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Starling family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-25 23:57:57 UTC</tt>.<br>
: The original revision id was <tt>154006001</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The head of the starling family is starling, which tempers out 126/125, the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent marvel tuning, is given by [[77edo]].  Other possible tunings are [[108edo]] and [[185edo]].


In starling, (6/5)^3 = 126/125 * 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.</pre></div>
The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Starling family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The head of the starling family is starling, which tempers out 126/125, the starling comma or septimal semicomma. Starling has a normal list basis of [2, 3, 5]; hence a 5-limit scale can be converted to starling simply by tempering it. One way to do that, and an excellent marvel tuning, is given by &lt;a class="wiki_link" href="/77edo"&gt;77edo&lt;/a&gt;. Other possible tunings are &lt;a class="wiki_link" href="/108edo"&gt;108edo&lt;/a&gt; and &lt;a class="wiki_link" href="/185edo"&gt;185edo&lt;/a&gt;. &lt;br /&gt;
== Starling ==
&lt;br /&gt;
{{Main| Starling and thrush }}
In starling, (6/5)^3 = 126/125 * 12/7, and minor thirds/major sixths are low complexity intervals. A suitable 5-limit scale to temper via starling will be one where there are chains of minor thirds. Starling has a 6/5-6/5-6/5-7/6 versions of the diminished seventh chord which is very characteristic of it. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as three stacked minor thirds and an augmented second, which is what it is in meantone, than as the modern version of four stacked very flat minor thirds.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a [[5-limit]] scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.
 
Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: [[126/125]]
 
{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5
 
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]
 
[[Minkowski lattice basis]]:
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}
 
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}
 
[[Badness]] (Sintel): 0.308
 
[[Projection pair]]: <code>7 125/18</code>
 
{{Databox|[[Minkowski blocks]]|
* 7: 25/24, 81/80
* 8: 16/15, 648/625
* 9: 27/25, 128/125
* 11: 16/15, 15625/15552
* 12: 128/125, 628/625
* 15: 128/125, 250/243
* 16: 648/625, 3125/3072
* 17: 25/24, 20480/19683
* 19: 81/80, 3125/3072
* 27: 128/125, 78732/78125
* 28: 648/625, 16875/16384
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
}}
 
=== Overview to extensions ===
Temperaments discussed elsewhere include
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]
 
Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.
 
== Thrush ==
{{Main| Starling and thrush }}
 
Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 176/175
 
{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}
 
Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]
 
Lattice basis:
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}
 
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7
 
{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}
 
[[Badness]] (Sintel): 0.424
 
[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 176/175, 196/195
 
Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}
 
Badness (Sintel): 0.633
 
=== Bluebird ===
{{Distinguish| Bluebirds }}
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}
 
Optimal tunings:  
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}
 
{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}
 
Badness (Sintel): 0.856
 
Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>
 
=== Nightingale ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 66/65, 126/125, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}
 
Optimal tunings:
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}
 
Badness (Sintel): 0.783
 
=== Veery ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 176/175
 
Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}
 
{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}
 
Badness (Sintel): 0.927
 
== Starnova ==
Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 385/384
 
{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}
 
{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}
 
[[Badness]] (Sintel): 0.813
 
[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>
 
Scales: [[Star and Nova]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 196/195, 385/384
 
Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}
 
{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}
 
Badness (Sintel): 1.07
 
== Thrasher ==
{{See also| Ptolemismic clan }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 100/99
 
{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}
 
Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]
 
Lattice basis:
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: angle (~6/5, ~5/4) = 98.8447°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9
 
{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]
 
[[Badness]] (Sintel): 0.577
 
Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 91/90, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *
 
<nowiki>*</nowiki> optimal patent val: [[34edo|34]]
 
Badness (Sintel): 0.820
 
=== Mockingbird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 40/39, 56/55, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}
 
Optimal tunings:
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}
 
Badness (Sintel): 0.804
 
=== Catbird ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 56/55, 78/77, 100/99
 
Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}
 
Optimal tunings:
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}
 
{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}
 
Badness (Sintel): 0.846
 
== Aplonis ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 540/539
 
{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}
 
{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}
 
[[Badness]] (Sintel): 0.778
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 126/125, 144/143, 196/195
 
Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}
 
Badness (Sintel): 0.768
 
== Treecreeper ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 126/125, 1232/1215
 
{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}
 
{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}
 
[[Badness]] (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 352/351
 
Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}
 
{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}
 
Badness (Sintel): 1.48
 
[[Category:Starling family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]
Retrieved from "https://en.xen.wiki/w/Starling_family"