TAMNAMS/Appendix: Difference between revisions

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=== Derivation of the step ratio names ===

The idea is to start with the simplest ratios ({{nowrap|L/s {{=}} 1/0}} and {{nowrap|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 ''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.)

* Applying the mediant to the starting intervals 1/0 and 1/1 gives {{nowrap|(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 ''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 {{nowrap|L/s {{=}} 1/1}} represents L and s being equal in size, it is called ''equalized''.

* As {{nowrap|L/s {{=}} 1/0}} represents {{nowrap|s {{=}} 0}}, it is called ''collapsed'', as the small scale steps collapse to zero cents and disappear.

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 === Derivation of the step ratio names ===
 The idea is to start with the simplest ratios ({{nowrap|L/s {{=}} 1/0}} and {{nowrap|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 ''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.)
+* Applying the mediant to the starting intervals 1/0 and 1/1 gives {{nowrap|(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 ''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 {{nowrap|L/s {{=}} 1/1}} represents L and s being equal in size, it is called ''equalized''.
 * As {{nowrap|L/s {{=}} 1/0}} represents {{nowrap|s {{=}} 0}}, it is called ''collapsed'', as the small scale steps collapse to zero cents and disappear.