Star and Nova

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[[toc|flat]]

=Star=
==Star transversal==
Star is an eight-note tempered scale, with something of the sound of the [[octatonic scale]]; however it is less regular and closer to just intonation. Star can be reached from various starting points, one of which is a 5-limit [[Fokker blocks|Fokker block]] which serves as a [[transversal]] for star. The block in question is [[smithgw_star]], 25/24-6/5-5/4-36/25-3/2-5/3-9/5-2/1, which consists of a chain of minor thirds 5/3-1-6/5-36/25 and that chain translated by 5/4, and which has three major and three minor triads.

==Notes of star==
By tempering out 126/125, leading to [[Starling family#x7-limit starling|7-limit starling temperament]] we add the 7-limit intervals 8/7, 7/6, 7/5 and their inversions to the mix, and by tempering by 385/384 also, leading to [[Starling family#x7-limit starling-11-limit|11-limit starling]], the 11/8 interval also. There is no significant tuning advantage to 11-limit starling, a planar temperament, over the linear temperament [[Starling temperaments#Valentine temperament-11-limit|valentine]], so we may consider star to be a scale of valentine. In valentine, with generator a tempered 21/20 of 78 cents, the notes of star are -4, 0, 1, 4, 5, 8, 9 and 13 generators; in the ordering of the scale itself, that's 0, 1, 4, 5, 8, 9, -4, 13, 0.

Since there are a finite number of 11-limit 8-note [[lesfip scales]], we may consider the [[star-lesfip|lesfip version of star]], star-lesfip, to be canonical. It can be reached from a variety of starting points. Lesfipping the 5-limit transversal in the range 14 to 17 cents of tolerance leads to star; so does lesfipping the 46et version in the range from 9 to 17 cents or the 77et version in the range 6 to 16 cents. Another tuning for star, not very different, is [[star|star in 77et]]; 77 giving the optimal patent val for both 11-limit starling and valentine.

==Chords of star==
Star has sixteen [[dyadic chord|dyadic tetrads]], of eleven different types, which are among those listed on the [[chords of valentine]] page. In terms of what's listed on that page, star has one otonal tetrad, given in the form 1-8/7-10/7-12/7 but better known as 1-5/4-3/2-7/4; one 1-6/5-3/2-12/7 utonal tetrad; and one each of the [[keenanismic chords|keenanismic tetrads]] 1-8/7-11/8-12/7, 1-6/5-11/8-12/7, 1-5/4-10/7-12/7 and 1-5/4-3/2-12/7. It also has two each of the starling tetrad, the traditional diminished seventh chord of meantone; the ambitonal [[just added sixth chord]] 1-6/5-3/2-9/5; and the three 9-limit starling tetrads 1-6/5-10/7-9/5, 1-5/4-10/7-9/5 and 1-5/4-3/2-9/5.

==Transformations==
Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard [[http://en.wikipedia.org/wiki/Cycle_notation|cycle notation]] used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an [[http://en.wikipedia.org/wiki/Elementary_abelian_group|elementary abelian 2-group]] isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory [[Graph-theoretic properties of scales#Examples-Star|here]].

=Starr=
Star has a twin, Starr, with very similar characteristics. It to can be reached from various starting points. One is  a 5-limit [[Fokker blocks|Fokker block]] [[transversal]], [[smithgw_star2]], 27/25-6/5-5/4-36/25-3/2-5/3-9/5-2/1, which consists of a chain of minor thirds 5/3-1-6/5-36/25 and that chain translated by 3/2 rather than 5/4, and which also has three major and three minor triads.

Original HTML content:

<html><head><title>Star and Nova</title></head><body><!-- ws:start:WikiTextTocRule:12:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --><a href="#Star">Star</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --><!-- ws:end:WikiTextTocRule:15 --><!-- ws:start:WikiTextTocRule:16: --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Starr">Starr</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: -->
<!-- ws:end:WikiTextTocRule:19 --><br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Star"></a><!-- ws:end:WikiTextHeadingRule:0 -->Star</h1>
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Star-Star transversal"></a><!-- ws:end:WikiTextHeadingRule:2 -->Star transversal</h2>
Star is an eight-note tempered scale, with something of the sound of the <a class="wiki_link" href="/octatonic%20scale">octatonic scale</a>; however it is less regular and closer to just intonation. Star can be reached from various starting points, one of which is a 5-limit <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> which serves as a <a class="wiki_link" href="/transversal">transversal</a> for star. The block in question is <a class="wiki_link" href="/smithgw_star">smithgw_star</a>, 25/24-6/5-5/4-36/25-3/2-5/3-9/5-2/1, which consists of a chain of minor thirds 5/3-1-6/5-36/25 and that chain translated by 5/4, and which has three major and three minor triads.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Star-Notes of star"></a><!-- ws:end:WikiTextHeadingRule:4 -->Notes of star</h2>
By tempering out 126/125, leading to <a class="wiki_link" href="/Starling%20family#x7-limit starling">7-limit starling temperament</a> we add the 7-limit intervals 8/7, 7/6, 7/5 and their inversions to the mix, and by tempering by 385/384 also, leading to <a class="wiki_link" href="/Starling%20family#x7-limit starling-11-limit">11-limit starling</a>, the 11/8 interval also. There is no significant tuning advantage to 11-limit starling, a planar temperament, over the linear temperament <a class="wiki_link" href="/Starling%20temperaments#Valentine temperament-11-limit">valentine</a>, so we may consider star to be a scale of valentine. In valentine, with generator a tempered 21/20 of 78 cents, the notes of star are -4, 0, 1, 4, 5, 8, 9 and 13 generators; in the ordering of the scale itself, that's 0, 1, 4, 5, 8, 9, -4, 13, 0.<br />
<br />
Since there are a finite number of 11-limit 8-note <a class="wiki_link" href="/lesfip%20scales">lesfip scales</a>, we may consider the <a class="wiki_link" href="/star-lesfip">lesfip version of star</a>, star-lesfip, to be canonical. It can be reached from a variety of starting points. Lesfipping the 5-limit transversal in the range 14 to 17 cents of tolerance leads to star; so does lesfipping the 46et version in the range from 9 to 17 cents or the 77et version in the range 6 to 16 cents. Another tuning for star, not very different, is <a class="wiki_link" href="/star">star in 77et</a>; 77 giving the optimal patent val for both 11-limit starling and valentine.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Star-Chords of star"></a><!-- ws:end:WikiTextHeadingRule:6 -->Chords of star</h2>
Star has sixteen <a class="wiki_link" href="/dyadic%20chord">dyadic tetrads</a>, of eleven different types, which are among those listed on the <a class="wiki_link" href="/chords%20of%20valentine">chords of valentine</a> page. In terms of what's listed on that page, star has one otonal tetrad, given in the form 1-8/7-10/7-12/7 but better known as 1-5/4-3/2-7/4; one 1-6/5-3/2-12/7 utonal tetrad; and one each of the <a class="wiki_link" href="/keenanismic%20chords">keenanismic tetrads</a> 1-8/7-11/8-12/7, 1-6/5-11/8-12/7, 1-5/4-10/7-12/7 and 1-5/4-3/2-12/7. It also has two each of the starling tetrad, the traditional diminished seventh chord of meantone; the ambitonal <a class="wiki_link" href="/just%20added%20sixth%20chord">just added sixth chord</a> 1-6/5-3/2-9/5; and the three 9-limit starling tetrads 1-6/5-10/7-9/5, 1-5/4-10/7-9/5 and 1-5/4-3/2-9/5.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Star-Transformations"></a><!-- ws:end:WikiTextHeadingRule:8 -->Transformations</h2>
Star has many permutations of its notes which send dyadic chords to other dyadic chords. We may use the standard <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Cycle_notation" rel="nofollow">cycle notation</a> used with permutation groups to mean permutations of the pitch classes of star lifted to permutations of star as a periodic scale. For instance, the cycle (01) interchanges note 0 with note 1, which also means exchanging note 8 for note 9, and in general Star[n] with Star[n+1] whenever n is divisible by 8. The four involutions (elements of order two) (01), (23), (45) and (67) all preserve the dyadic harmony character of the chords of star, while changing the actual chords. Together, they generate an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Elementary_abelian_group" rel="nofollow">elementary abelian 2-group</a> isomorphic to (Z/2Z)^4, which means fifteen nontrivial transformations. To these may be added involutions exchanging the adjacent even-odd pairs of the previous group, so that [0 2] for instance would mean, for n divisible by 8, Star[n] changes places with Star[n+2], and Star[n+1] with Star[n+3]. These involutions generate a group of infinite order, but on pitch classes we obtain a group of order 384. This is discussed from the point of view of graph theory <a class="wiki_link" href="/Graph-theoretic%20properties%20of%20scales#Examples-Star">here</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Starr"></a><!-- ws:end:WikiTextHeadingRule:10 -->Starr</h1>
Star has a twin, Starr, with very similar characteristics. It to can be reached from various starting points. One is  a 5-limit <a class="wiki_link" href="/Fokker%20blocks">Fokker block</a> <a class="wiki_link" href="/transversal">transversal</a>, <a class="wiki_link" href="/smithgw_star2">smithgw_star2</a>, 27/25-6/5-5/4-36/25-3/2-5/3-9/5-2/1, which consists of a chain of minor thirds 5/3-1-6/5-36/25 and that chain translated by 3/2 rather than 5/4, and which also has three major and three minor triads.</body></html>