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Encyclopedia of Microtonal Music Theory

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Lumma stability

[Manuel Op de Coul, Scala file "tips.par"; Scala command "Help|Tips|Show all"]

A kind of refinement to Rothenberg stability [defined by Carl Lumma] that also takes the interval's sizes into account. It is defined as the portion of the pitch spectrum within one (formal) octave which is not "covered" by the difference of its intervals for each interval class.

For example, an equal tempered scale has stability 1 by definition because for each interval class there is only one interval, leaving the whole spectrum "uncovered". But take the pentatonic mode 2 3 2 2 3 [of 12-edo]:

1 step:  2  3  2  2  3
2 steps: 5  5  4  5  5
3 steps: 7  7  7  7  8
4 steps: 9 10  9 10 10
		

Now imagine the following line, each tick representing 100 cents:

|         <-1s->    <-2s->         <-3s->    <-4s->         |
|----|----|----|----|----|----|----|----|----|----|----|----|
0    1    2    3    4    5    6    7    8    9   10   11   12
		

So 8/12th of the line is not covered, giving stability 66.67%.

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