Gallery of omnitetrachordal scales: Difference between revisions
Wikispaces>spt3125 **Imported revision 590311800 - Original comment: ** |
Wikispaces>spt3125 **Imported revision 590813958 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2016- | : This revision was by author [[User:spt3125|spt3125]] and made on <tt>2016-09-01 21:50:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>590813958</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 137: | Line 137: | ||
||= ||= **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** ||= **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]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||= **1** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= **2** ||= ||= ||= [[OTC | ||= **2** ||= ||= ||= [[OTC 3L 2s|x]] ||= ||= [[OTC 5L 2s|x]] ||= ||= ||= [[OTC 8L 2s|x]] ||= ||= [[OTC 10L 2s|x]] ||= ||= [[OTC 12L 2s|x]] ||= ||= ||= ||= ||= [[OTC 17L 2s|x]] ||= ||= ||= ||= ||= [[OTC 22L 2s|x]] ||= ||= ||= || | ||
||= **3** ||= ||= [[OTC 2L 3s|x]] ||= ||= ||= ||= ||= [[OTC 7L 3s|x]] ||= ||= ||= ||= ||= [[OTC 12L 3s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||= **3** ||= ||= [[OTC 2L 3s|x]] ||= ||= ||= ||= ||= [[OTC 7L 3s|x]] ||= ||= ||= ||= ||= [[OTC 12L 3s|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||= **4** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||= **4** ||= ||= [[OTC 2L ns|x]] ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
| Line 181: | Line 181: | ||
The value of P is calculated as follows: | The value of P is calculated as follows: | ||
a | a = the number of L steps per 2/1 | ||
b | b = the number of s steps per 2/1 | ||
c | c = the number of L steps per 4/3 | ||
d | d = the number of s steps per 4/3 | ||
x | x = log<span style="vertical-align: sub;">2</span>(4/3) ``=`` ~0.41504 octaves ``=`` ~498.045 cents | ||
aL+bs | aL+bs = log<span style="vertical-align: sub;">2</span>(2/1) = 1 | ||
cL+ds | cL+ds = x | ||
s | s = (c-ax)/(bc-ad) | ||
L | L = (x-ds)/c | ||
P | 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. | 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. | ||
| Line 198: | Line 198: | ||
[[#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. | [[#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 [[OTC 2L 8s|OTC 2L+8s]] pajara MODMOS, LsssssLsss -- at P | Consider for example the [[OTC 2L 8s|OTC 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]] | [[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. | 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 | 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 | C = 2Q-P | ||
Q | Q = (C+P)/2 | ||
P | 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 | ...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 | **"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. | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>Gallery of omnitetrachordal scales</title></head><body><!-- ws:start:WikiTextTocRule: | <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"><html><head><title>Gallery of omnitetrachordal scales</title></head><body><!-- ws:start:WikiTextTocRule:59:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 1em;"><a href="#Scales with two step sizes">Scales with two step sizes</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-3 tones">3 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:61 --><!-- ws:start:WikiTextTocRule:62: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-4 tones">4 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:62 --><!-- ws:start:WikiTextTocRule:63: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-5 tones">5 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:63 --><!-- ws:start:WikiTextTocRule:64: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-6 tones">6 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:64 --><!-- ws:start:WikiTextTocRule:65: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-7 tones">7 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:65 --><!-- ws:start:WikiTextTocRule:66: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-8 tones">8 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:66 --><!-- ws:start:WikiTextTocRule:67: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-9 tones">9 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:67 --><!-- ws:start:WikiTextTocRule:68: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-10 tones">10 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:68 --><!-- ws:start:WikiTextTocRule:69: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-11 tones">11 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:69 --><!-- ws:start:WikiTextTocRule:70: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-12 tones">12 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:70 --><!-- ws:start:WikiTextTocRule:71: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-13 tones">13 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:71 --><!-- ws:start:WikiTextTocRule:72: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-14 tones">14 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:72 --><!-- ws:start:WikiTextTocRule:73: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-15 tones">15 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:73 --><!-- ws:start:WikiTextTocRule:74: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-16 tones">16 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:74 --><!-- ws:start:WikiTextTocRule:75: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-17 tones">17 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:75 --><!-- ws:start:WikiTextTocRule:76: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-18 tones">18 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:76 --><!-- ws:start:WikiTextTocRule:77: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-19 tones">19 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:77 --><!-- ws:start:WikiTextTocRule:78: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-20 tones">20 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:78 --><!-- ws:start:WikiTextTocRule:79: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-21 tones">21 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:79 --><!-- ws:start:WikiTextTocRule:80: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-22 tones">22 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-23 tones">23 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-24 tones">24 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-&gt;24 tones">&gt;24 tones</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><div style="margin-left: 2em;"><a href="#Scales with two step sizes-L/s matrix">L/s matrix</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><div style="margin-left: 1em;"><a href="#Scales with 3 step sizes">Scales with 3 step sizes</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><div style="margin-left: 1em;"><a href="#Definitions and formulas">Definitions and formulas</a></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --></div> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:87 --><br /> | ||
An (incomplete) list of <a class="wiki_link" href="/Omnitetrachordality">omnitetrachordal</a> (OTC) scales.<br /> | An (incomplete) list of <a class="wiki_link" href="/Omnitetrachordality">omnitetrachordal</a> (OTC) scales.<br /> | ||
<br /> | <br /> | ||
<hr /> | <hr /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:5:&lt;h1&gt; --><h1 id="toc0"><a name="Scales with two step sizes"></a><!-- ws:end:WikiTextHeadingRule:5 -->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 /> | (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 /> | 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 /> | ||
| Line 253: | Line 254: | ||
Follow the links for more detailed info on each scale! <em>(Some links may not be working yet...)</em><br /> | Follow the links for more detailed info on each scale! <em>(Some links may not be working yet...)</em><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc1"><a name="Scales with two step sizes-3 tones"></a><!-- ws:end:WikiTextHeadingRule:7 -->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 /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+s</a> (the simplest possible OTC scale; s=9/8, L=4/3)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:9:&lt;h2&gt; --><h2 id="toc2"><a name="Scales with two step sizes-4 tones"></a><!-- ws:end:WikiTextHeadingRule:9 -->4 tones</h2> | ||
(no OTC scales possible)<br /> | (no OTC scales possible)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:11:&lt;h2&gt; --><h2 id="toc3"><a name="Scales with two step sizes-5 tones"></a><!-- ws:end:WikiTextHeadingRule:11 -->5 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%203s">2L+3s</a> - sLsLs (MOS)<br /> | <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 /> | <a class="wiki_link" href="/OTC%203L%202s">3L+2s</a> - LsLsL (MOS)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc4"><a name="Scales with two step sizes-6 tones"></a><!-- ws:end:WikiTextHeadingRule:13 -->6 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+4s</a> - LsssLs<br /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+4s</a> - LsssLs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:15:&lt;h2&gt; --><h2 id="toc5"><a name="Scales with two step sizes-7 tones"></a><!-- ws:end:WikiTextHeadingRule:15 -->7 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%205s">2L+5s</a> - sLsssLs (MOS)<br /> | <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 /> | <a class="wiki_link" href="/OTC%205L%202s">5L+2s</a> - LsLLLsL (MOS)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:17:&lt;h2&gt; --><h2 id="toc6"><a name="Scales with two step sizes-8 tones"></a><!-- ws:end:WikiTextHeadingRule:17 -->8 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+6s</a> - LssssLss<br /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+6s</a> - LssssLss<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:19:&lt;h2&gt; --><h2 id="toc7"><a name="Scales with two step sizes-9 tones"></a><!-- ws:end:WikiTextHeadingRule:19 -->9 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+7s</a> - LsssssLss<br /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+7s</a> - LsssssLss<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:21:&lt;h2&gt; --><h2 id="toc8"><a name="Scales with two step sizes-10 tones"></a><!-- ws:end:WikiTextHeadingRule:21 -->10 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%208s">2L+8s</a> - LsssssLsss<br /> | <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%203L%207s">3L+7s</a> - LsssLsLsss<br /> | ||
| Line 283: | Line 284: | ||
<a class="wiki_link" href="/OTC%208L%202s">8L+2s</a> - LLsLLLsLLL<br /> | <a class="wiki_link" href="/OTC%208L%202s">8L+2s</a> - LLsLLLsLLL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:23:&lt;h2&gt; --><h2 id="toc9"><a name="Scales with two step sizes-11 tones"></a><!-- ws:end:WikiTextHeadingRule:23 -->11 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+9s</a> - LssssssLsss<br /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+9s</a> - LssssssLsss<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc10"><a name="Scales with two step sizes-12 tones"></a><!-- ws:end:WikiTextHeadingRule:25 -->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%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%205L%207s">5L+7s</a> - ssLsLssLsLsL (MOS)<br /> | ||
| Line 292: | Line 293: | ||
<a class="wiki_link" href="/OTC%2010L%202s">10L+2s</a> - LLLsLLLLsLLL<br /> | <a class="wiki_link" href="/OTC%2010L%202s">10L+2s</a> - LLLsLLLLsLLL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:27:&lt;h2&gt; --><h2 id="toc11"><a name="Scales with two step sizes-13 tones"></a><!-- ws:end:WikiTextHeadingRule:27 -->13 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+11s</a> - LsssssssLssss<br /> | <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 /> | <a class="wiki_link" href="/OTC%207L%206s">7L+6s</a> - sLLLssLssLLLs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:29:&lt;h2&gt; --><h2 id="toc12"><a name="Scales with two step sizes-14 tones"></a><!-- ws:end:WikiTextHeadingRule:29 -->14 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+12s</a> - LssssssssLssss<br /> | <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%205L%209s">5L+9s</a> - sssLsLsssLsLsL<br /> | ||
| Line 302: | Line 303: | ||
<a class="wiki_link" href="/OTC%2012L%202s">12L+2s</a> - LLLLLLLsLLLLLs<br /> | <a class="wiki_link" href="/OTC%2012L%202s">12L+2s</a> - LLLLLLLsLLLLLs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:31:&lt;h2&gt; --><h2 id="toc13"><a name="Scales with two step sizes-15 tones"></a><!-- ws:end:WikiTextHeadingRule:31 -->15 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+13s</a> - LssssssssLsssss<br /> | <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%207L%208s">7L+8s</a> - LsLLssLsLLssLss, LssLLsLssLLsLss<br /> | ||
<a class="wiki_link" href="/OTC%2012L%203s">12L+3s</a> - LsLLLLLsLLLLLsL<br /> | <a class="wiki_link" href="/OTC%2012L%203s">12L+3s</a> - LsLLLLLsLLLLLsL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:33:&lt;h2&gt; --><h2 id="toc14"><a name="Scales with two step sizes-16 tones"></a><!-- ws:end:WikiTextHeadingRule:33 -->16 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+14s</a> - LsssssssssLsssss<br /> | <a class="wiki_link" href="/OTC%202L%20ns">2L+14s</a> - LsssssssssLsssss<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:35:&lt;h2&gt; --><h2 id="toc15"><a name="Scales with two step sizes-17 tones"></a><!-- ws:end:WikiTextHeadingRule:35 -->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%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%205L%2012s">5L+12s</a> - sLsssLssLssLsssLs (MOS)<br /> | ||
| Line 317: | Line 318: | ||
<a class="wiki_link" href="/OTC%2012L%205s">12L+5s</a> - LsLLLsLLsLLsLLLsL (MOS)<br /> | <a class="wiki_link" href="/OTC%2012L%205s">12L+5s</a> - LsLLLsLLsLLsLLLsL (MOS)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:37:&lt;h2&gt; --><h2 id="toc16"><a name="Scales with two step sizes-18 tones"></a><!-- ws:end:WikiTextHeadingRule:37 -->18 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+16s</a> - LssssssssssLssssss<br /> | <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 /> | <a class="wiki_link" href="/OTC%207L%2011s">7L+11s</a> - LLsssLsLLsssLsssLs, LsssLLsLsssLLsLsss<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:39:&lt;h2&gt; --><h2 id="toc17"><a name="Scales with two step sizes-19 tones"></a><!-- ws:end:WikiTextHeadingRule:39 -->19 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+17s</a> - LsssssssssssLssssss<br /> | <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%205L%2014s">5L+14s</a> - sLssssLssLssLssssLs<br /> | ||
| Line 330: | Line 331: | ||
<a class="wiki_link" href="/OTC%2017L%202s">17L+2s</a> - LLLLLsLLLLLLLsLLLLL<br /> | <a class="wiki_link" href="/OTC%2017L%202s">17L+2s</a> - LLLLLsLLLLLLLsLLLLL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:41:&lt;h2&gt; --><h2 id="toc18"><a name="Scales with two step sizes-20 tones"></a><!-- ws:end:WikiTextHeadingRule:41 -->20 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+18s</a> - LsssssssssssLsssssss<br /> | <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%207L%2013s">7L+13s</a> - sssLsssLssLLsssLssLL, sssLsssLsLsLsssLsLsL, sssLsssLLssLsssLLssL<br /> | ||
<a class="wiki_link" href="/OTC%2012L%208s">12L+8s</a> - sLLssLLsLLLssLLsLLLs, sLLssLLLsLLssLLLsLLs, sLsLsLLLsLsLsLLLsLsL<br /> | <a class="wiki_link" href="/OTC%2012L%208s">12L+8s</a> - sLLssLLsLLLssLLsLLLs, sLLssLLLsLLssLLLsLLs, sLsLsLLLsLsLsLLLsLsL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:43:&lt;h2&gt; --><h2 id="toc19"><a name="Scales with two step sizes-21 tones"></a><!-- ws:end:WikiTextHeadingRule:43 -->21 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+19s</a> - LssssssssssssLsssssss<br /> | <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%205L%2016s">5L+16s</a> - sLsssssLssLssLsssssLs<br /> | ||
<a class="wiki_link" href="/OTC%207L%2014s">7L+14s</a> - ssssLssssLsLLssssLsLL, ssssLssssLLsLssssLLsL<br /> | <a class="wiki_link" href="/OTC%207L%2014s">7L+14s</a> - ssssLssssLsLLssssLsLL, ssssLssssLLsLssssLLsL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:45:&lt;h2&gt; --><h2 id="toc20"><a name="Scales with two step sizes-22 tones"></a><!-- ws:end:WikiTextHeadingRule:45 -->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%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%205L%2017s">5L+17s</a> - ssssLsssLssssLsssLsssL (MOS)<br /> | ||
| Line 349: | Line 350: | ||
<a class="wiki_link" href="/OTC%2017L%205s">17L+5s</a> - sLLLsLLLsLLLLsLLLsLLLL (MOS)<br /> | <a class="wiki_link" href="/OTC%2017L%205s">17L+5s</a> - sLLLsLLLsLLLLsLLLsLLLL (MOS)<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:47:&lt;h2&gt; --><h2 id="toc21"><a name="Scales with two step sizes-23 tones"></a><!-- ws:end:WikiTextHeadingRule:47 -->23 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+21s</a> - LsssssssssssssLssssssss<br /> | <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 /> | <a class="wiki_link" href="/OTC%207L%2016s">7L+16s</a> - ssssLssssLssLLssssLssLL, ssLsLsLssssLssssLsLsLss, ssssLssssLLssLssssLLssL<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:49:&lt;h2&gt; --><h2 id="toc22"><a name="Scales with two step sizes-24 tones"></a><!-- ws:end:WikiTextHeadingRule:49 -->24 tones</h2> | ||
<a class="wiki_link" href="/OTC%202L%20ns">2L+22s</a> - LssssssssssssssLssssssss<br /> | <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%205L%2019s">5L+19s</a> - sssssLsssLsssssLsssLsssL<br /> | ||
| Line 364: | Line 365: | ||
<a class="wiki_link" href="/OTC%2019L%205s">19L+5s</a> - LLLsLLLLLsLLLsLLLLLsLLLs<br /> | <a class="wiki_link" href="/OTC%2019L%205s">19L+5s</a> - LLLsLLLLLsLLLsLLLLLsLLLs<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:51:&lt;h2&gt; --><h2 id="toc23"><a name="Scales with two step sizes-&gt;24 tones"></a><!-- ws:end:WikiTextHeadingRule:51 -->&gt;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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:53:&lt;h2&gt; --><h2 id="toc24"><a name="Scales with two step sizes-L/s matrix"></a><!-- ws:end:WikiTextHeadingRule:53 -->L/s matrix</h2> | ||
| Line 486: | Line 487: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%203L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%205L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 496: | Line 497: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%208L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%2010L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%2012L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 514: | Line 515: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%2017L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 524: | Line 525: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><a class="wiki_link" href="/OTC% | <td style="text-align: center;"><a class="wiki_link" href="/OTC%2022L%202s">x</a><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 1,780: | Line 1,781: | ||
<hr /> | <hr /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:55:&lt;h1&gt; --><h1 id="toc25"><a name="Scales with 3 step sizes"></a><!-- ws:end:WikiTextHeadingRule:55 -->Scales with 3 step sizes</h1> | ||
<ul><li><a class="wiki_link" href="/OTC%20JI%2022%20sruti%20scale">22-tone Indian 'sruti' scale</a></li></ul><br /> | <ul><li><a class="wiki_link" href="/OTC%20JI%2022%20sruti%20scale">22-tone Indian 'sruti' scale</a></li></ul><br /> | ||
<hr /> | <hr /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:57:&lt;h1&gt; --><h1 id="toc26"><a name="Definitions and formulas"></a><!-- ws:end:WikiTextHeadingRule:57 -->Definitions and formulas</h1> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextAnchorRule:88:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@dual&quot; title=&quot;Anchor: dual&quot;/&gt; --><a name="dual"></a><!-- ws:end:WikiTextAnchorRule:88 --><strong>&quot;Dual&quot;</strong> refers to the &quot;inverse&quot; 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 /> | <br /> | ||
<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextAnchorRule:89:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@perfect&quot; title=&quot;Anchor: perfect&quot;/&gt; --><a name="perfect"></a><!-- ws:end:WikiTextAnchorRule:89 --><strong>&quot;Perfect&quot;</strong> means that values for L and s exist such that L &gt; 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 /> | <br /> | ||
In this case the value <strong>P</strong> is given, where P = L/s. For a perfect scale, P &gt; 1. Note that if a scale &quot;a&quot; is perfect (P<span style="vertical-align: sub;">a</span> <!-- ws:start:WikiTextRawRule:00:``=`` -->=<!-- ws:end:WikiTextRawRule:00 --> L/s), its dual &quot;b&quot; 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> &gt; 1, then P<span style="vertical-align: sub;">b</span> &lt; 1 ).<br /> | In this case the value <strong>P</strong> is given, where P = L/s. For a perfect scale, P &gt; 1. Note that if a scale &quot;a&quot; is perfect (P<span style="vertical-align: sub;">a</span> <!-- ws:start:WikiTextRawRule:00:``=`` -->=<!-- ws:end:WikiTextRawRule:00 --> L/s), its dual &quot;b&quot; 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> &gt; 1, then P<span style="vertical-align: sub;">b</span> &lt; 1 ).<br /> | ||
| Line 1,796: | Line 1,797: | ||
The value of P is calculated as follows:<br /> | The value of P is calculated as follows:<br /> | ||
<br /> | <br /> | ||
a | a = the number of L steps per 2/1<br /> | ||
b | b = the number of s steps per 2/1<br /> | ||
c | c = the number of L steps per 4/3<br /> | ||
d | d = the number of s steps per 4/3<br /> | ||
x | x = log<span style="vertical-align: sub;">2</span>(4/3) <!-- ws:start:WikiTextRawRule:03:``=`` -->=<!-- ws:end:WikiTextRawRule:03 --> ~0.41504 octaves <!-- ws:start:WikiTextRawRule:04:``=`` -->=<!-- ws:end:WikiTextRawRule:04 --> ~498.045 cents<br /> | ||
<br /> | <br /> | ||
aL+bs | aL+bs = log<span style="vertical-align: sub;">2</span>(2/1) = 1<br /> | ||
cL+ds | cL+ds = x<br /> | ||
<br /> | <br /> | ||
s | s = (c-ax)/(bc-ad)<br /> | ||
L | L = (x-ds)/c<br /> | ||
P | P = L/s<br /> | ||
<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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextAnchorRule: | <!-- ws:start:WikiTextAnchorRule:90:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@Q&quot; title=&quot;Anchor: Q&quot;/&gt; --><a name="Q"></a><!-- ws:end:WikiTextAnchorRule:90 --><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 /> | <br /> | ||
Consider for example the <a class="wiki_link" href="/OTC%202L%208s">OTC 2L+8s</a> pajara MODMOS, LsssssLsss -- at P | Consider for example the <a class="wiki_link" href="/OTC%202L%208s">OTC 2L+8s</a> 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:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:1504:&lt;img src=&quot;/file/view/P_Q_C_LsssssLsss.png/590309824/P_Q_C_LsssssLsss.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><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:1504 --><br /> | ||
<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 /> | 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 /> | <br /> | ||
We can also consider a point C, where the number of steps &quot;crosses&quot; the just 4/3 entirely - in the example above, this corresponds to 4/3 | We can also consider a point C, where the number of steps &quot;crosses&quot; 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.<br /> | ||
C | C = 2Q-P<br /> | ||
Q | Q = (C+P)/2<br /> | ||
P | P = 2Q-C<br /> | ||
<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 | ...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 :)<br /> | ||
<br /> | <br /> | ||
<strong>&quot;L/s range&quot;</strong>: For any L+s scale pattern, the ratio L/s may range from 1 (L | <strong>&quot;L/s range&quot;</strong>: 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 &quot;full L/s range is good&quot; simply means that the approximation of 3/2 or 4/3 is reasonable across the entire range; no other assessment of the scale's &quot;goodness&quot; is intended.</body></html></pre></div> | ||