Gallery of omnitetrachordal scales
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[[toc]]
An (incomplete) list of [[Omnitetrachordality|omnitetrachordal]] (OTC) scales.
----
=Scales with two step sizes=
(not necessarily [[MOS]] or [[Distributional Evenness|DE]])
These are mostly tempered (irrational) scales. For a few patterns (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit [[Just intonation|JI]]) version with only two step sizes is possible.
In some cases, multiple OTC patterns with the same number of large and small steps exist.
Follow the links for more detailed info on each scale! //(Some links may not be working yet...)//
==3 tones==
[[OTC 2L ns|2L+s]] (the simplest possible OTC scale; s=9/8, L=4/3)
==4 tones==
(no OTC scales possible)
==5 tones==
[[OTC 2L 3s|2L+3s]] - sLsLs (MOS)
[[OTC 3L 2s|3L+2s]] - LsLsL (MOS)
==6 tones==
[[OTC 2L ns|2L+4s]] - LsssLs
==7 tones==
[[OTC 2L 5s|2L+5s]] - sLsssLs (MOS)
[[OTC 5L 2s|5L+2s]] - LsLLLsL (MOS)
==8 tones==
[[OTC 2L ns|2L+6s]] - LssssLss
==9 tones==
[[OTC 2L ns|2L+7s]] - LsssssLss
==10 tones==
[[OTC 2L 8s|2L+8s]] - LsssssLsss
[[OTC 3L 7s|3L+7s]] - LsssLsLsss
[[OTC 5L 5s|5L+5s]] - LsLsLsLsLs (MOS)
[[OTC 7L 3s|7L+3s]] - sLLLsLLLsL
[[OTC 8L 2s|8L+2s]] - LLsLLLsLLL
==11 tones==
[[OTC 2L ns|2L+9s]] - LssssssLsss
==12 tones==
[[OTC 2L 10s|2L+10s]] - LsssLsssssss (4+4+4), sssLssssLsss (5+2+5)
[[OTC 5L 7s|5L+7s]] - ssLsLssLsLsL (MOS)
[[OTC 7L 5s|7L+5s]] - LLsLsLLsLsLs (MOS)
[[OTC 10L 2s|10L+2s]] - LLLsLLLLsLLL
==13 tones==
[[OTC 2L ns|2L+11s]] - LsssssssLssss
[[OTC 7L 6s|7L+6s]] - sLLLssLssLLLs
==14 tones==
[[OTC 2L ns|2L+12s]] - LssssssssLssss
[[OTC 5L 9s|5L+9s]] - sssLsLsssLsLsL
[[OTC 7L 7s|7L+7s]] - LsLsLsLsLsLsLs (MOS)
[[OTC 12L 2s|12L+2s]] - LLLLLLLsLLLLLs
==15 tones==
[[OTC 2L ns|2L+13s]] - LssssssssLsssss
[[OTC 7L 8s|7L+8s]] - LsLLssLsLLssLss, LssLLsLssLLsLss
[[OTC 12L 3s|12L+3s]] - LsLLLLLsLLLLLsL
==16 tones==
[[OTC 2L ns|2L+14s]] - LsssssssssLsssss
==17 tones==
[[OTC 2L ns|2L+15s]] - LsssssssssLssssss (7+3+7), LssssssssssLsssss (6+5+6)
[[OTC 5L 12s|5L+12s]] - sLsssLssLssLsssLs (MOS)
[[OTC 7L 10s|7L+10s]] - sLsLsLssLssLsLsLs, sLssLLssLssLLssLs
[[OTC 10L 7s|10L+7s]] - LsLsLsLLsLLsLsLsL, LsLLssLLsLLssLLsL
[[OTC 12L 5s|12L+5s]] - LsLLLsLLsLLsLLLsL (MOS)
==18 tones==
[[OTC 2L ns|2L+16s]] - LssssssssssLssssss
[[OTC 7L 11s|7L+11s]] - LLsssLsLLsssLsssLs, LsssLLsLsssLLsLsss
==19 tones==
[[OTC 2L ns|2L+17s]] - LsssssssssssLssssss
[[OTC 5L 14s|5L+14s]] - sLssssLssLssLssssLs
[[OTC 7L 12s|7L+12s]] - sLssLsLssLssLsLssLs (MOS)
[[OTC 10L 9s|10L+9s]] - LsLLsssLLsLLsssLLsL
[[OTC 12L 7s|12L+7s]] - LsLLsLsLLsLLsLsLLsL (MOS)
[[OTC 14L 5s|14L+5s]] - LLLLsLLsLLLLsLLsLLs
[[OTC 17L 2s|17L+2s]] - LLLLLsLLLLLLLsLLLLL
==20 tones==
[[OTC 2L ns|2L+18s]] - LsssssssssssLsssssss
[[OTC 7L 13s|7L+13s]] - sssLsssLssLLsssLssLL, sssLsssLsLsLsssLsLsL, sssLsssLLssLsssLLssL
[[OTC 12L 8s|12L+8s]] - sLLssLLsLLLssLLsLLLs, sLLssLLLsLLssLLLsLLs, sLsLsLLLsLsLsLLLsLsL
==21 tones==
[[OTC 2L ns|2L+19s]] - LssssssssssssLsssssss
[[OTC 5L 16s|5L+16s]] - sLsssssLssLssLsssssLs
[[OTC 7L 14s|7L+14s]] - ssssLssssLsLLssssLsLL, ssssLssssLLsLssssLLsL
==22 tones==
[[OTC 2L ns|2L+20s]] - LssssssssssssLssssssss (9+4+9), LsssssssssssssLsssssss (8+6+8)
[[OTC 5L 17s|5L+17s]] - ssssLsssLssssLsssLsssL (MOS)
[[OTC 7L 15s|7L+15s]] - LssLsLsssLssLsLsssLsss, LsssLsLssLsssLsLssLsss, LsssLLsssLsssLLsssLsss
[[OTC 10L 12s|10L+12s]] - LsLsLsLsLsLssLsLsLsLss
[[OTC 12L 10s|12L+10s]] - sLsLsLsLsLsLLsLsLsLsLL, sLLsLsLLssLLssLLsLsLLs, sLLssLLLssLLssLLLssLLs
[[OTC 15L 7s|15L+7s]] - sLLLssLLLsLLLssLLLsLLL
[[OTC 17L 5s|17L+5s]] - sLLLsLLLsLLLLsLLLsLLLL (MOS)
==23 tones==
[[OTC 2L ns|2L+21s]] - LsssssssssssssLssssssss
[[OTC 7L 16s|7L+16s]] - ssssLssssLssLLssssLssLL, ssLsLsLssssLssssLsLsLss, ssssLssssLLssLssssLLssL
==24 tones==
[[OTC 2L ns|2L+22s]] - LssssssssssssssLssssssss
[[OTC 5L 19s|5L+19s]] - sssssLsssLsssssLsssLsssL
[[OTC 7L 17s|7L+17s]] - sssLsssLssLssLsssLssLssL, sssLsssLsssLsLsssLsssLsL
[[OTC 10L 14s|10L+14s]] - LssLLssssLLssLLssssLLssL, sssLsLsLsLsssLsLsLsLsLsL
[[OTC 12L 12s|12L+12s]] - LsLsLsLsLsLsLsLsLsLsLsLs (MOS), ssLsLLssLLssLsLLssLLssLL, ssLLsLssLLssLLsLssLLssLL
[[OTC 14L 10s|14L+10s]] - LssLLLLssLLssLLssLLLLssL, LLsLsLsLsLsLsLLLsLsLsLsL
[[OTC 15L 9s|15L+9s]] - LLsssLLLsLLLsLLLsssLLLsL
[[OTC 17L 7s|17L+7s]] - LLLsLLsLLsLLLsLLsLLsLLLs, LLLsLsLLLsLLLsLLLsLsLLLs
[[OTC 19L 5s|19L+5s]] - LLLsLLLLLsLLLsLLLLLsLLLs
==>24 tones==
Many larger OTC L+s scales are believed to exist, but due to exponentially increasing computation time, these have not yet been studied in detail. Scales of the form [[OTC 2L ns|2L+ns]] are known to exist up to size 53, and assumed to exist for any larger size.
==L/s matrix==
||= ||= **1** ||= **2** ||= **3** ||= **4** ||= **5** ||= **6** ||= **7** ||= **8** ||= **9** ||= **10** ||= **11** ||= **12** ||= **13** ||= **14** ||= **15** ||= **16** ||= **17** ||= **18** ||= **19** ||= **20** ||= **21** ||= **22** ||= **23** ||= **24** ||= **L** ||
||= **1** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **2** ||= ||= ||= [[OTC 2L 3s|x]] ||= ||= [[OTC 2L 5s|x]] ||= ||= ||= [[OTC 2L 8s|x]] ||= ||= [[OTC 2L 10s|x]] ||= ||= [[OTC 2L 12s|x]] ||= ||= ||= ||= ||= [[OTC 2L 17s|x]] ||= ||= ||= ||= ||= [[OTC 2L 22s|x]] ||= ||= ||= ||
||= **3** ||= ||= [[OTC 2L 3s|x]] ||= ||= ||= ||= ||= [[OTC 7L 3s|x]] ||= ||= ||= ||= ||= [[OTC 12L 3s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **4** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **5** ||= ||= [[OTC 2L 5s|x]] ||= ||= ||= [[OTC 5L 5s|x]] ||= ||= [[OTC 7L 5s|x]] ||= ||= ||= ||= ||= [[OTC 12L 5s|x]] ||= ||= [[OTC 14L 5s|x]] ||= ||= ||= [[OTC 17L 5s|x]] ||= ||= [[OTC 19L 5s|x]] ||= ||= ||= ||= ||= ||= ||
||= **6** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= [[OTC 7L 6s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **7** ||= ||= [[OTC 2L ns|x]] ||= [[OTC 3L 7s|x]] ||= ||= [[OTC 5L 7s|x]] ||= ||= [[OTC 7L 7s|x]] ||= ||= ||= [[OTC 10L 7s|x]] ||= ||= [[OTC 12L 7s|x]] ||= ||= ||= [[OTC 15L 7s|x]] ||= ||= [[OTC 17L 7s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **8** ||= ||= [[OTC 2L 8s|x]] ||= ||= ||= ||= ||= [[OTC 7L 8s|x]] ||= ||= ||= ||= ||= [[OTC 12L 8s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **9** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 9s|x]] ||= ||= ||= ||= ||= [[OTC 10L 9s|x]] ||= ||= ||= ||= ||= [[OTC 15L 9s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **10** ||= ||= [[OTC 2L 10s|x]] ||= ||= ||= ||= ||= [[OTC 7L 10s|x]] ||= ||= ||= ||= ||= [[OTC 12L 10s|x]] ||= ||= [[OTC 14L 10s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **11** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= [[OTC 7L 11s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **12** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 12s|x]] ||= ||= [[OTC 7L 12s|x]] ||= ||= ||= [[OTC 10L 12s|x]] ||= ||= [[OTC 12L 12s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **13** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= [[OTC 7L 13s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **14** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 14s|x]] ||= ||= [[OTC 7L 14s|x]] ||= ||= ||= [[OTC 10L 14s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **15** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= [[OTC 7L 15s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **16** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 16s|x]] ||= ||= [[OTC 7L 16s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **17** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 17s|x]] ||= ||= [[OTC 7L 17s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **18** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **19** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= [[OTC 5L 19s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **20** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **21** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **22** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **23** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **24** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||
||= **s** ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||= {{ }} ||
----
=Scales with 3 step sizes=
* [[OTC 22 sruti|22-tone Indian 'sruti' scale]]
----
=Definitions and formulas=
[[#dual]]**"Dual"** refers to the "inverse" of a L+s scale pattern, where every L is replaced by s, and vice versa. For example, {{ sLssL }} and {{ LsLLs }} are duals. If a scale is OTC, its dual is often OTC as well, but not always!
[[#perfect]]**"Perfect"** means that values for L and s exist such that L > s and that every mode of the scale will contain a perfect (just) 3/2 or 4/3 (or both). (See also [[Eigenmonzo subgroup]].)
In this case the value **P** is given, where P = L/s. For a perfect scale, P > 1. Note that if a scale "a" is perfect (P<span style="vertical-align: sub;">a</span> ``=`` L/s), its dual "b" will have the value P<span style="vertical-align: sub;">b</span> ``=`` s/L ``=`` 1/P<span style="vertical-align: sub;">a</span>, and therefore must be imperfect (if P<span style="vertical-align: sub;">a</span> > 1, then P<span style="vertical-align: sub;">b</span> < 1 ).
In some cases, P may be less than zero. I'm not yet sure what this means :)
The value of P is calculated as follows:
a ``=`` the number of L steps per 2/1
b ``=`` the number of s steps per 2/1
c ``=`` the number of L steps per 4/3
d ``=`` the number of s steps per 4/3
x ``=`` log<span style="vertical-align: sub;">2</span>(4/3) ``=`` ~0.41504 octaves ``=`` ~498.045 cents
aL+bs ``=`` log<span style="vertical-align: sub;">2</span>(2/1) = 1
cL+ds ``=`` x
s ``=`` (c-ax)/(bc-ad)
L ``=`` (x-ds)/c
P ``=`` L/s
Note that the same procedure could be used to calculate the L/s ratio necessary to give any other just interval, such as 5/4, 11/8, etc.
[[#Q]]**Q** is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.
Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P ``=`` L/s ``=`` 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q ``=`` L/s ``=`` 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:
[[image:P_Q_C_LsssssLsss.png]]
For some scales, Q will not exist. For others, a second Q may exist that is less than P, placing a lower bound on the L/s ratio as well as an upper one.
We can also consider a point C, where the number of steps "crosses" the just 4/3 entirely - in the example above, this corresponds to 4/3 ``=`` L+2s. Q is then halfway between P and C, i.e.
C ``=`` 2Q-P
Q ``=`` (C+P)/2
P ``=`` 2Q-C
...which leads to the curious result that, although P is undefined when calculated by the normal method (due to division by zero) for a scale such as blackwood[10] (LsLsLsLsLs), C and Q can be calculated, and thus a value of P ``=`` -1 can be found anyway, even though it seems not to be of any use :)
**"L/s range"**: For any L+s scale pattern, the ratio L/s may range from 1 (L``=``s, in which case the scale is (L+s)edo ) to ~infinity (s``=``0, in which case the scale is (L)edo ). A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.
Original HTML content:
<html><head><title>Gallery of omnitetrachordal scales</title></head><body><!-- ws:start:WikiTextTocRule:80:<img id="wikitext@@toc@@normal" class="WikiMedia WikiMediaToc" title="Table of Contents" src="/site/embedthumbnail/toc/normal?w=225&h=100"/> --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 1em;"><a href="#Scales with two step sizes">Scales with two step sizes</a></div>
<!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-3 tones">3 tones</a></div>
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<!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes->24 tones">>24 tones</a></div>
<!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-L/s matrix">L/s matrix</a></div>
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An (incomplete) list of <a class="wiki_link" href="/Omnitetrachordality">omnitetrachordal</a> (OTC) scales.<br />
<br />
<hr />
<br />
<!-- ws:start:WikiTextHeadingRule:26:<h1> --><h1 id="toc0"><a name="Scales with two step sizes"></a><!-- ws:end:WikiTextHeadingRule:26 -->Scales with two step sizes</h1>
(not necessarily <a class="wiki_link" href="/MOS">MOS</a> or <a class="wiki_link" href="/Distributional%20Evenness">DE</a>)<br />
These are mostly tempered (irrational) scales. For a few patterns (usually where the generator is close to a perfect 4/3 or 3/2), a Pythagorean (3-limit <a class="wiki_link" href="/Just%20intonation">JI</a>) version with only two step sizes is possible.<br />
<br />
In some cases, multiple OTC patterns with the same number of large and small steps exist. <br />
<br />
Follow the links for more detailed info on each scale! <em>(Some links may not be working yet...)</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:<h2> --><h2 id="toc1"><a name="Scales with two step sizes-3 tones"></a><!-- ws:end:WikiTextHeadingRule:28 -->3 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+s</a> (the simplest possible OTC scale; s=9/8, L=4/3)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:<h2> --><h2 id="toc2"><a name="Scales with two step sizes-4 tones"></a><!-- ws:end:WikiTextHeadingRule:30 -->4 tones</h2>
(no OTC scales possible)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:<h2> --><h2 id="toc3"><a name="Scales with two step sizes-5 tones"></a><!-- ws:end:WikiTextHeadingRule:32 -->5 tones</h2>
<a class="wiki_link" href="/OTC%202L%203s">2L+3s</a> - sLsLs (MOS)<br />
<a class="wiki_link" href="/OTC%203L%202s">3L+2s</a> - LsLsL (MOS)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:<h2> --><h2 id="toc4"><a name="Scales with two step sizes-6 tones"></a><!-- ws:end:WikiTextHeadingRule:34 -->6 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+4s</a> - LsssLs<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:<h2> --><h2 id="toc5"><a name="Scales with two step sizes-7 tones"></a><!-- ws:end:WikiTextHeadingRule:36 -->7 tones</h2>
<a class="wiki_link" href="/OTC%202L%205s">2L+5s</a> - sLsssLs (MOS)<br />
<a class="wiki_link" href="/OTC%205L%202s">5L+2s</a> - LsLLLsL (MOS)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:<h2> --><h2 id="toc6"><a name="Scales with two step sizes-8 tones"></a><!-- ws:end:WikiTextHeadingRule:38 -->8 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+6s</a> - LssssLss<br />
<br />
<!-- ws:start:WikiTextHeadingRule:40:<h2> --><h2 id="toc7"><a name="Scales with two step sizes-9 tones"></a><!-- ws:end:WikiTextHeadingRule:40 -->9 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+7s</a> - LsssssLss<br />
<br />
<!-- ws:start:WikiTextHeadingRule:42:<h2> --><h2 id="toc8"><a name="Scales with two step sizes-10 tones"></a><!-- ws:end:WikiTextHeadingRule:42 -->10 tones</h2>
<a class="wiki_link" href="/OTC%202L%208s">2L+8s</a> - LsssssLsss<br />
<a class="wiki_link" href="/OTC%203L%207s">3L+7s</a> - LsssLsLsss<br />
<a class="wiki_link" href="/OTC%205L%205s">5L+5s</a> - LsLsLsLsLs (MOS)<br />
<a class="wiki_link" href="/OTC%207L%203s">7L+3s</a> - sLLLsLLLsL<br />
<a class="wiki_link" href="/OTC%208L%202s">8L+2s</a> - LLsLLLsLLL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:44:<h2> --><h2 id="toc9"><a name="Scales with two step sizes-11 tones"></a><!-- ws:end:WikiTextHeadingRule:44 -->11 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+9s</a> - LssssssLsss<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:<h2> --><h2 id="toc10"><a name="Scales with two step sizes-12 tones"></a><!-- ws:end:WikiTextHeadingRule:46 -->12 tones</h2>
<a class="wiki_link" href="/OTC%202L%2010s">2L+10s</a> - LsssLsssssss (4+4+4), sssLssssLsss (5+2+5)<br />
<a class="wiki_link" href="/OTC%205L%207s">5L+7s</a> - ssLsLssLsLsL (MOS)<br />
<a class="wiki_link" href="/OTC%207L%205s">7L+5s</a> - LLsLsLLsLsLs (MOS)<br />
<a class="wiki_link" href="/OTC%2010L%202s">10L+2s</a> - LLLsLLLLsLLL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:<h2> --><h2 id="toc11"><a name="Scales with two step sizes-13 tones"></a><!-- ws:end:WikiTextHeadingRule:48 -->13 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+11s</a> - LsssssssLssss<br />
<a class="wiki_link" href="/OTC%207L%206s">7L+6s</a> - sLLLssLssLLLs<br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:<h2> --><h2 id="toc12"><a name="Scales with two step sizes-14 tones"></a><!-- ws:end:WikiTextHeadingRule:50 -->14 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+12s</a> - LssssssssLssss<br />
<a class="wiki_link" href="/OTC%205L%209s">5L+9s</a> - sssLsLsssLsLsL<br />
<a class="wiki_link" href="/OTC%207L%207s">7L+7s</a> - LsLsLsLsLsLsLs (MOS)<br />
<a class="wiki_link" href="/OTC%2012L%202s">12L+2s</a> - LLLLLLLsLLLLLs<br />
<br />
<!-- ws:start:WikiTextHeadingRule:52:<h2> --><h2 id="toc13"><a name="Scales with two step sizes-15 tones"></a><!-- ws:end:WikiTextHeadingRule:52 -->15 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+13s</a> - LssssssssLsssss<br />
<a class="wiki_link" href="/OTC%207L%208s">7L+8s</a> - LsLLssLsLLssLss, LssLLsLssLLsLss<br />
<a class="wiki_link" href="/OTC%2012L%203s">12L+3s</a> - LsLLLLLsLLLLLsL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:54:<h2> --><h2 id="toc14"><a name="Scales with two step sizes-16 tones"></a><!-- ws:end:WikiTextHeadingRule:54 -->16 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+14s</a> - LsssssssssLsssss<br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:<h2> --><h2 id="toc15"><a name="Scales with two step sizes-17 tones"></a><!-- ws:end:WikiTextHeadingRule:56 -->17 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+15s</a> - LsssssssssLssssss (7+3+7), LssssssssssLsssss (6+5+6)<br />
<a class="wiki_link" href="/OTC%205L%2012s">5L+12s</a> - sLsssLssLssLsssLs (MOS)<br />
<a class="wiki_link" href="/OTC%207L%2010s">7L+10s</a> - sLsLsLssLssLsLsLs, sLssLLssLssLLssLs<br />
<a class="wiki_link" href="/OTC%2010L%207s">10L+7s</a> - LsLsLsLLsLLsLsLsL, LsLLssLLsLLssLLsL<br />
<a class="wiki_link" href="/OTC%2012L%205s">12L+5s</a> - LsLLLsLLsLLsLLLsL (MOS)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:58:<h2> --><h2 id="toc16"><a name="Scales with two step sizes-18 tones"></a><!-- ws:end:WikiTextHeadingRule:58 -->18 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+16s</a> - LssssssssssLssssss<br />
<a class="wiki_link" href="/OTC%207L%2011s">7L+11s</a> - LLsssLsLLsssLsssLs, LsssLLsLsssLLsLsss<br />
<br />
<!-- ws:start:WikiTextHeadingRule:60:<h2> --><h2 id="toc17"><a name="Scales with two step sizes-19 tones"></a><!-- ws:end:WikiTextHeadingRule:60 -->19 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+17s</a> - LsssssssssssLssssss<br />
<a class="wiki_link" href="/OTC%205L%2014s">5L+14s</a> - sLssssLssLssLssssLs<br />
<a class="wiki_link" href="/OTC%207L%2012s">7L+12s</a> - sLssLsLssLssLsLssLs (MOS)<br />
<a class="wiki_link" href="/OTC%2010L%209s">10L+9s</a> - LsLLsssLLsLLsssLLsL<br />
<a class="wiki_link" href="/OTC%2012L%207s">12L+7s</a> - LsLLsLsLLsLLsLsLLsL (MOS)<br />
<a class="wiki_link" href="/OTC%2014L%205s">14L+5s</a> - LLLLsLLsLLLLsLLsLLs<br />
<a class="wiki_link" href="/OTC%2017L%202s">17L+2s</a> - LLLLLsLLLLLLLsLLLLL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:62:<h2> --><h2 id="toc18"><a name="Scales with two step sizes-20 tones"></a><!-- ws:end:WikiTextHeadingRule:62 -->20 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+18s</a> - LsssssssssssLsssssss<br />
<a class="wiki_link" href="/OTC%207L%2013s">7L+13s</a> - sssLsssLssLLsssLssLL, sssLsssLsLsLsssLsLsL, sssLsssLLssLsssLLssL<br />
<a class="wiki_link" href="/OTC%2012L%208s">12L+8s</a> - sLLssLLsLLLssLLsLLLs, sLLssLLLsLLssLLLsLLs, sLsLsLLLsLsLsLLLsLsL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:64:<h2> --><h2 id="toc19"><a name="Scales with two step sizes-21 tones"></a><!-- ws:end:WikiTextHeadingRule:64 -->21 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+19s</a> - LssssssssssssLsssssss<br />
<a class="wiki_link" href="/OTC%205L%2016s">5L+16s</a> - sLsssssLssLssLsssssLs<br />
<a class="wiki_link" href="/OTC%207L%2014s">7L+14s</a> - ssssLssssLsLLssssLsLL, ssssLssssLLsLssssLLsL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:66:<h2> --><h2 id="toc20"><a name="Scales with two step sizes-22 tones"></a><!-- ws:end:WikiTextHeadingRule:66 -->22 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+20s</a> - LssssssssssssLssssssss (9+4+9), LsssssssssssssLsssssss (8+6+8)<br />
<a class="wiki_link" href="/OTC%205L%2017s">5L+17s</a> - ssssLsssLssssLsssLsssL (MOS)<br />
<a class="wiki_link" href="/OTC%207L%2015s">7L+15s</a> - LssLsLsssLssLsLsssLsss, LsssLsLssLsssLsLssLsss, LsssLLsssLsssLLsssLsss<br />
<a class="wiki_link" href="/OTC%2010L%2012s">10L+12s</a> - LsLsLsLsLsLssLsLsLsLss<br />
<a class="wiki_link" href="/OTC%2012L%2010s">12L+10s</a> - sLsLsLsLsLsLLsLsLsLsLL, sLLsLsLLssLLssLLsLsLLs, sLLssLLLssLLssLLLssLLs<br />
<a class="wiki_link" href="/OTC%2015L%207s">15L+7s</a> - sLLLssLLLsLLLssLLLsLLL<br />
<a class="wiki_link" href="/OTC%2017L%205s">17L+5s</a> - sLLLsLLLsLLLLsLLLsLLLL (MOS)<br />
<br />
<!-- ws:start:WikiTextHeadingRule:68:<h2> --><h2 id="toc21"><a name="Scales with two step sizes-23 tones"></a><!-- ws:end:WikiTextHeadingRule:68 -->23 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+21s</a> - LsssssssssssssLssssssss<br />
<a class="wiki_link" href="/OTC%207L%2016s">7L+16s</a> - ssssLssssLssLLssssLssLL, ssLsLsLssssLssssLsLsLss, ssssLssssLLssLssssLLssL<br />
<br />
<!-- ws:start:WikiTextHeadingRule:70:<h2> --><h2 id="toc22"><a name="Scales with two step sizes-24 tones"></a><!-- ws:end:WikiTextHeadingRule:70 -->24 tones</h2>
<a class="wiki_link" href="/OTC%202L%20ns">2L+22s</a> - LssssssssssssssLssssssss<br />
<a class="wiki_link" href="/OTC%205L%2019s">5L+19s</a> - sssssLsssLsssssLsssLsssL<br />
<a class="wiki_link" href="/OTC%207L%2017s">7L+17s</a> - sssLsssLssLssLsssLssLssL, sssLsssLsssLsLsssLsssLsL<br />
<a class="wiki_link" href="/OTC%2010L%2014s">10L+14s</a> - LssLLssssLLssLLssssLLssL, sssLsLsLsLsssLsLsLsLsLsL<br />
<a class="wiki_link" href="/OTC%2012L%2012s">12L+12s</a> - LsLsLsLsLsLsLsLsLsLsLsLs (MOS), ssLsLLssLLssLsLLssLLssLL, ssLLsLssLLssLLsLssLLssLL<br />
<a class="wiki_link" href="/OTC%2014L%2010s">14L+10s</a> - LssLLLLssLLssLLssLLLLssL, LLsLsLsLsLsLsLLLsLsLsLsL<br />
<a class="wiki_link" href="/OTC%2015L%209s">15L+9s</a> - LLsssLLLsLLLsLLLsssLLLsL<br />
<a class="wiki_link" href="/OTC%2017L%207s">17L+7s</a> - LLLsLLsLLsLLLsLLsLLsLLLs, LLLsLsLLLsLLLsLLLsLsLLLs<br />
<a class="wiki_link" href="/OTC%2019L%205s">19L+5s</a> - LLLsLLLLLsLLLsLLLLLsLLLs<br />
<br />
<!-- ws:start:WikiTextHeadingRule:72:<h2> --><h2 id="toc23"><a name="Scales with two step sizes->24 tones"></a><!-- ws:end:WikiTextHeadingRule:72 -->>24 tones</h2>
Many larger OTC L+s scales are believed to exist, but due to exponentially increasing computation time, these have not yet been studied in detail. Scales of the form <a class="wiki_link" href="/OTC%202L%20ns">2L+ns</a> are known to exist up to size 53, and assumed to exist for any larger size.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:74:<h2> --><h2 id="toc24"><a name="Scales with two step sizes-L/s matrix"></a><!-- ws:end:WikiTextHeadingRule:74 -->L/s matrix</h2>
<table class="wiki_table">
<tr>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><strong>1</strong><br />
</td>
<td style="text-align: center;"><strong>2</strong><br />
</td>
<td style="text-align: center;"><strong>3</strong><br />
</td>
<td style="text-align: center;"><strong>4</strong><br />
</td>
<td style="text-align: center;"><strong>5</strong><br />
</td>
<td style="text-align: center;"><strong>6</strong><br />
</td>
<td style="text-align: center;"><strong>7</strong><br />
</td>
<td style="text-align: center;"><strong>8</strong><br />
</td>
<td style="text-align: center;"><strong>9</strong><br />
</td>
<td style="text-align: center;"><strong>10</strong><br />
</td>
<td style="text-align: center;"><strong>11</strong><br />
</td>
<td style="text-align: center;"><strong>12</strong><br />
</td>
<td style="text-align: center;"><strong>13</strong><br />
</td>
<td style="text-align: center;"><strong>14</strong><br />
</td>
<td style="text-align: center;"><strong>15</strong><br />
</td>
<td style="text-align: center;"><strong>16</strong><br />
</td>
<td style="text-align: center;"><strong>17</strong><br />
</td>
<td style="text-align: center;"><strong>18</strong><br />
</td>
<td style="text-align: center;"><strong>19</strong><br />
</td>
<td style="text-align: center;"><strong>20</strong><br />
</td>
<td style="text-align: center;"><strong>21</strong><br />
</td>
<td style="text-align: center;"><strong>22</strong><br />
</td>
<td style="text-align: center;"><strong>23</strong><br />
</td>
<td style="text-align: center;"><strong>24</strong><br />
</td>
<td style="text-align: center;"><strong>L</strong><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>1</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
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<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
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<td style="text-align: center;"><br />
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<td style="text-align: center;"><br />
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<td style="text-align: center;"><br />
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<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>2</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%203s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%208s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%2010s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%2012s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%2017s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%2022s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>3</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%203s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%203s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%203s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>4</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>5</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2014L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2017L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2019L%205s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>6</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%206s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>7</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%203L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2010L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2015L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2017L%207s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>8</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%208s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%208s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%208s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>9</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%209s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2010L%209s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2015L%209s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>10</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%2010s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2010s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%2010s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2014L%2010s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>11</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2011s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>12</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%2012s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2012s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2010L%2012s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%2012s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>13</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2013s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>14</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%2014s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2014s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%2010L%2014s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>15</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2015s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>16</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%2016s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2016s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>17</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%2017s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%207L%2017s">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td style="text-align: center;"><strong>18</strong><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
</td>
<td style="text-align: center;"><br />
</td>
<td style="text-align: center;"><br />
</td>
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<td style="text-align: center;"><strong>19</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%2019s">x</a><br />
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<td style="text-align: center;"><strong>20</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><strong>21</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><strong>22</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><strong>23</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><strong>24</strong><br />
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<td style="text-align: center;"><a class="wiki_link" href="/OTC%202L%20ns">x</a><br />
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<td style="text-align: center;"><strong>s</strong><br />
</td>
<td style="text-align: center;"><tt> </tt><br />
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<td style="text-align: center;"><tt> </tt><br />
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<td style="text-align: center;"><tt> </tt><br />
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<td style="text-align: center;"><tt> </tt><br />
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<td style="text-align: center;"><tt> </tt><br />
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<hr />
<br />
<!-- ws:start:WikiTextHeadingRule:76:<h1> --><h1 id="toc25"><a name="Scales with 3 step sizes"></a><!-- ws:end:WikiTextHeadingRule:76 -->Scales with 3 step sizes</h1>
<ul><li><a class="wiki_link" href="/OTC%2022%20sruti">22-tone Indian 'sruti' scale</a></li></ul><br />
<hr />
<br />
<!-- ws:start:WikiTextHeadingRule:78:<h1> --><h1 id="toc26"><a name="Definitions and formulas"></a><!-- ws:end:WikiTextHeadingRule:78 -->Definitions and formulas</h1>
<br />
<!-- ws:start:WikiTextAnchorRule:109:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@dual" title="Anchor: dual"/> --><a name="dual"></a><!-- ws:end:WikiTextAnchorRule:109 --><strong>"Dual"</strong> refers to the "inverse" of a L+s scale pattern, where every L is replaced by s, and vice versa. For example, <tt> sLssL </tt> and <tt> LsLLs </tt> are duals. If a scale is OTC, its dual is often OTC as well, but not always!<br />
<br />
<!-- ws:start:WikiTextAnchorRule:110:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@perfect" title="Anchor: perfect"/> --><a name="perfect"></a><!-- ws:end:WikiTextAnchorRule:110 --><strong>"Perfect"</strong> means that values for L and s exist such that L > s and that every mode of the scale will contain a perfect (just) 3/2 or 4/3 (or both). (See also <a class="wiki_link" href="/Eigenmonzo%20subgroup">Eigenmonzo subgroup</a>.)<br />
<br />
In this case the value <strong>P</strong> is given, where P = L/s. For a perfect scale, P > 1. Note that if a scale "a" is perfect (P<span style="vertical-align: sub;">a</span> <!-- ws:start:WikiTextRawRule:00:``=`` -->=<!-- ws:end:WikiTextRawRule:00 --> L/s), its dual "b" will have the value P<span style="vertical-align: sub;">b</span> <!-- ws:start:WikiTextRawRule:01:``=`` -->=<!-- ws:end:WikiTextRawRule:01 --> s/L <!-- ws:start:WikiTextRawRule:02:``=`` -->=<!-- ws:end:WikiTextRawRule:02 --> 1/P<span style="vertical-align: sub;">a</span>, and therefore must be imperfect (if P<span style="vertical-align: sub;">a</span> > 1, then P<span style="vertical-align: sub;">b</span> < 1 ).<br />
<br />
In some cases, P may be less than zero. I'm not yet sure what this means :)<br />
<br />
The value of P is calculated as follows:<br />
<br />
a <!-- ws:start:WikiTextRawRule:03:``=`` -->=<!-- ws:end:WikiTextRawRule:03 --> the number of L steps per 2/1<br />
b <!-- ws:start:WikiTextRawRule:04:``=`` -->=<!-- ws:end:WikiTextRawRule:04 --> the number of s steps per 2/1<br />
c <!-- ws:start:WikiTextRawRule:05:``=`` -->=<!-- ws:end:WikiTextRawRule:05 --> the number of L steps per 4/3<br />
d <!-- ws:start:WikiTextRawRule:06:``=`` -->=<!-- ws:end:WikiTextRawRule:06 --> the number of s steps per 4/3<br />
x <!-- ws:start:WikiTextRawRule:07:``=`` -->=<!-- ws:end:WikiTextRawRule:07 --> log<span style="vertical-align: sub;">2</span>(4/3) <!-- ws:start:WikiTextRawRule:08:``=`` -->=<!-- ws:end:WikiTextRawRule:08 --> ~0.41504 octaves <!-- ws:start:WikiTextRawRule:09:``=`` -->=<!-- ws:end:WikiTextRawRule:09 --> ~498.045 cents<br />
<br />
aL+bs <!-- ws:start:WikiTextRawRule:010:``=`` -->=<!-- ws:end:WikiTextRawRule:010 --> log<span style="vertical-align: sub;">2</span>(2/1) = 1<br />
cL+ds <!-- ws:start:WikiTextRawRule:011:``=`` -->=<!-- ws:end:WikiTextRawRule:011 --> x<br />
<br />
s <!-- ws:start:WikiTextRawRule:012:``=`` -->=<!-- ws:end:WikiTextRawRule:012 --> (c-ax)/(bc-ad)<br />
L <!-- ws:start:WikiTextRawRule:013:``=`` -->=<!-- ws:end:WikiTextRawRule:013 --> (x-ds)/c<br />
P <!-- ws:start:WikiTextRawRule:014:``=`` -->=<!-- ws:end:WikiTextRawRule:014 --> L/s<br />
<br />
Note that the same procedure could be used to calculate the L/s ratio necessary to give any other just interval, such as 5/4, 11/8, etc.<br />
<br />
<!-- ws:start:WikiTextAnchorRule:111:<img src="/i/anchor.gif" class="WikiAnchor" alt="Anchor" id="wikitext@@anchor@@Q" title="Anchor: Q"/> --><a name="Q"></a><!-- ws:end:WikiTextAnchorRule:111 --><strong>Q</strong> is calculated similarly to P, but indicates a limit of sorts -- a point on the L/s continuum beyond which the omnitetrachordality of the scale can be considered to 'break down' in some way.<br />
<br />
Consider for example the 2L+8s pajara MODMOS, LsssssLsss -- at P <!-- ws:start:WikiTextRawRule:015:``=`` -->=<!-- ws:end:WikiTextRawRule:015 --> L/s <!-- ws:start:WikiTextRawRule:016:``=`` -->=<!-- ws:end:WikiTextRawRule:016 --> 1.885, L+3s forms a just 4/3. As L/s increases, L gets larger and s smaller; at Q <!-- ws:start:WikiTextRawRule:017:``=`` -->=<!-- ws:end:WikiTextRawRule:017 --> L/s <!-- ws:start:WikiTextRawRule:018:``=`` -->=<!-- ws:end:WikiTextRawRule:018 --> 4.827, a just 4/3 is not L+3s, but L+2.5s. Past this point, L+2s will therefore be closer to a just 4/3 than L+3s:<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:1525:<img src="/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png" alt="" title="" /> --><img src="/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png" alt="P_Q_C_LsssssLsss.png" title="P_Q_C_LsssssLsss.png" /><!-- ws:end:WikiTextLocalImageRule:1525 --><br />
<br />
For some scales, Q will not exist. For others, a second Q may exist that is less than P, placing a lower bound on the L/s ratio as well as an upper one.<br />
<br />
We can also consider a point C, where the number of steps "crosses" the just 4/3 entirely - in the example above, this corresponds to 4/3 <!-- ws:start:WikiTextRawRule:019:``=`` -->=<!-- ws:end:WikiTextRawRule:019 --> L+2s. Q is then halfway between P and C, i.e.<br />
C <!-- ws:start:WikiTextRawRule:020:``=`` -->=<!-- ws:end:WikiTextRawRule:020 --> 2Q-P<br />
Q <!-- ws:start:WikiTextRawRule:021:``=`` -->=<!-- ws:end:WikiTextRawRule:021 --> (C+P)/2<br />
P <!-- ws:start:WikiTextRawRule:022:``=`` -->=<!-- ws:end:WikiTextRawRule:022 --> 2Q-C<br />
<br />
...which leads to the curious result that, although P is undefined when calculated by the normal method (due to division by zero) for a scale such as blackwood[10] (LsLsLsLsLs), C and Q can be calculated, and thus a value of P <!-- ws:start:WikiTextRawRule:023:``=`` -->=<!-- ws:end:WikiTextRawRule:023 --> -1 can be found anyway, even though it seems not to be of any use :)<br />
<br />
<strong>"L/s range"</strong>: For any L+s scale pattern, the ratio L/s may range from 1 (L<!-- ws:start:WikiTextRawRule:024:``=`` -->=<!-- ws:end:WikiTextRawRule:024 -->s, in which case the scale is (L+s)edo ) to ~infinity (s<!-- ws:start:WikiTextRawRule:025:``=`` -->=<!-- ws:end:WikiTextRawRule:025 -->0, in which case the scale is (L)edo ). A note such as "full L/s range is good" simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's "goodness" is intended.</body></html>