Mathematics of MOS: Difference between revisions
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These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and fail to be a MOS, meaning that not all non-unison classes have only two specific intervals. | These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and fail to be a MOS, meaning that not all non-unison classes have only two specific intervals. | ||
=== Characterizations === | === Characterizations === | ||
There are several equivalent definitions of MOS scales: | There are several equivalent definitions of MOS scales: | ||
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# [[Binary]] and has a generator | # [[Binary]] and has a generator | ||
# Binary and [[distributionally even]] | # Binary and [[distributionally even]] | ||
# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L − s = c) | # Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|L − s {{=}} c}}) | ||
# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line ''y'' = ''p''/''q'' | # Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap|''y'' {{=}} ''p''/''q'' ''x''}} that stays as close to the line without crossing it.) | ||
While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent: | While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent: | ||
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== Properties == | == Properties == | ||
Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period | Let us represent the period as 1. This would be the logarithm base 2 of 2 if the period is an octave, or in general we can measure intervals by the log base P when P is the period. | ||
Suppose the fractions {{frac|''a''|''b''}} and {{frac|''c''|''d''}} are a [[Wikipedia:Farey sequence#Farey neighbours|Farey pair]], meaning that {{nowrap|{{frac|''a''|''b''}} < {{frac|''c''|''d''}}}} and {{nowrap|''bc'' − ''ad'' {{=}} 1}}. If {{nowrap|''g'' {{=}} (1 − ''t''){{frac|''a''|''b''}} + (''t''){{frac|''c''|''d''}}}} for {{nowrap|0 ≤ ''t'' ≤ 1}}, then when {{nowrap|''t'' {{=}} 0}}, the scale generated by ''g'' will consist of an equal division of 1 (representing P) into steps of size {{frac|1|''b''}}, and when {{nowrap|''t'' {{=}} 1}} into steps of size {{frac|1|''d''}}. In between, when {{nowrap|''t'' {{=}} {{sfrac|''b''|''b'' + ''d''}}}}, we obtain a generator equal to the [[Wikipedia:Mediant_%28mathematics%29|mediant]] {{nowrap|''m'' {{=}} {{sfrac|''a'' + ''c''|''b'' + ''d''}}}} and which will divide the period into {{nowrap|''b'' + ''d''}} equal steps. For all other values {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} we obtain two different sizes of steps, the small steps ''s'', and the large steps ''L'', with the total number of steps {{nowrap|''b'' + ''d''}}, and these scales are the MOS associated to the Farey pair. When ''g'' is between {{frac|''a''|''b''}} and ''m'', there will be ''b'' large steps and ''d'' small steps, and when it is between ''m'' and {{frac|''c''|''d''}}, ''d'' large steps and ''b'' small ones. | |||
While all the scales constructed by generators ''g'' with {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}} with the exception of the mediant which gives an equal tuning are MOS, not all the scales are [[Wikipedia:Rothenberg_propriety|proper]] in the sense of Rothenberg. The ''range of propriety'' for MOS is {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} ≤ ''g''}} ≤ {{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, where MOS coming from a Farey pair ({{frac|''a''|''b''}}, {{frac|''c''|''d''}}) are proper when in this range, and improper (unless the MOS has only one small step) when out of it. If {{nowrap|{{sfrac|2''a'' + ''c''|2''b'' + ''d''}} < ''g''}} < {{sfrac|''a'' + 2''c''|''b'' + 2''d''}}, then the scales are strictly proper. Hence, the diatonic scale in 12et, with generator 7/12, is proper but not strictly proper since starting from the pair ({{frac|1|2}}, {{frac|3|5}}) we find the range of propriety for these seven-note MOS to be [{{frac|5|9}}, {{frac|7|12}}]. | |||
Given a generator ''g'', we can find an MOS for ''g'' with period 1 by means of the [[Wikipedia:Continued_fraction#Semiconvergents|semiconvergents]] to ''g''. A pair of successive semiconvergents have the property that they define a Farey pair, and when ''g'' is contained in the pair (that is, {{nowrap|{{frac|''a''|''b''}} < ''g'' < {{frac|''c''|''d''}}}}), we have defined a MOS for ''g'' with {{nowrap|''b'' + ''d''}} as the number of notes in the MOS, with ''b'' notes of one size and ''d'' of the other. | |||
For example, suppose we want an MOS for 1/4-comma meantone. The generator will then be {{sfrac|log<sub>2</sub>(5)|4}}, which has semiconvergents 1/2, 2/3, 3/5, 4/7, 7/12, 11/19, 18/31, 29/50, 47/81, 65/112... If we settle on 31 as a good size for our MOS, we see 18/31 is the mediant between the Farey pair 11/19 and 7/12, for which the range of strict propriety is {{nowrap|29/50 < ''x'' < 25/43}}. Since ''g'' is in that range and not equal to 18/31, we will get a strictly proper MOS. | |||
== Visualizing MOS: Generator chains, pitch spaces, and hierarchies == | |||
As MOS Scales are generated by repeated iterations of a single interval, the generator, it is useful to visualize a contiguous "generator chain" that organizes the scale. For example, if the generator is some kind of perfect fifth, then the generator chain is a chain of fifths: {{dash|F, C, G, D, A, E, B|med}}. We know that adjacent tones in the chain are a perfect fifth apart (possibly with octave-displacement), such as F to C, and that tones two spaces away are some kind of second or ninth apart, such as F to G. It is clear from the chain that B to F is *not* a perfect fifth, but must be something else (unless the chain closes to form a circle, as would be the case in [[7edo]]). The generator chain shows us that every interval of the MOS scale represents a move on the generator chain some number of generators up or down. | |||
Since the generator chain and logarithmic pitch space are both 1-dimensional, it may be helpful to graph them together in 2 dimensions. Here is a diagram for sensi[8], an octatonic [[ | Another common way to view the tones of an MOS scale is as points in logarithmic pitch space, with larger gaps between points representing larger intervals and smaller gaps between points representing smaller intervals. Then we see that our scale has large steps and small steps and intervals that are composed of some stackings of large and small steps. It is not obvious, looking at the generator chain or looking at the tones in pitch space, what the relationship is. Indeed, it is different for different MOS scales—an "L" will not always represent the same number of generators up or down when we move to a different scale. | ||
Since the generator chain and logarithmic pitch space are both 1-dimensional, it may be helpful to graph them together in 2 dimensions. Here is a diagram for sensi[8], an octatonic [[3L 5s]] MOS scale with a generator of between 442 and 445{{nbhsp}}¢. The x-axis shows the generator chain and the y-axis shows the nine tones (eight plus octave) in logarithmic pitch space. You can see that the vertical lines are evenly-spaced (since every generator is the same), while the horizontal lines have large and small gaps, representing the large and small steps of sensi[8]. | |||
[[File:map_of_sensi-8-.png|alt=Map of Sensi[8].png]] | [[File:map_of_sensi-8-.png|alt=Map of Sensi[8].png]] | ||
And another way to visualize MOS scales is hierarchically. Every MOS scale with 3 or more tones: | And another way to visualize MOS scales is hierarchically. Every MOS scale with 3 or more tones: | ||
* contains at least one MOS scale with fewer tones (and in fact, more than one instance of it), and | |||
* is contained (more than once) in either | |||
** an MOS scale with more tones, or | |||
** an equal scale (when {{nowrap|L:s {{=}} 2:1}}). | |||
Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where L:s=2:1, which means that there are no more MOS scales to be had -- the next stopping point is at a complete chromatic scale of [[37edo]]. | Below is a diagram showing four MOS scales in logarithmic pitch space generated by 7\[[37edo]] (approx. 227¢). Each contains the ones above it and is contained by the ones below it. There are additional MOS scales that would appear above the first line with 1, 2, 3, 4, and 5 tones, but they have been omitted. The bottom line is a case where L:s=2:1, which means that there are no more MOS scales to be had -- the next stopping point is at a complete chromatic scale of [[37edo]]. | ||