Possible Sagittal notation for starling temperament

Here are a few candidates for Sagittal representation of starling temperament (rank 3, tempers out 126/125)

Generator mapping (2/1, 4/3, 16/15):
[<1, 2, 2, 1], <0, -1, 1, 5], <0, 0, -1, -3]>

(-1, +2, +2) |( [10, -6, 1, -1>
(+2, -5, +1) /| [-4, 4, -1>
(+1, -3, +3) |) [6, -2, 0, -1>
(+2, -6, +6) (/| [12, -4, 0, -2>
(+1, -1, -6) |\) [-23, 11, 0, 2>
(+0, +1, -4) (|\ [-13, 5, 1, 1>
(-1, +3, -2) )/|| [-2, 1, -1, 1>
(+2, -4, -3) ||) [-17, 9, 0, 1>
(+1, -2, -1) ||\ [-7, 3, 1>
(+3, -7, +0) /||\ [-11, 7>

The big gap in this is anything with +5 or -5 of the 16/15 generator. There's )/|\ [-3, 4, 1, -2> which maps to (+5, -13, +5), and even )/|||\ [-14, 11, 1, -2> which is (+8, -20, +5), but nothing else.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

* Just some quick questions about this post:

> Here are a few candidates for Sagittal representation of starling
> temperament (rank 3, tempers out 126/125)
>
> Generator mapping (2/1, 4/3, 16/15):
> [<1, 2, 2, 1], <0, -1, 1, 5], <0, 0, -1, -3]>

* I see this is M^-1, except you leave off <0,0,0,1](Last column)
How is this 4 X 3 matrix used? This is assuming M is the three
generators plus 126/125. (4 X 4).

> (-1, +2, +2) |( [10, -6, 1, -1>
> (+2, -5, +1) /| [-4, 4, -1>
> (+1, -3, +3) |) [6, -2, 0, -1>
> (+2, -6, +6) (/| [12, -4, 0, -2>
> (+1, -1, -6) |\) [-23, 11, 0, 2>
> (+0, +1, -4) (|\ [-13, 5, 1, 1>
> (-1, +3, -2) )/|| [-2, 1, -1, 1>
> (+2, -4, -3) ||) [-17, 9, 0, 1>
> (+1, -2, -1) ||\ [-7, 3, 1>
> (+3, -7, +0) /||\ [-11, 7>

* I see 126/125 gets tempered out, as stated. Remind me please,
why M is the three generators (not tempered) plus the tempered
comma. I used to know! Thanks. I remember Mv converts a val,
and uM^-1 converts a monzo.

> The big gap in this is anything with +5 or -5 of the 16/15
generator.
> There's )/|\ [-3, 4, 1, -2> which maps to (+5, -13, +5), and
even )/|||\
> [-14, 11, 1, -2> which is (+8, -20, +5), but nothing else.
>

Paul G Hjelmstad wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> > * Just some quick questions about this post:
> >> Here are a few candidates for Sagittal representation of starling >> temperament (rank 3, tempers out 126/125)
>>
>> Generator mapping (2/1, 4/3, 16/15):
>> [<1, 2, 2, 1], <0, -1, 1, 5], <0, 0, -1, -3]>
> > * I see this is M^-1, except you leave off <0,0,0,1](Last column)
> How is this 4 X 3 matrix used? This is assuming M is the three > generators plus 126/125. (4 X 4). This is a generator mapping, one of many possible for the temperament. You take the prime exponent vector of the interval you want to approximate (as a row vector) and multiply by the generator mapping, resulting in a row vector with 3 elements. Then you'll want a tuning map (I believe it's called, someone correct me if the terminology isn't quite right) -- a column vector that specifies the tuning of each of the generators: e.g. <1199.01, 497.63, 107.04]. Take the inner product -- each element of the tuning map is multiplied by the corresponding element of the result and added to produce the tempered interval.

E.g. say you want to approximate 5/4. The monzo is [-2 0 1 0>, resulting in [0, 1, -1> -- i.e., the approximation of 5/4 is the approximation of 4/3 minus the approximation of 16/15 (as you might expect). From <1199.01, 497.63, 107.04 | 0, 1, -1>, you get 390.59 cents as the approximation of 5/4.

You can find the starling approximation of 7/4 the same way. From [-2, 0, 0, 1>, when you multiply by the generator mapping you get [-1, 5, -3>: down an octave, up 5 perfect fourths, and then down 3 semitones, or 968.03 cents. This is the same as the approximation of 125/72. Then of course if you put in [1, 2, -3, 1>, which represents 126/125, the result is [0, 0, 0> or 0 cents.

Another way to explain this is that the generator mapping specifies the tuning of the prime intervals 2/1, 3/1, 5/1, and 7/1. The advantage of describing it that way is that you don't need to explain about matrices and how to multiply them (which isn't a subject I'd expect most musicians to be familiar with). For that purpose, the form used in Paul Erlich's paper and Graham Breed's temperament finder is more useful, but one can easily be converted to the other.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Paul G Hjelmstad wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@>
wrote:
> >
> > * Just some quick questions about this post:
> >
> >> Here are a few candidates for Sagittal representation of
starling
> >> temperament (rank 3, tempers out 126/125)
> >>
> >> Generator mapping (2/1, 4/3, 16/15):
> >> [<1, 2, 2, 1], <0, -1, 1, 5], <0, 0, -1, -3]>
> >
> > * I see this is M^-1, except you leave off <0,0,0,1](Last column)
> > How is this 4 X 3 matrix used? This is assuming M is the three
> > generators plus 126/125. (4 X 4).
>
> This is a generator mapping, one of many possible for the
temperament.
> You take the prime exponent vector of the interval you want to
> approximate (as a row vector) and multiply by the generator
mapping,
> resulting in a row vector with 3 elements. Then you'll want a
tuning map
> (I believe it's called, someone correct me if the terminology isn't
> quite right) -- a column vector that specifies the tuning of each
of the
> generators: e.g. <1199.01, 497.63, 107.04]. Take the inner product -
-
> each element of the tuning map is multiplied by the corresponding
> element of the result and added to produce the tempered interval.
>
> E.g. say you want to approximate 5/4. The monzo is [-2 0 1 0>,
resulting
> in [0, 1, -1> -- i.e., the approximation of 5/4 is the
approximation of
> 4/3 minus the approximation of 16/15 (as you might expect). From
> <1199.01, 497.63, 107.04 | 0, 1, -1>, you get 390.59 cents as the
> approximation of 5/4.
>
> You can find the starling approximation of 7/4 the same way. From [-
2,
> 0, 0, 1>, when you multiply by the generator mapping you get [-1,
5,
> -3>: down an octave, up 5 perfect fourths, and then down 3
semitones, or
> 968.03 cents. This is the same as the approximation of 125/72. Then
of
> course if you put in [1, 2, -3, 1>, which represents 126/125, the
result
> is [0, 0, 0> or 0 cents.
>
> Another way to explain this is that the generator mapping specifies
the
> tuning of the prime intervals 2/1, 3/1, 5/1, and 7/1. The advantage
of
> describing it that way is that you don't need to explain about
matrices
> and how to multiply them (which isn't a subject I'd expect most
> musicians to be familiar with). For that purpose, the form used in
Paul
> Erlich's paper and Graham Breed's temperament finder is more
useful, but
> one can easily be converted to the other.

Thanks. I have a big spreadsheet that has grown over the years,
where I use matrices, adjutants, inverses, determinants etc. I have
always wondered how 1 0 0 0 was added, to make a matrix square, now
it is a question of how 0 0 0 1 is dropped off, but it works, and not
too mysterious actually. (Linear Algebra isn't bad, and it is real
important with Group Theory and Character Tables and such, so I am
being forced to learn it), but it is cool you have a way for people
to do these tuning things without having to know it, directly.

PGH