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|2:1 ≤ L:s ≤ 1:0
|2:1 ≤ L:s ≤ 1:0
|}
|}
=== 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 [[mediant]] (aka Farey addition) 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.
* 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".
There are also tertiary names beyond the above:
* Anything softer than supersoft is "ultrasoft," and anything 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".
* 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.
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".
This results in the "central 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 notes per period in an [[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.


=== Central spectrum ===
=== Central spectrum ===
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'''Paucitonic''': L/s = 1/0 = infinity (trivial/pathological)
'''Paucitonic''': L/s = 1/0 = infinity (trivial/pathological)
=== Extending the spectrum's edges ===
Extending the spectrum builds on the central spectrum and relies on a few key observations. Firstly, as periods and MOSSes come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of "simple" tunings for the step ratio for a given MOS pattern and period, the notion of "simple" used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to MOS patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s. The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the "hard" end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: MOS patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.
The most obvious adjustment to the edges is to draw a distinction between "ultrasoft" and "pseudoequalized" by adding a step ratio corresponding to "semiequalized", and between "ultrahard" and "pseudopaucitonic" by adding a step ratio corresponding to "semipaucitonic". Thus:
'''Ultrasoft''' is between '''supersoft''' and '''semiequalized''' and '''pseudoequalized''' is between '''semiequalized''' and '''equalized'''.
'''Ultrahard''' is between '''superhard''' and '''semipaucitonic''', and '''pseudopaucitonic''' is between '''semipaucitonic''' and '''paucitonic'''.
Then all that's left is to decide what the step ratios for semipaucitonic and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for '''semipaucitonic''' we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as '''extrahard'''. Although L/s = 10/1 for '''semipaucitonic''' may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most "equally spaced" continuing on from 6/1 for the same reason that L/s = 6/5 is the most "equally spaced". Note that while the range from '''superhard''' to '''semipaucitonic''' is '''ultrahard''', the region may be split into two sub-ranges:
'''superhard''' (L/s=4/1) to '''extrahard''' (L/s=6/1) is '''hyperhard''' (4 < L/s < 6).
'''extrahard''' (L/s=6/1) to '''semipaucitonic''' (L/s=10/1) is '''clustered''' (6 < L/s < 10).
With the inclusion of these 3 new L/s rations nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed below, just for the regions affected to avoid repetition.


=== Extended spectrum ===
=== Extended spectrum ===
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| [[9L 1s]] || || || ||
| [[9L 1s]] || || || ||
|}
|}
<references/>
== More on step ratio names ==
=== 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 [[mediant]] (aka Farey addition) 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.
 
* 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".
 
There are also tertiary names beyond the above:
 
* Anything softer than supersoft is "ultrasoft," and anything 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".
 
* 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.
 
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".
 
This results in the "central 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 notes per period in an [[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.
=== Extending the spectrum's edges ===
Extending the spectrum builds on the central spectrum and relies on a few key observations. Firstly, as periods and MOSSes come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of "simple" tunings for the step ratio for a given MOS pattern and period, the notion of "simple" used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to MOS patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s. The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the "hard" end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: MOS patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.
 
The most obvious adjustment to the edges is to draw a distinction between "ultrasoft" and "pseudoequalized" by adding a step ratio corresponding to "semiequalized", and between "ultrahard" and "pseudopaucitonic" by adding a step ratio corresponding to "semipaucitonic". Thus:
 
'''Ultrasoft''' is between '''supersoft''' and '''semiequalized''' and '''pseudoequalized''' is between '''semiequalized''' and '''equalized'''.
 
'''Ultrahard''' is between '''superhard''' and '''semipaucitonic''', and '''pseudopaucitonic''' is between '''semipaucitonic''' and '''paucitonic'''.
 
Then all that's left is to decide what the step ratios for semipaucitonic and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for '''semipaucitonic''' we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as '''extrahard'''. Although L/s = 10/1 for '''semipaucitonic''' may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most "equally spaced" continuing on from 6/1 for the same reason that L/s = 6/5 is the most "equally spaced". Note that while the range from '''superhard''' to '''semipaucitonic''' is '''ultrahard''', the region may be split into two sub-ranges:
 
'''superhard''' (L/s=4/1) to '''extrahard''' (L/s=6/1) is '''hyperhard''' (4 < L/s < 6).
 
'''extrahard''' (L/s=6/1) to '''semipaucitonic''' (L/s=10/1) is '''clustered''' (6 < L/s < 10).
 
With the inclusion of these 3 new L/s rations nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed below, just for the regions affected to avoid repetition.
 
[[Category:MOS]]
[[Category:MOS]]
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