3/2 = 705.~~145~~{{c}}, ~7/4 = 969.~~111~~{{c}}

* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}

* [[CWE]]: ~5/4 = 400.~~000~~{{c}}, ~3/2 = 705.~~349~~{{c}}, ~7/4 = 968.~~440~~{{c}}

: [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }}

* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}

: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }}

{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *

<nowiki/>* [[optimal patent val]]: [[36edo|36]]

[[Badness]] (Sintel): 0.735

@@ Line 13: / Line 13: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.311{{c}}, ~5/4 = 389.540{{c}}, ~7/4 = 971.553{{c}}
+* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.393{{c}}, ~7/4 = 971.520{{c}}
+: [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
+: error map: {{val| 0.000 +3.080 +2.694 -1.011 }}
 {{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
-[[Badness]] (Sintel): 0.079
+[[Badness]] (Sintel): 0.0793
 === Overview to extensions ===
-The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Julius, a.k.a. varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have neat extensions to the 13-limit via tempering out both [[351/350]] and [[352/351]].
+The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]].
 Discussed elsewhere are:
@@ Line 49: / Line 51: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.360{{c}}, ~3/2 = 702.695{{c}}, ~5/4 = 389.312{{c}}
+* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.592{{c}}, ~5/4 = 389.376{{c}}
+: [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}
+: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }}
-{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}
+{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}
 [[Badness]] (Sintel): 0.816
@@ Line 64: / Line 68: @@
 Optimal tunings:
-* WE: ~2 = 1199.435{{c}}, ~3/2 = 702.506{{c}}, ~5/4 = 389.233{{c}}
+* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.703{{c}}, ~5/4 = 389.430{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}
-{{Optimal ET sequence|legend=0| 31, 46, 77, 80, 111 }}
+{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}
-Badness (Sintel): 1.073
+Badness (Sintel): 1.07
 == Augenic ==
@@ Line 80: / Line 84: @@
 [[Optimal tuning]]s:
-* [[WE]]: ~5/4 = 398.924{{c}}, ~3/2 = 705.145{{c}}, ~7/4 = 969.111{{c}}
+* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
-* [[CWE]]: ~5/4 = 400.000{{c}}, ~3/2 = 705.349{{c}}, ~7/4 = 968.440{{c}}
+: [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }}
+* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}
+: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }}
-{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e }}
+{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *
+<nowiki/>* [[optimal patent val]]: [[36edo|36]]
 [[Badness]] (Sintel): 0.735