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Meantone family: Difference between revisions

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* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]


==== Tridecimal meantone ====
==== Fokkertone ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 169: Line 169:
Badness (Sintel): 0.746
Badness (Sintel): 0.746


===== Meantonic =====
===== 17-limit =====
Dubbed ''meantonic'' here, this extension maps the 17/16 to the octave-reduced triple-augmented seventh (C–B𝄪♯), and 19/16 to the quadruple-augmented unison (C–C𝄪𝄪). The major second is now 19/17, and 17/16 is conflated with 19/18, as do all the other extensions discussed below. 31edo also conflates 17/16~19/18 with 16/15 whereas 50edo conflates all of 17/16, 18/17, 19/18, and 20/19, so a good tuning would be somewhere in this range.
This extension is perhaps the most practical, as it maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.  
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 66/65, 81/80, 99/98, 105/104, 121/119
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 | 0 1 4 10 18 15 26 }}
 
Optimal tunings:
* WE: ~2 = 1201.2376{{c}}, ~3/2 = 697.0954{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4563{{c}}
 
{{Optimal ET sequence|legend=0| 12fg, 19eg, 31, 50e }}
 
Badness (Sintel): 0.970
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 66/65, 77/76, 81/80, 99/98, 105/104, 121/119
 
Mapping: {{mapping| 1 0 -4 -13 -25 -20 -37 -40 | 0 1 4 10 18 15 26 28 }}
 
Optimal tunings:
* WE: ~2 = 1201.4134{{c}}, ~3/2 = 697.0933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.3526{{c}}
 
{{Optimal ET sequence|legend=0| 12fghh, 19egh, 31, 50e }}
 
Badness (Sintel): 1.09
 
===== Huygens =====
Dubbed ''huygens'' here, this extension is perhaps the most practical, as it maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.  


Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Line 218: Line 186:
Badness (Sintel): 1.02
Badness (Sintel): 1.02


====== 19-limit ======
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 309: Line 277:
Badness (Sintel): 1.09
Badness (Sintel): 1.09


===== Meridetonic =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 78/77, 81/80, 99/98, 126/125, 273/272
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 | 0 1 4 10 18 27 38 }}
 
Optimal tunings:
* WE: ~2 = 1199.9428{{c}}, ~3/2 = 697.4804{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5113{{c}}
 
{{Optimal ET sequence|legend=0| 12fg, 31fg, 43 }}
 
Badness (Sintel): 1.41
 
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 78/77, 81/80, 99/98, 126/125, 153/152, 273/272
 
Mapping: {{mapping| 1 0 -4 -13 -25 -39 -56 -59 | 0 1 4 10 18 27 38 40 }}
 
Optimal tunings:
* WE: ~2 = 1200.0089{{c}}, ~3/2 = 697.4864{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.4815{{c}}
 
{{Optimal ET sequence|legend=0| 12fghh, 31fgh, 43 }}
 
Badness (Sintel): 1.54
 
===== Sauveuric =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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Badness (Sintel): 1.22
Badness (Sintel): 1.22


====== 19-limit ======
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 581: Line 519:


Badness (Sintel): 1.25
Badness (Sintel): 1.25
==== Meanplop ====
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 78/77, 81/80, 91/90
Mapping: {{mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }}
Optimal tunings:
* WE: ~2 = 1202.3237{{c}}, ~3/2 = 697.5502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2135{{c}}
Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 11/13 0 0 0 -1/13 }}
: unchanged-interval (eigenmonzo) basis: 2.11
{{Optimal ET sequence|legend=0| 12e, 19, 31f }}
Badness (Sintel): 1.14
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 52/51, 65/64, 78/77, 81/80, 91/90
Mapping: {{mapping| 1 0 -4 -13 24 10 12 | 0 1 4 10 -13 -4 -5 }}
Optimal tunings:
* WE: ~2 = 1201.4737{{c}}, ~3/2 = 697.2690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4129{{c}}
{{Optimal ET sequence|legend=0| 12e, 19 }}
Badness (Sintel): 1.37
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 39/38, 52/51, 65/64, 77/76, 81/80, 91/90
Mapping: {{mapping| 1 0 -4 -13 24 10 12 9 | 0 1 4 10 -13 -4 -5 -3 }}
Optimal tunings:
* WE: ~2 = 1200.8839{{c}}, ~3/2 = 697.0104{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}}
{{Optimal ET sequence|legend=0| 12e, 19 }}
Badness (Sintel): 1.43


=== Meanenneadecal ===
=== Meanenneadecal ===
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Badness (Sintel): 1.36
Badness (Sintel): 1.36
==== Meanundec ====
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 40/39, 45/44, 56/55
Mapping: {{mapping| 1 0 -4 -13 -6 -1 | 0 1 4 10 6 3 }}
Optimal tunings:
* WE: ~2 = 1196.0359{{c}}, ~3/2 = 694.9504{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7474{{c}}
{{Optimal ET sequence|legend=0| 7d, 12f, 19f }}
Badness (Sintel): 1.00
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 27/26, 34/33, 40/39, 45/44, 56/55
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 | 0 1 4 10 6 3 7 }}
Optimal tunings:
* WE: ~2 = 1196.8604{{c}}, ~3/2 = 695.7613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1744{{c}}
{{Optimal ET sequence|legend=0| 7dg, 12f }}
Badness (Sintel): 1.09
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 27/26, 34/33, 40/39, 45/44, 56/55, 57/55
Mapping: {{mapping| 1 0 -4 -13 -6 -1 -7 -10 | 0 1 4 10 6 3 7 9 }}
Optimal tunings:
* WE: ~2 = 1196.9296{{c}}, ~3/2 = 696.3321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7122{{c}}
{{Optimal ET sequence|legend=0| 7dgh, 12f }}
Badness (Sintel): 1.16
=== Meanundeci ===
Meanundeci is a low-complexity low-accuracy entry that maps the 11/8 to the perfect fourth (C–F), and tridecimal meanundeci maps the 13/8 to the minor sixth (C–A♭).
Subgroup: 2.3.5.7.11
Comma list: 33/32, 55/54, 77/75
Mapping: {{mapping| 1 0 -4 -13 5 | 0 1 4 10 -1 }}
Optimal tunings:
* WE: ~2 = 1205.7146{{c}}, ~3/2 = 697.9977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.1805{{c}}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
Badness (Sintel): 1.04
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 33/32, 55/54, 65/64, 77/75
Mapping: {{mapping| 1 0 -4 -13 5 10 | 0 1 4 10 -1 -4 }}
Optimal tunings:
* WE: ~2 = 1205.5631{{c}}, ~3/2 = 697.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.0144{{c}}
{{Optimal ET sequence|legend=0| 7d, 12e, 19e }}
Badness (Sintel): 1.09


=== Bimeantone ===
=== Bimeantone ===
Line 1,143: Line 955:


Badness (Sintel): 0.727
Badness (Sintel): 0.727
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 45/44, 52/49, 64/63
Mapping: {{mapping| 1 0 -4 6 -6 10 | 0 1 4 -2 6 -4 }}
Optimal tunings:
* WE: ~2 = 1198.1958{{c}}, ~3/2 = 694.7159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.6809{{c}}
{{Optimal ET sequence|legend=0| 7, 12 }}
Badness (Sintel): 1.12
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
Mapping: {{mapping| 1 0 -4 6 -6 10 12 | 0 1 4 -2 6 -4 -5 }}
Optimal tunings:
* WE: ~2 = 1197.7959{{c}}, ~3/2 = 694.8362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.0834{{c}}
{{Optimal ET sequence|legend=0| 7, 12 }}
Badness (Sintel): 1.25
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Mapping: {{mapping| 1 0 -4 6 -6 10 12 9 | 0 1 4 -2 6 -4 -5 -3 }}
Optimal tunings:
* WE: ~2 = 1197.6198{{c}}, ~3/2 = 694.8362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2075{{c}}
{{Optimal ET sequence|legend=0| 5ef, 7, 12, 19d, 31def }}
Badness (Sintel): 1.24
==== Dominatrix ====
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 64/63
Mapping: {{mapping| 1 0 -4 6 -6 -1 | 0 1 4 -2 6 3 }}
Optimal tunings:
* WE: ~2 = 1193.1574{{c}}, ~3/2 = 694.5610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.7268{{c}}
{{Optimal ET sequence|legend=0| 5e, 7, 12f }}
Badness (Sintel): 0.756


=== Domination ===
=== Domination ===
Line 1,248: Line 1,000:


Badness (Sintel): 0.864
Badness (Sintel): 0.864
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 22/21, 27/26, 33/32, 36/35
Mapping: {{mapping| 1 0 -4 6 5 -1 | 0 1 4 -2 -1 3 }}
Optimal tunings:
* WE: ~2 = 1197.8123{{c}}, ~3/2 = 695.4727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5713{{c}}
{{Optimal ET sequence|legend=0| 5, 7 }}
Badness (Sintel): 0.963
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
Mapping: {{mapping| 1 0 -4 6 5 -1 12 | 0 1 4 -2 -1 3 -5 }}
Optimal tunings:
* WE: ~2 = 1197.6327{{c}}, ~3/2 = 695.6030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9316{{c}}
{{Optimal ET sequence|legend=0| 5, 7 }}
Badness (Sintel): 1.25
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Mapping: {{mapping| 1 0 -4 6 5 -1 12 9 | 0 1 4 -2 -1 3 -5 -3 }}
Optimal tunings:
* WE: ~2 = 1197.5295{{c}}, ~3/2 = 695.6325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0579{{c}}
{{Optimal ET sequence|legend=0| 5, 7, 12ef, 19def }}
Badness (Sintel): 1.28


== Sharptone ==
== Sharptone ==
Retrieved from "https://en.xen.wiki/w/Meantone_family"