Some temperament generators with interestiong beat ratios

Here are some which can go with the previously discussed meantone tuning. The first number is the "brat", or beat ratio, and the polynomial can either be solved and the solution used as a generator (the simplest method) or it can be used as a characteristic polynomial for a carefully chosen linear recurrence. I may return to this, starting with miracle, if there is interest; all the ones I give could certainly work with 24 notes to the octave, or even 12.

Mavila -1 x^4 + x^3 - 8
Kemun -1 x^6 - 2x^5 + 2
Porcupine 4 2x^5 - x^2 - 2
Beatles -1 x^11 + x^9 - 16
Magic 2 5x^5 + 4x - 20
Myna 5/2 5x^10 + 4x^9 - 50
Myna 3 5x^10 + 6x^9 - 60
Sensi 2/3 x^9 - 3x^7 + 8
Sensi 1 x^9 - 5x^7 + 20
Orwell -1 x^10 + 2x^3 - 8

Hi Gene,
I'm glad to see my polynomials find some echo here ! ;)

I haven't check them all, but at least four of them are already part
of my collection :

What you called Mavila is the octave complement of Erv Wilson'Mavila,
which I found years ago and named respectively Kamas and Samak (know
they are'nt 4th degree polynomials). Samak is the convergent form and
I resolved its fractal waveform and applied it to scales in my
earliest photosonic disks.
Samak = 1.3532099641993

Then your brat 2 version of Magic I found also and named Septimage
(1,2459967949887)
here are a few other Equal-beating recurrent versions of Magic :

Astrid = 1.2435963905735
Sulimage = 1.2448792286718
Egeria = 1.2453440266787
Terzbirat = 1.2456780612142
Septimage = 1.2459967949887
Aswafée = 1.2467941052842

Then I have Porcupine "brat 4" known here as
Dlotkotier = 1.0991125987978397125
of which we can resume the -c by 77 - 48 = 29
in the series 192 211 232 255 280 308
I had no idea it was related to Porcupine temperament and it's interesting, as it is part of some of my musical projects.

- - - - - - - -
Jacques

Gene wrote :

> Here are some which can go with the previously discussed meantone > tuning. The first number is the "brat", or beat ratio, and the > polynomial can either be solved and the solution used as a > generator (the simplest method) or it can be used as a > characteristic polynomial for a carefully chosen linear recurrence. > I may return to this, starting with miracle, if there is interest; > all the ones I give could certainly work with 24 notes to the > octave, or even 12.
>
> Mavila -1 x^4 + x^3 - 8
> Kemun -1 x^6 - 2x^5 + 2
> Porcupine 4 2x^5 - x^2 - 2
> Beatles -1 x^11 + x^9 - 16
> Magic 2 5x^5 + 4x - 20
> Myna 5/2 5x^10 + 4x^9 - 50
> Myna 3 5x^10 + 6x^9 - 60
> Sensi 2/3 x^9 - 3x^7 + 8
> Sensi 1 x^9 - 5x^7 + 20
> Orwell -1 x^10 + 2x^3 - 8

... Of course also "Keemun", known as :
Gretel = 1.20160781285
(which I thought was "Hanson") ?
and others of the same family :
Lagunag = 1.200511272385
Carmine = 1.201045931042892
- - - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> What you called Mavila is the octave complement of Erv Wilson'Mavila,
> which I found years ago and named respectively Kamas and Samak (know
> they are'nt 4th degree polynomials). Samak is the convergent form and
> I resolved its fractal waveform and applied it to scales in my
> earliest photosonic disks.
> Samak = 1.3532099641993

The way I think of temperaments, the two generators together with an octave give the same tuning of the same temperament, and I'd call them all mavila. A brat -1 tuning of mavila using the fourth as a generator is what you call Sarnak if I understand correctly.

> Then your brat 2 version of Magic I found also and named Septimage
> (1,2459967949887)
> here are a few other Equal-beating recurrent versions of Magic :

This is very interesting as it allows us to see what you consider some good "magic numbers" for the brat:

> Astrid = 1.2435963905735 brat = -1
> Sulimage = 1.2448792286718 brat = 1/4
> Egeria = 1.2453440266787 brat = 7/8
> Terzbirat = 1.2456780612142
> Septimage = 1.2459967949887 brat = 2
> Aswafée = 1.2467941052842 brat = 4

Terzbirat is the only one without a rational brat, and I'd be interested to know the polynomial for it. For the other brats, looking at [minor/major, major/fifth, fifth/minor] tells the story of why the brat is "magic"; we have this:

[-1, 1, -1]
[1/4, 2, 2]
[7/8, 4, 2/7]
[2, 5, -1/10]
[4, -1, -1/4]

Some brats which it seems to me might have been included are
1 and 3:

[1, 5, 1/5]
[3, -5/3, -1/5]

leading to polynomials

5x^5 - 4x - 10
5x^5 + 12x - 30

> Then I have Porcupine "brat 4" known here as
> Dlotkotier = 1.0991125987978397125
> of which we can resume the -c by 77 - 48 = 29
> in the series 192 211 232 255 280 308
> I had no idea it was related to Porcupine temperament and it's
> interesting, as it is part of some of my musical projects.

Twelve notes worth of it would certainly be a useful addition to your list of scales; I don't see why we need worry overmuch about the fact that 12 is not a semiconvergent for log2(Dlotkotier).

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Gene wrote :
>
> > Here are some which can go with the previously discussed meantone
> > tuning. The first number is the "brat", or beat ratio, and the
> > polynomial can either be solved and the solution used as a
> > generator (the simplest method) or it can be used as a
> > characteristic polynomial for a carefully chosen linear recurrence.
> > I may return to this, starting with miracle, if there is interest;
> > all the ones I give could certainly work with 24 notes to the
> > octave, or even 12.
> >
> > Mavila -1 x^4 + x^3 - 8
> > Kemun -1 x^6 - 2x^5 + 2
> > Porcupine 4 2x^5 - x^2 - 2
> > Beatles -1 x^11 + x^9 - 16
> > Magic 2 5x^5 + 4x - 20
> > Myna 5/2 5x^10 + 4x^9 - 50
> > Myna 3 5x^10 + 6x^9 - 60
> > Sensi 2/3 x^9 - 3x^7 + 8
> > Sensi 1 x^9 - 5x^7 + 20
> > Orwell -1 x^10 + 2x^3 - 8
>
>
> ... Of course also "Keemun", known as :
> Gretel = 1.20160781285 brat = -1
> (which I thought was "Hanson") ?

If you don't care about the 7 limit, you can tune the 5 limit more accurately, giving Hanson. More complex mappings to the 7 limit now give more complex temperaments in the Hanson family, notably catakleismic.

> and others of the same family :
> Lagunag = 1.200511272385 brat = 1/4
> Carmine = 1.201045931042892 brat = 4

Carmine makes for a good Hanson generator, though brats 2 or 3 might be preferred by some. For catakleismic, a brat of 2/3 or 1 would be better:

3x^6 - 2x^5 - 4, brat = 2/3
5x^6 - 2x^5 - 10, brat = 1

Catakliesmic is a relatively complex temperament and 24 notes of it would make much more sense than 12.

Just wondering, if Bohlen Pierce is considered great for "maximum consonance for mostly odd-harmonic-only instruments"...are there any special scales build in a similar fashion for use with even-harmonic intensive instruments?

genewardsmith wrote:
> Here are some which can go with the previously discussed meantone
> tuning. The first number is the "brat", or beat ratio, and the
> polynomial can either be solved and the solution used as a generator
> (the simplest method) or it can be used as a characteristic
> polynomial for a carefully chosen linear recurrence. I may return to
> this, starting with miracle, if there is interest; all the ones I
> give could certainly work with 24 notes to the octave, or even 12.
> > Mavila -1 x^4 + x^3 - 8
> Keemun -1 x^6 - 2x^5 + 2
> Porcupine 4 2x^5 - x^2 - 2
> Beatles -1 x^11 + x^9 - 16
> Magic 2 5x^5 + 4x - 20
> Myna 5/2 5x^10 + 4x^9 - 50
> Myna 3 5x^10 + 6x^9 - 60
> Sensi 2/3 x^9 - 3x^7 + 8
> Sensi 1 x^9 - 5x^7 + 20
> Orwell -1 x^10 + 2x^3 - 8

If I figured this right, I get around 356.358 for a beatles generator. Some preliminary experimentation shows that this results in some nice smooth-sounding tetrads compared with the more usual TOP-based version (which, although pleasant sounding, have more noticeable beating). I'd like to look into this further with other temperaments if I can find the time.

I know there was a discussion on brats some years ago, but I don't recall anything about it. Could you summarize the method for finding these polynomials and the beat ratios that go with them?

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

> > I may return to
> > this, starting with miracle, if there is interest; all the ones I
> > give could certainly work with 24 notes to the octave, or even 12.

Here are some more of these:

Miracle 1/4 4x^13 - x^7 - 8
Valentine 2 5x^9 + 2x^5 - 20
Valentine 3 5x^9 + 6x^5 - 30
Valentine 4 x^9 + 2x^5 - 8
Mothra 3 3x^12 + 10x^3 - 30
Mothra 4 x^12 + 2x^3 - 8
Rodan 4 x^17 + 4x^3 - 16
Hemiwuerschmidt 1 5x^16 - 8x^2 - 20
Hemiwuerschmidt 2 5x^16 + 8x^2 - 40

> I know there was a discussion on brats some years ago, but I don't
> recall anything about it. Could you summarize the method for finding
> these polynomials and the beat ratios that go with them?

If f is the fifth of a triad in close root position, and t is the major third, then

b = (6t-5f)/(4t-5)

is the beat ratio, or "brat", between the major and minor thirds. Other beat ratios in the triad are given by 5/(3-2b) and (3-2b)/5b and their inverses. If x is a generator, so that f is 2^a x^u and t is 2^b x^v, then we may substitute into the definition of b and solve for specific brats.

An "even harmonic instrument" that has only harmonics 2, 4, 6, 8, 10,
12, 14, 16, etc is the same thing as a "total harmonic instrument"
that has 1, 2, 3, 4, 5, 6, 7, 8, except viewed from the perspective of
the fundamental being an octave down.

-Mike

On Fri, May 7, 2010 at 9:04 PM, Michael <djtrancendance@...> wrote:
>
>
>
>     Just wondering, if Bohlen Pierce is considered great for "maximum consonance for mostly odd-harmonic-only instruments"...are there any special scales build in a similar fashion for use with even-harmonic intensive instruments?

On 7th of May 2010 Gene wrote :

(Jacques) :
> > What you called Mavila is the octave complement of Erv
> Wilson'Mavila,
> > which I found years ago and named respectively Kamas and Samak (know
> > they are'nt 4th degree polynomials). Samak is the convergent form
> and
> > I resolved its fractal waveform and applied it to scales in my
> > earliest photosonic disks.
> > Samak = 1.3532099641993
>
> The way I think of temperaments, the two generators together with
> an octave give the same tuning of the same temperament, and I'd
> call them all mavila. A brat -1 tuning of mavila using the fourth
> as a generator is what you call Sarnak if I understand correctly.

Samak. It's useful with recurrent sequences sometimes to
differentiate the 2 directions, but enventually one name will
designate the whole thing.
I don't understand fully where you find a "brat - 1" here, between the thirds (-3g for the major and +4g for the minor I guess ?) I have
to study that, I arrived here for other reasons.

> > Then your brat 2 version of Magic I found also and named Septimage
> > (1,2459967949887)
> > here are a few other Equal-beating recurrent versions of Magic :
>
> This is very interesting as it allows us to see what you consider
> some good "magic numbers" for the brat:
>
> > Astrid = 1.2435963905735 brat = -1
> > Sulimage = 1.2448792286718 brat = 1/4
> > Egeria = 1.2453440266787 brat = 7/8
> > Terzbirat = 1.2456780612142
> > Septimage = 1.2459967949887 brat = 2
> > Aswafée = 1.2467941052842 brat = 4
>
> Terzbirat is the only one without a rational brat, and I'd be
> interested to know the polynomial for it.

Yes, it's a good one :
9x2 = 8x + 4
Obviously it doesn't belong to the same story. But it has an
interesting -c property and gives a x^5 really close to 3.

> For the other brats, looking at [minor/major, major/fifth, fifth/
> minor] tells the story of why the brat is "magic"; we have this:
>
> [-1, 1, -1]
> [1/4, 2, 2]
> [7/8, 4, 2/7]
> [2, 5, -1/10]
> [4, -1, -1/4]

Yes, that's convincing. Also these are only major triads eq-beatings
and I apparently focused on the central line, which looks to be the
ratio between the fifth and the third beatings.

Here are three eq-b minor triads in Magic for a change :
8x^5 = 5x^4 +12 1.246443127504163
2x^5 = 5x^4 - 8 1.2429063450259
4x^5 = 24 - 5x^4 1.24525700056762234

very sensitive.
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Samak. It's useful with recurrent sequences sometimes to
> differentiate the 2 directions, but enventually one name will
> designate the whole thing.
> I don't understand fully where you find a "brat - 1" here, between
> the thirds (-3g for the major and +4g for the minor I guess ?)

I defined the brat in terms of the major third t and the fifth f, and sticking t=Samak^3/2 and f=2/Samak into the definition yields -1, the most equal of all the equal beat ratios.

> > Terzbirat is the only one without a rational brat, and I'd be
> > interested to know the polynomial for it.
>
>
> Yes, it's a good one :
> 9x2 = 8x + 4
> Obviously it doesn't belong to the same story. But it has an
> interesting -c property and gives a x^5 really close to 3.

I don't know what a -c property is, but apparently this is another magic generator, and has one. It has a brat of
(980 sqrt(13)+5704)/6561, but some other beat properties are rational.
I think I'll give Magic[16] with both, and someone may be inspired to decide which makes for the better magic generator.

On 7th of May 2010 Gene wrote :

> (Jacques) :
> > > What you called Mavila is the octave complement of Erv
> > Wilson'Mavila,
> > > which I found years ago and named respectively Kamas and Samak > (know
> > > they are'nt 4th degree polynomials). Samak is the convergent form
> > and
> > > I resolved its fractal waveform and applied it to scales in my
> > > earliest photosonic disks.
> > > Samak = 1.3532099641993
> >
> > The way I think of temperaments, the two generators together with
> > an octave give the same tuning of the same temperament, and I'd
> > call them all mavila. A brat -1 tuning of mavila using the fourth
> > as a generator is what you call Sarnak if I understand correctly.
>
> Samak. It's useful with recurrent sequences sometimes to
> differentiate the 2 directions, but eventually one name will
> designate the whole thing.
> I don't understand fully where you find a "brat - 1" here, between
> the thirds (-3g for the major and +4g for the minor I guess ?) I have
> to study that, I arrived here for other reasons.
>
> I defined the brat in terms of the major third t and the fifth f, > and sticking t=Samak^3/2 and f=2/Samak into the definition yields > -1, the most equal of all the equal beat ratios.

It's different from the main usual "brat" then, it has to be precised.
I still don't understand and even less, since the fifth beating is quite heavy in here.
The "minor brat" (on effectively perceived maj6th/maj3rd beats) is doing good results with Kamas (or the Samak fifth) :
mb = -1> 24 - 5x^4 = 5x3 - 16 > Kamas (x^4 = 8 - x^3) or x = 1.477967243009
and suggests as well other good ones :
mb = -1/2> 5x^4 = 56 - 10x^3 (Saptamak) > 1.4768727445
mb = -1> x^4 = 8 - x^3 (Kamas-Samak) > 1.477967243009
mb = -2> 10x^4 = 64 - 5x^3 (Dhamak) > 1.4788452037
mb = 2>10x^4 = 5x^3 + 32 (Mismak) > 1,4823738118
mb = 1> 5x^4 = 5x^3 + 8 (Samakmin) > 1.4868066476
but curiously, unless a mistake, Susak =
mb = 1/2> 5x^4 = 10x^3 - 8 > 1.3506928321696 or 1.6317070309441
gives a very strange result but ALSO a correct Pelog with the 1st solution !
(2/Susak = 1.480721561828...)

> > > Terzbirat is the only one without a rational brat, and I'd be
> > > interested to know the polynomial for it.
> >
> >
> > Yes, it's a good one :
> > 9x2 = 8x + 4
> > Obviously it doesn't belong to the same story. But it has an
> > interesting -c property and gives a x^5 really close to 3.
>
> I don't know what a -c property is, but apparently this is another > magic generator, and has one. It has a brat of
> (980 sqrt(13)+5704)/6561, but some other beat properties are rational.
> I think I'll give Magic[16] with both, and someone may be inspired > to decide which makes for the better magic generator.

Is this the usual brat as well ? There is still something I don't understand since the algorithm has only a maximal power of 2 of the generator, so I don't know where you get the minor 3rd for example.
- - - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Is this the usual brat as well ? There is still something I don't
> understand since the algorithm has only a maximal power of 2 of the
> generator, so I don't know where you get the minor 3rd for example.

The minor third is the third between the third and the fifth of a major triad in close root position.