Minimal filled scale

Suppose we have a linear temperament with octave period. A chord-type
in this temperament is a set of generators. A question we might ask is
what is the what is the cardinality of the smallest set of contiguous
generators which arise from contiguous generator translates of the
chord--the minimal filled scale for the chord. This doesn't depend on
the temperament, but only on the chord, considered as a set or list.

Maple code for it is this:

mfs := proc(l)
# minimal filled scale from chord l
local i, s, u;
u := sort(convert(l,list));
s := -1;
for i to nops(u)-1 do
s := max(s,u[i+1]-u[i]) od;
s+u[nops(u)]-u[1] end:

This program sorts the scale (converting it first to a list if it
starts out as a set or array Maple data type.) It then finds the
maximum chord interval in terms of generator steps, and adds that to
the span of the scale--the difference between the least and greatest
element in terms of generator steps. Extending it to non-octave
periods involves deciding what definition is then best.

The minimal filled scale for septimal miracle is Miracle[19], and for
11-limit miracle is Canasta (Miracle[31].) For 5-limit meantone we get
the diatonic scale (Meantone[7]), and in the 7-limit, Meantone[16].
And so on and so forth...

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Suppose we have a linear temperament with octave period. A chord-type
> in this temperament is a set of generators. A question we might ask is
> what is the what is the cardinality of the smallest set of contiguous
> generators which arise from contiguous generator translates of the
> chord--the minimal filled scale for the chord. This doesn't depend on
> the temperament, but only on the chord, considered as a set or list.
>

Something like a transitive hull?

For some reason this didn't get posted; I'll try again.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The minimal filled scale for septimal miracle is Miracle[19], and for
> 11-limit miracle is Canasta (Miracle[31].) For 5-limit meantone we get
> the diatonic scale (Meantone[7]), and in the 7-limit, Meantone[16].

This was less than clear, because I forgot to say I was using the
major p-limit chord as my defining chord for this. Probably I should
have used odd limit for 11-limit meantone, including 9, which would
have given me Miracle[28].

One definition for fractional-octave periods is simply to multiply
each generator by the number of periods in an octave, so the 7-limit
minimal filled scale for pajara would be what you get by putting
{0,2,-4,-4} into the program I gave, which would be Paul's favorite
Pajara[10]; going to the 9-limit would give Pajara[12].

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > Suppose we have a linear temperament with octave period. A chord-type
> > in this temperament is a set of generators. A question we might ask is
> > what is the what is the cardinality of the smallest set of contiguous
> > generators which arise from contiguous generator translates of the
> > chord--the minimal filled scale for the chord. This doesn't depend on
> > the temperament, but only on the chord, considered as a set or list.
> >
>
> Something like a transitive hull?

I don't see a connection.

Greetings from beautiful Portland!

>Suppose we have a linear temperament with octave period. A
>chord-type in this temperament is a set of generators. A
>question we might ask is what is the what is the cardinality
>of the smallest set of contiguous generators which arise from
>contiguous generator translates of the chord--the minimal
>filled scale for the chord.

"Continuous generator translates"??

>This doesn't depend on the temperament, but only on the chord,
>considered as a set or list.

Huh? It must depend on the mapping.

>The minimal filled scale for septimal miracle is Miracle[19],
>and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit
>meantone we get the diatonic scale (Meantone[7]), and in the
>7-limit, Meantone[16]. And so on and so forth...

How is this different from Graham complexity?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> Greetings from beautiful Portland!
>
> >Suppose we have a linear temperament with octave period. A
> >chord-type in this temperament is a set of generators. A
> >question we might ask is what is the what is the cardinality
> >of the smallest set of contiguous generators which arise from
> >contiguous generator translates of the chord--the minimal
> >filled scale for the chord.
>
> "Continuous generator translates"??

Contiguous. I mean if we have for example a chord [0 1 4 10], we take
[1 2 5 11] [1 3 6 12] etc. until we've filled all the holes, and every
note is harmonizable by at least one such chord.

> >This doesn't depend on the temperament, but only on the chord,
> >considered as a set or list.
>
> Huh? It must depend on the mapping.

See above--it depends on what is mapped to (say [0 1 4 10] for 7-limit
meantone) but not at all on where it is mapped from.

> >The minimal filled scale for septimal miracle is Miracle[19],
> >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit
> >meantone we get the diatonic scale (Meantone[7]), and in the
> >7-limit, Meantone[16]. And so on and so forth...
>
> How is this different from Graham complexity?

How is it the same?

>> Greetings from beautiful Portland!
>>
>> >Suppose we have a linear temperament with octave period. A
>> >chord-type in this temperament is a set of generators. A
>> >question we might ask is what is the what is the cardinality
>> >of the smallest set of contiguous generators which arise from
>> >contiguous generator translates of the chord--the minimal
>> >filled scale for the chord.
>>
>> "Continuous generator translates"??
>
>Contiguous.

Weird; high-level typo; I read it correctly.

>I mean if we have for example a chord [0 1 4 10], we take
>[1 2 5 11] [1 3 6 12] etc.

You mean [2 3 6 12]?

>until we've filled all the holes,

I still don't get it. You're harmonizing every note of the
original chord?

>and every
>note is harmonizable by at least one such chord.

The original chord has this property...

>> >The minimal filled scale for septimal miracle is Miracle[19],
>> >and for 11-limit miracle is Canasta (Miracle[31].) For 5-limit
>> >meantone we get the diatonic scale (Meantone[7]), and in the
>> >7-limit, Meantone[16]. And so on and so forth...
>>
>> How is this different from Graham complexity?
>
>How is it the same?

I was hoping an explanation of the difference would help us
understand it.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >I mean if we have for example a chord [0 1 4 10], we take
> >[1 2 5 11] [1 3 6 12] etc.
>
> You mean [2 3 6 12]?

Right.

> >until we've filled all the holes,
>
> I still don't get it. You're harmonizing every note of the
> original chord?

No, I'm harmonizing everything with translates of the chord in a
minimal contiguous-generator scale containing the chord.

> >and every
> >note is harmonizable by at least one such chord.
>
> The original chord has this property...

No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10.
2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the
numbers from 0 to 15, every one of them has a major tetrad to
harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from
0 to 5 is {0..15}; no smaller value than 5 will work.

>> >I mean if we have for example a chord [0 1 4 10], we take
>> >[1 2 5 11] [1 3 6 12] etc.
>>
>> You mean [2 3 6 12]?
>
>Right.
>
>> >until we've filled all the holes,
>>
>> I still don't get it. You're harmonizing every note of the
>> original chord?
>
>No, I'm harmonizing everything with translates of the chord in a
>minimal contiguous-generator scale containing the chord.
>
>> >and every
>> >note is harmonizable by at least one such chord.
>>
>> The original chord has this property...
>
>No, the numbers from 0 to 10 only find harmonies for 0, 1, 4 and 10.
>2, 3, 5, 6, 7, 8 and 9 have no major tetrad. If, however, I take the
>numbers from 0 to 15, every one of them has a major tetrad to
>harmonize it. The union of the sets {i,i+1,i+4,i+10} as i ranges from
>0 to 5 is {0..15}; no smaller value than 5 will work.

Got it. Cool.

-Carl