A nearly-proper, nearly-stable scale

Using Carl's definitions, "Lumma stability" is 1 when a scale is
proper, and "Lumma propriety" is positive for proper scales. Below is
a "nearly stable" scale, in the sense that it has Lumma stability of
515/531, or 97% stability. Scala therefore rates it as having a 3%
"impropriety factor". It has Lumma propriety of -2/531, not very much
negative.

I found it by checking all 1716 scales you get by taking six major
thirds, seven minor thirds and a subminor third in a circle, in 531 et.
This gave two inversely related scales with maximum stability, and
this one is the more major-oriented one.

! hemifamcyc.scl
Hemifamity cycle of thirds scale, nearest to proper
14
!
85.875706
180.790960
291.525424
386.440678
402.259887
497.175141
607.909605
702.824859
788.700565
883.615819
969.491525
994.350282
1105.084746
1200.000000
! [38 80 129 171 178 220 269 311 349 391 429 440 489 531]

At 11:23 PM 6/13/2006, you wrote:
>Using Carl's definitions, "Lumma stability" is 1 when a scale is
>proper,

Yes.

>and "Lumma propriety" is positive for proper scales.

Lumma propriety should lie between 0 and 1, as it is the portion
of the octave which is not "covered" by scale degrees.

I should note that I've never checked Scala's output for correctness.
I've seen a lot of the same values for a lot of different scales,
which troubles me.

>I found it by checking all 1716 scales you get by taking six major
>thirds, seven minor thirds and a subminor third in a circle, in 531 et.
>This gave two inversely related scales with maximum stability, and
>this one is the more major-oriented one.
>
>! hemifamcyc.scl
>Hemifamity cycle of thirds scale, nearest to proper
>14
>!
>85.875706
>180.790960
>291.525424
>386.440678
>402.259887
>497.175141
>607.909605
>702.824859
>788.700565
>883.615819
>969.491525
>994.350282
>1105.084746
>1200.000000
>! [38 80 129 171 178 220 269 311 349 391 429 440 489 531]

Interesting.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 11:23 PM 6/13/2006, you wrote:
> >Using Carl's definitions, "Lumma stability" is 1 when a scale is
> >proper,
>
> Yes.
>
> >and "Lumma propriety" is positive for proper scales.
>
> Lumma propriety should lie between 0 and 1, as it is the portion
> of the octave which is not "covered" by scale degrees.

Lumma propriety, for proper scales, is found by summing the lengths of
all the interval classes, dividing by the number of scale steps, and
subtracting that from 1. Since that is propriety for proper scales, it
arguably ought to be propriety for improper scales, on the grounds
that it's the simplest formula and tells us something interesting in
all cases. But I suppose one solution is to call the function above by
some other name, and then take note of the fact that when the scales
in question are proper, it is the same as Lumma propriety. But I may
not bother with Lumma propriety unless you will explain why you think
it is important--it would be nice if you put your paper up on the web.

>> >Using Carl's definitions, "Lumma stability" is 1 when a scale is
>> >proper,
>>
>> Yes.
>>
>> >and "Lumma propriety" is positive for proper scales.
>>
>> Lumma propriety should lie between 0 and 1, as it is the portion
>> of the octave which is not "covered" by scale degrees.
>
>Lumma propriety, for proper scales, is found by summing the lengths of
>all the interval classes, dividing by the number of scale steps, and
>subtracting that from 1. Since that is propriety for proper scales, it
>arguably ought to be propriety for improper scales, on the grounds
>that it's the simplest formula and tells us something interesting in
>all cases. But I suppose one solution is to call the function above by
>some other name, and then take note of the fact that when the scales
>in question are proper, it is the same as Lumma propriety. But I may
>not bother with Lumma propriety unless you will explain why you think
>it is important--it would be nice if you put your paper up on the web.

http://lumma.org/tuning/FMP/paper.txt
http://lumma.org/tuning/FMP/chart.txt

-Carl