A equal-beating temperament question

Hello all,

Is there a general formula for determining the rate at which all fifths would
beat if they distribute the syntonic comma (or commas) equally, but 'equally'
in the sense of 'equal-beating' and not 'logarithmically'.

In other words, if I tune, in pure fifths, C-G-D-A-E(pure), and A-E ends up
beating at x/sec, and I want to know how fast a tempered C-G, G-D as a
fourth, D-A, and A-E as a fourth will be if they distribute the comma in an
equal beating way, whose rate we'll call 'y', is there a formula for getting
'y' from 'x'? Know that I already know how to find the basic beat rate for
any interval given their hertz frequencies...

This opens up another question. It appears scala defines an equal beating
meantone as one where the basis-pitch fifth's (i.e. C-G) beat rate in a
mathematically correct tuning is copied up and down the circle of fifths,
where I had always understood it to mean 'distribute the syntonic comma
equally among x fifths such that each fifth or fourth beats equally'. From
some basic calculations, I can see that the two definitions do not come up
with the same beat rate. In any event, if I'm wrong about my understanding of
what makes an 'equal-beating meantone', one still can define a whole family
of meantone tunings based on THIS definition.....

All best,
Aaron.

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

Hello Aaron,

>Is there a general formula for determining the rate at which all fifths
would
>beat if they distribute the syntonic comma (or commas) equally, but
'equally'
>in the sense of 'equal-beating' and not 'logarithmically'.

Finding equal beating temperaments fifths involves finding roots of a
polynomial
of some order. So, in the general case when the order of the polynomial is
higher than three, there is no formula in closed form and you need to find
the root numerically. (Somebody correct me if I'm wrong).

>It appears scala defines an equal beating
>meantone as one where the basis-pitch fifth's (i.e. C-G) beat rate in a
>mathematically correct tuning is copied up and down the circle of fifths,

I believe the word "meantone" is mentioned nowhere in relation to this
type of scale.
Equal beating meantones can be calculated with Tools->Solve equal beating
equation. The scales you refer to are created by PYTHAGOREAN/EQUALBEATS and
are irregular temperaments.

Manuel

On Tuesday 07 October 2003 09:37 am, Manuel Op de Coul wrote:
> Hello Aaron,
>
> >Is there a general formula for determining the rate at which all fifths
>
> would
>
> >beat if they distribute the syntonic comma (or commas) equally, but
>
> 'equally'
>
> >in the sense of 'equal-beating' and not 'logarithmically'.
>
> Finding equal beating temperaments fifths involves finding roots of a
> polynomial
> of some order. So, in the general case when the order of the polynomial is
> higher than three, there is no formula in closed form and you need to find
> the root numerically. (Somebody correct me if I'm wrong).
>
> >It appears scala defines an equal beating
> >meantone as one where the basis-pitch fifth's (i.e. C-G) beat rate in a
> >mathematically correct tuning is copied up and down the circle of fifths,
>
> I believe the word "meantone" is mentioned nowhere in relation to this
> type of scale.
> Equal beating meantones can be calculated with Tools->Solve equal beating
> equation. The scales you refer to are created by PYTHAGOREAN/EQUALBEATS and
> are irregular temperaments.
>
> Manuel

Manuel-

Hi! Yes, it's not explicitly mentioned as such, but looking at your data, when
you say compare, in scala 'showbeats', the beat rates of the fifths of 1/4
comma meantone with 1/4-eqbeating meantone (meanquareq.scl), we see that the
beatrate for the C-G of the 1/4 comma normal meantone is applied as the
beatrate for all of the fifths of the 1/4 comma equal-beating meantone.

What I was getting at was taking a just C-E, and making the fifths (and
fourths) of C-G-D-A-E all equal beat the syntonic comma. From what you've
described, I'm not sure how scala would do this..at least, you'd need to give
more detailed instructions about how to do it. Would one change the formal
octave to a 17th, or a 3rd? It's not clear how to proceed... From there, I'd
tune the usual meantone bearing plan....up and down major thirds from C, G,
D, A, and up a third from E......

Best,
Aaron.

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> Finding equal beating temperaments fifths involves finding roots of
a
> polynomial
> of some order. So, in the general case when the order of the
polynomial is
> higher than three, there is no formula in closed form and you need
to find
> the root numerically. (Somebody correct me if I'm wrong).

I'm fairly sure this is off-topic, but here goes. Degree four and
less polynomials are solveable, so their roots can be expressed
purely algebraically (for instance in terms of radicals, though this
really isn't the best choice.) Degree five and higher requires
introduction of new functions, which could even be algebraic but
which in practice have mostly been transcendental. Degree five, for
instance, can be solved by adding the Dedekind eta function to the
list of allowable functions. However, there are always special cases
in all degrees which are solvable (when the Galois group of the
splitting field over the base field is a solvable group.)

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
>
> Hello all,
>
> Is there a general formula for determining the rate at which all
fifths would
> beat if they distribute the syntonic comma (or commas) equally,
but 'equally'
> in the sense of 'equal-beating' and not 'logarithmically'.
>
> In other words, if I tune, in pure fifths, C-G-D-A-E(pure), and A-E
ends up
> beating at x/sec, and I want to know how fast a tempered C-G, G-D
as a
> fourth, D-A, and A-E as a fourth will be if they distribute the
comma in an
> equal beating way, whose rate we'll call 'y', is there a formula
for getting
> 'y' from 'x'? Know that I already know how to find the basic beat
rate for
> any interval given their hertz frequencies...

it depends on what your bearing plan is. an interval will beat twice
as fast if it's played an octave higher.

On Tuesday 07 October 2003 02:50 pm, Paul Erlich wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
>
> wrote:
> > Hello all,
> >
> > Is there a general formula for determining the rate at which all
>
> fifths would
>
> > beat if they distribute the syntonic comma (or commas) equally,
>
> but 'equally'
>
> > in the sense of 'equal-beating' and not 'logarithmically'.
> >
> > In other words, if I tune, in pure fifths, C-G-D-A-E(pure), and A-E
>
> ends up
>
> > beating at x/sec, and I want to know how fast a tempered C-G, G-D
>
> as a
>
> > fourth, D-A, and A-E as a fourth will be if they distribute the
>
> comma in an
>
> > equal beating way, whose rate we'll call 'y', is there a formula
>
> for getting
>
> > 'y' from 'x'? Know that I already know how to find the basic beat
>
> rate for
>
> > any interval given their hertz frequencies...
>
> it depends on what your bearing plan is. an interval will beat twice
> as fast if it's played an octave higher.

Yes, I know. Although I should have stated it explicitly, I was assuming that
the bearing plan folds fifths that extend beyond the first octave into
fourths....

Anyway, apparantly, there is no formula?

-Aaron.

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> Yes, I know. Although I should have stated it explicitly, I was
assuming that
> the bearing plan folds fifths that extend beyond the first octave
into
> fourths....

right, but do you start with C at the bottom of the first octave,
or . . .

give the exact specs for what you want, and i'm sure gene can find
the answer.

oh yeah, for johnny's sake, i will adopt correct capitalization, and
a softer, gentler tone, beginning with the next message . . .

Aaron wrote:
>the beat rates of the fifths of 1/4
>comma meantone with 1/4-eqbeating meantone (meanquareq.scl),

Ah, you mean that scale. Well I put "1/4-comma" in quotes to
indicate that it's a variation. I'll change the comment to make
it clearer and it's not a real meantone.

>What I was getting at was taking a just C-E, and making the fifths (and
>fourths) of C-G-D-A-E all equal beat the syntonic comma. From what you've
>described, I'm not sure how scala would do this.

Me neither... there's no instruction to directly achieve this.
So I did it by hand. A bit of trial and error. I used
PYTHAGOREAN/GIVENBEATS. The octave must be 2. This is the result
with exact ratios.

0: 1/1 0.000 unison, perfect prime
1: 71/68 74.741
2: 19/17 192.558 quasi-meantone
3: 508/425 308.839
4: 5/4 386.314 major third
5: 568/425 502.114
6: 95/68 578.871
7: 127/85 695.153
8: 25/16 772.627 classic augmented fifth
9: 142/85 888.427
10: 152/85 1006.244
11: 127/68 1081.466
12: 2/1 1200.000 octave

Manuel

On Wednesday 08 October 2003 05:09 am, Manuel Op de Coul wrote:
> Aaron wrote:
> >the beat rates of the fifths of 1/4
> >comma meantone with 1/4-eqbeating meantone (meanquareq.scl),
>
> Ah, you mean that scale. Well I put "1/4-comma" in quotes to
> indicate that it's a variation. I'll change the comment to make
> it clearer and it's not a real meantone.
>
> >What I was getting at was taking a just C-E, and making the fifths (and
> >fourths) of C-G-D-A-E all equal beat the syntonic comma. From what you've
> >described, I'm not sure how scala would do this.
>
> Me neither... there's no instruction to directly achieve this.
> So I did it by hand. A bit of trial and error. I used
> PYTHAGOREAN/GIVENBEATS. The octave must be 2. This is the result
> with exact ratios.
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 71/68 74.741
> 2: 19/17 192.558 quasi-meantone
> 3: 508/425 308.839
> 4: 5/4 386.314 major third
> 5: 568/425 502.114
> 6: 95/68 578.871
> 7: 127/85 695.153
> 8: 25/16 772.627 classic augmented fifth
> 9: 142/85 888.427
> 10: 152/85 1006.244
> 11: 127/68 1081.466
> 12: 2/1 1200.000 octave
>
> Manuel

Thanks Manuel!

I independently verified your data. Looks right. Are all equal beating schemes
like this expressible as exact ratios? Or did you do a 'nearest ratio' fit?

-Aaron.

Aaron wrote:
>Are all equal beating schemes
>like this expressible as exact ratios?

Yes, if the octave is also rational. But the ratios get
large pretty quickly, see temp12ebfr.scl for example.

>Or did you do a 'nearest ratio' fit?

Yes and no, the Scala routine doesn't produce the ratios,
so knowing that they are rational, one can safely do that
without changing the values. Use the FAREY command.

Manuel