MOS scale: Difference between revisions

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Sometimes, scales are defined with respect to a period and an additional "equivalence interval," considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOS's'''. MOS's in which the equivalence interval is equal to the period are sometimes called '''Strict MOS's'''. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of ~~Wilsons~~' concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.

As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where ''s'' = 0. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. On this basis, a system has been proposed by a few users, detailed in the next section.

== Step ratio spectrum ==

=== Derivation ===

The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the [[Wikipedia: Mediant (mathematics)|mediant]] to adjacent fractions. Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the "quintessential" ("quintess." or "essential" for short) or "basic" tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.) As L/s = 1/1 represents L and s being equal in size, it is called "equalized". As L/s = 1/0 represents s = 0, it is called "paucitonic", meaning "few tones", as the resulting scale is also equalized but with fewer tones per period than expected. The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the "soft" tuning, and analogously, the mediant of 2/1 and 1/0, 3/1, is called the "hard" tuning. Thus you can say that a step ratio tuning is "hard of" or "soft of" another step ratio tuning. To get something between soft and basic we take the mediant again and get 5/3 for "semisoft", and analogously 5/2 for "semihard". To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for "superhard" and analogously 4/3 for "supersoft". Something softer than supersoft is "ultrasoft". Something harder than superhard is "ultrahard". Something between soft and supersoft is "parasoft", as "para-" means both "beyond" and "next to". Something between hard and superhard is "parahard". The reasoning for the "para- super- ultra-" progression (note that "super-" is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because "parapythagorean" is between "pythagorean" and "superpythagorean". Something between soft and basic is "hyposoft" as it is less soft than soft. Something between hard and basic is "hypohard" for the same reason. Between semisoft and quintessential is "minisoft" and between semihard and quintessential is "minihard". Finally, between soft and semisoft is "quasisoft" as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix "quasi-", and between hard and semihard is "quasihard" for the same reason. This results in the "basic spectrum" below - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding [[EPD||Equal Pitch Division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum, detailed after. (Note: Extended spectrum not yet written here - soon. Just wanted to save first. This note will be removed after.)

=== Basic spectrum ===

'''Equalized''': L/s = 1/1 (trivial/pathological)

::: ('''Ultrasoft''' range here, may also be called "pseudoequalized" if especially close to paucitonic.)

:: '''Supersoft''': L/s = 4/3

::: ('''Parasoft''' range here.)

: '''Soft''': L/s = 3/2

::: (Beginning of '''hyposoft''' range here.)

::: ('''Quasisoft''' range here.)

:: '''Semisoft''': L/s = 5/3

::: ('''Minisoft''' range here.)

::: (End of '''hyposoft''' range here.)

'''Quintesssential''': L/s = 2/1

::: (Beginning of '''hypohard''' range here.)

::: ('''Minihard''' range here.)

:: '''Semihard''': L/s = 5/2

::: ('''Quasihard''' range here.)

::: (End of '''hypohard''' range here.)

: '''Hard''': L/s = 3/1

::: ('''Parahard''' range here.)

:: '''Superhard''': L/s = 4/1

::: ('''Ultrahard''' range here, may also be called "pseudopaucitonic" if especially close to paucitonic.)

'''Paucitonic''': L/s = 1/0 = infinity (trivial/pathological)

== Mathematics ==

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 Sometimes, scales are defined with respect to a period and an additional "equivalence interval," considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOS's'''. MOS's in which the equivalence interval is equal to the period are sometimes called '''Strict MOS's'''. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.
-With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilsons' concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.
+With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional Evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.
+As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where ''s'' = 0. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. On this basis, a system has been proposed by a few users, detailed in the next section.
+== Step ratio spectrum ==
+=== Derivation ===
+The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the [[Wikipedia: Mediant (mathematics)|mediant]] to adjacent fractions. Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the "quintessential" ("quintess." or "essential" for short) or "basic" tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.) As L/s = 1/1 represents L and s being equal in size, it is called "equalized". As L/s = 1/0 represents s = 0, it is called "paucitonic", meaning "few tones", as the resulting scale is also equalized but with fewer tones per period than expected. The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the "soft" tuning, and analogously, the mediant of 2/1 and 1/0, 3/1, is called the "hard" tuning. Thus you can say that a step ratio tuning is "hard of" or "soft of" another step ratio tuning. To get something between soft and basic we take the mediant again and get 5/3 for "semisoft", and analogously 5/2 for "semihard". To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for "superhard" and analogously 4/3 for "supersoft". Something softer than supersoft is "ultrasoft". Something harder than superhard is "ultrahard". Something between soft and supersoft is "parasoft", as "para-" means both "beyond" and "next to". Something between hard and superhard is "parahard". The reasoning for the "para- super- ultra-" progression (note that "super-" is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because "parapythagorean" is between "pythagorean" and "superpythagorean". Something between soft and basic is "hyposoft" as it is less soft than soft. Something between hard and basic is "hypohard" for the same reason. Between semisoft and quintessential is "minisoft" and between semihard and quintessential is "minihard". Finally, between soft and semisoft is "quasisoft" as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix "quasi-", and between hard and semihard is "quasihard" for the same reason. This results in the "basic spectrum" below - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding [[EPD||Equal Pitch Division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum, detailed after. (Note: Extended spectrum not yet written here - soon. Just wanted to save first. This note will be removed after.)
+=== Basic spectrum ===
+'''Equalized''': L/s = 1/1 (trivial/pathological)
+::: ('''Ultrasoft''' range here, may also be called "pseudoequalized" if especially close to paucitonic.)
+:: '''Supersoft''': L/s = 4/3
+::: ('''Parasoft''' range here.)
+: '''Soft''': L/s = 3/2
+::: (Beginning of '''hyposoft''' range here.)
+::: ('''Quasisoft''' range here.)
+:: '''Semisoft''': L/s = 5/3
+::: ('''Minisoft''' range here.)
+::: (End of '''hyposoft''' range here.)
+'''Quintesssential''': L/s = 2/1
+::: (Beginning of '''hypohard''' range here.)
+::: ('''Minihard''' range here.)
+:: '''Semihard''': L/s = 5/2
+::: ('''Quasihard''' range here.)
+::: (End of '''hypohard''' range here.)
+: '''Hard''': L/s = 3/1
+::: ('''Parahard''' range here.)
+:: '''Superhard''': L/s = 4/1
+::: ('''Ultrahard''' range here, may also be called "pseudopaucitonic" if especially close to paucitonic.)
+'''Paucitonic''': L/s = 1/0 = infinity (trivial/pathological)
 == Mathematics ==