TAMNAMS/Appendix: Difference between revisions

Line 22:

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 {{nowrap|[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 {{nowrap|L + s}}.

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 {{nowrap|''x''L + ''y''s}}, where ''x'' and ''y'' are the number of large and small steps in the scale, and where L and s are from the step ratio L/s (where L and s 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 {{nowrap|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.

@@ Line 22: / Line 22: @@
 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 {{nowrap|[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 {{nowrap|L + s}}.
+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 {{nowrap|''x''L + ''y''s}}, where ''x'' and ''y'' are the number of large and small steps in the scale, and where L and s are from the step ratio L/s (where L and s 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 {{nowrap|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&mdash;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&mdash;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.