MOS scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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===Blackwood R constant===

In the context of the "recognizable diatonic" scales deriving from the Farey pair (1/2, 3/5) [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 <= R <= 2.

In the context of the "recognizable diatonic" scales deriving from the Farey pair [1/2, 3/5] [[http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr.|Easley Blackwood Jr.]] defined a characterizing constant R which we may generalize to any MOS as follows. If a/b < g < c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 <= R <= 2.

When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R>1 (or R<1 if we prefer.)

==Catalog of MOS==

Below is a list of MOS with number of elements from 5 to 10.

Below is a list of MOS with number of elements from 5 to 10, plus some of the more significant larger MOS.

~~Not all mathematical possibilities are listed - solutions~~ of the equation that would yield too "exotic" scale steps (too small/too big diffference between s and L) are excluded. (The concrete - sort of arbitrary - restrictions applied were: a solution appears if 7/6 < L/s < 5.)

|| [[PentatonicMOS|Pentatonic MOS]] || || || || || || [[1L 4s]] || || [[2L 3s]] || || [[3L 2s]] || || [[4L 1s]] || || || || || ||

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|| [[DecatonicMOS|Decatonic MOS]] || [[1L 9s]] || || [[2L 8s]] || || [[3L 7s]] || || [[4L 6s]] || || [[5L 5s]] || || [[6L 4s]] || || [[7L 3s]] || || [[8L 2s]] || || [[9L 1s]] ||

Keemun[11] [4L 7s]

Sensi[11] [8L 3s]

Meantone[12] [7L 5s]

Superpyth[12] [5L 7s]

Pajara[12] [10L 2s]

Injera/Doublewide[12] [2L 10s]

Augene[12] [3L 9s]

Godzilla[14] [9L 5s]

Porcupine[15] [7L 8s]

Myna[15] [4L 11s]

Valentine[15] [1L 14s]

Mothra[16] [5L 11s]

Wizard[16] [6L 10s]

Garibaldi[17] [5L 12s]

Mohajira[17] [7L 10s]

Beatles[17] [10L 7s]

Magic[19] [3L 16s]

Myna[19] [4L 15s]

Sensi[19] [8L 11s]

Miracle[21] [10L 11s]

Magic[22] [19L 3s]

Orwell[22] [9L 13s]

Wizard[22] [6L 16s]

==MOS As Applied To Rhythms==

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<h3 id="toc4"><a name="MOS scales-Classification of MOS-Blackwood R constant"></a>Blackwood R constant</h3>

In the context of the &quot;recognizable diatonic&quot; scales deriving from the Farey pair (1/2, 3/5) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow">Easley Blackwood Jr.</a> defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.

In the context of the &quot;recognizable diatonic&quot; scales deriving from the Farey pair [1/2, 3/5] <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr." rel="nofollow">Easley Blackwood Jr.</a> defined a characterizing constant R which we may generalize to any MOS as follows. If a/b &lt; g &lt; c/d is a generator with the given Farey pair, take the ratio of relative errors R = (bg - a)/(c - dg). Since this is a ratio of positive numbers, it is positive. As g tends towards a/b it tends to zero, and as g goes to c/d R goes to infinity. When g equals (a + c)/(b + d) it takes the value 1, and the range of propriety is 1/2 &lt;= R &lt;= 2.

When R is less than 1, it represents the ratio in (logarithmic) size between the smaller and the larger step. When it is greater than 1, it is larger/smaller. By replacing g with 1 - g if necessary, we can reduce always to the case where R&gt;1 (or R&lt;1 if we prefer.)

<h2 id="toc5"><a name="MOS scales-Catalog of MOS"></a>Catalog of MOS</h2>

Below is a list of MOS with number of elements from 5 to 10~~. ~~