Lumma stability

a kind of refinement to Rothenberg stability 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 :

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%.

[from Manuel Op de Coul, Scala file "tips.par"; view in Scala with command "SHOW DATA"]

See also:

Rothenberg stability
Rothenberg efficiency.
Rothenberg redundancy
Lumma impropriety

Updated: 2002.2.4