Algebraic generators

Okay, curiosity has got the better of me.

I understand everything on the xenwiki temperament pages now except ONE thing, and that's these "algebraic generators". They are algebraic numbers, that are in the range of temperament generators, and they have these really fanciful-sounding names like "Cybozem" or "Radieubiz" or "Terzbirat". What I don't understand is why these specific algebraic numbers are chosen. Algebraic numbers are dense! There's infinitely many arbitrarily close to any temperament generator you can think of.

Why are these specific ones mentioned and given names that sound like sci-fi characters or medieval Islamic scholars or what have you?

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Why are these specific ones mentioned and given names that sound like sci-fi characters or medieval Islamic scholars or what have you?

Jacques Dudon is responsible for the names. The polynomials for the generators make good characteristic polynomials for recurrence relationships.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Jacques Dudon is responsible for the names. The polynomials for the generators make good characteristic polynomials for recurrence relationships.

What does that mean, recurrence relationships? Why does it make those specific generators interesting in terms of temperaments?

Keenan

>What does that mean, recurrence relationships? Why does it make those
>specific generators interesting in terms of temperaments?

I'm still trying to figure that out. -Carl

On Thu, Aug 25, 2011 at 9:11 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> > Jacques Dudon is responsible for the names. The polynomials for the generators make good characteristic polynomials for recurrence relationships.
>
> What does that mean, recurrence relationships? Why does it make those specific generators interesting in terms of temperaments?

It has to do with isoharmonicity between the chords, or at least a
generalized version of isoharmonicity that deals with "proportional
beating" rather than "equal beating." If you play a chord like this
with sine waves, you'll end up getting the "periodicity buzz" effect,
which should really be called "isoharmonicity buzz," when you play it.
For harmonic timbres that's not going to work unless the isoharmonic
chord happens to be rationally intoned as well, and the effect tapers
off as you get to higher-RI chords and with more complex
isoharmonicity ratios.

In general, the effect should be stronger with timbres that are less harsh.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> It has to do with isoharmonicity between the chords, or at least a
> generalized version of isoharmonicity that deals with "proportional
> beating" rather than "equal beating." If you play a chord like this
> with sine waves, you'll end up getting the "periodicity buzz" effect,
> which should really be called "isoharmonicity buzz," when you play it.
> For harmonic timbres that's not going to work unless the isoharmonic
> chord happens to be rationally intoned as well, and the effect tapers
> off as you get to higher-RI chords and with more complex
> isoharmonicity ratios.
>
> In general, the effect should be stronger with timbres that are less harsh.

So these generators which are a certain kind of algebraic number are supposed to give you chords where the difference tones between frequencies have some small-number integer relationships? Like the actual notes of the temperament are not JI but their first-order difference tones are?

How is this actually achieved? What kind of algebraic numbers work best?

Keenan

On Fri, Aug 26, 2011 at 2:47 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> So these generators which are a certain kind of algebraic number are supposed to give you chords where the difference tones between frequencies have some small-number integer relationships? Like the actual notes of the temperament are not JI but their first-order difference tones are?

Yes, except it's not the "difference tones" that are important, but
the beating. Difference tones implies that we're dealing with a
nonlinearity in the ear. This has to do with critical band
interactions in the cochlea. This thread lays a lot of the theory out:

/tuning/topicId_95699.html#95699

If you can be pedantic about the term "orthogonal," then I'll damn
well not have people calling things difference tones that aren't
tones!

> How is this actually achieved? What kind of algebraic numbers work best?

I'm not sure if Gene's programmed in a 1:1 isoharmonicity ratio. But
at any rate, let's consider a 1:1 ratio for meantone. This means that
we want, in the chord ~4:5:6, the ~6 - ~5 = ~5 - ~4. If we're going to
call the meantone generator "g," and assuming it's a type of 3/2,
here's an algebraic expression to reflect that:

~6 - ~5 = ~5 - ~4
4g - g^4 = g^4 - 4, which simplifies to
g^4 - 2g - 2 = 0

The solution to this is going to be an algebraic number. I don't feel
like working out the specific solution right now, but by plugging it
into an equation calculator I get 1.4945301804796691 as the only
positive real root. This corresponds to a fifth of 695.630437 cents.
Load this into scala, put the GM 80 Ocarina patch on, play some major
triads, and enjoy the isoharmonicity. Don't double the octave though.
If you have a soundfont that can play sine waves even better.

The effect is more noticeable with high-error temperaments. For
example, let's consider mavila, where we'll consider our generator to
again be a 3/2:

~6 - ~5 = ~5 - ~4
4g - 16/g^3 = 16/g^3 - 4
g^4 + g^3 - 8 = 0

This yields 1.4779672430090125, which is 676.337154 cents. Load it up
in Scala and pay attention to the quality of the major triads. Now
compare to 16-equal and 23-equal to hear the resulting warbliness.

OK, one more. You said you hated father temperament before. I'm going
to instead use Uncle temperament instead, where 6 ~4/3's gets you to
5/4. I'll skip the math this time, but if you solve you end up with
467.48 cents. Load it up, put the Ocarina patch on and watch the magic
unfold. Now make the generator 1 cent sharper, aka at 468.48 cents
instead, and watch it all fall apart, a broken, beating, battered mess
of a temperament only half its former self. Well, OK, I actually like
the warbling, but you get the point.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Aug 26, 2011 at 2:47 AM, Keenan Pepper <keenanpepper@...> wrote:
> >
> > So these generators which are a certain kind of algebraic number are supposed to give you chords where the difference tones between frequencies have some small-number integer relationships? Like the actual notes of the temperament are not JI but their first-order difference tones are?
>
> Yes, except it's not the "difference tones" that are important, but
> the beating. Difference tones implies that we're dealing with a
> nonlinearity in the ear. This has to do with critical band
> interactions in the cochlea. This thread lays a lot of the theory out:
>
> /tuning/topicId_95699.html#95699
>
> If you can be pedantic about the term "orthogonal," then I'll damn
> well not have people calling things difference tones that aren't
> tones!

Fair enough. I guess I should have just said "differences between frequencies".

> > How is this actually achieved? What kind of algebraic numbers work best?
>
> I'm not sure if Gene's programmed in a 1:1 isoharmonicity ratio. But
> at any rate, let's consider a 1:1 ratio for meantone. This means that
> we want, in the chord ~4:5:6, the ~6 - ~5 = ~5 - ~4. If we're going to
> call the meantone generator "g," and assuming it's a type of 3/2,
> here's an algebraic expression to reflect that:
>
> ~6 - ~5 = ~5 - ~4
> 4g - g^4 = g^4 - 4, which simplifies to
> g^4 - 2g - 2 = 0
>
> The solution to this is going to be an algebraic number. I don't feel
> like working out the specific solution right now, but by plugging it
> into an equation calculator I get 1.4945301804796691 as the only
> positive real root. This corresponds to a fifth of 695.630437 cents.
> Load this into scala, put the GM 80 Ocarina patch on, play some major
> triads, and enjoy the isoharmonicity. Don't double the octave though.
> If you have a soundfont that can play sine waves even better.
>
> The effect is more noticeable with high-error temperaments. For
> example, let's consider mavila, where we'll consider our generator to
> again be a 3/2:
>
> ~6 - ~5 = ~5 - ~4
> 4g - 16/g^3 = 16/g^3 - 4
> g^4 + g^3 - 8 = 0
>
> This yields 1.4779672430090125, which is 676.337154 cents. Load it up
> in Scala and pay attention to the quality of the major triads. Now
> compare to 16-equal and 23-equal to hear the resulting warbliness.
>
> OK, one more. You said you hated father temperament before. I'm going
> to instead use Uncle temperament instead, where 6 ~4/3's gets you to
> 5/4. I'll skip the math this time, but if you solve you end up with
> 467.48 cents. Load it up, put the Ocarina patch on and watch the magic
> unfold. Now make the generator 1 cent sharper, aka at 468.48 cents
> instead, and watch it all fall apart, a broken, beating, battered mess
> of a temperament only half its former self. Well, OK, I actually like
> the warbling, but you get the point.

Huh, this is actually really simple. Thanks!

Keenan

Mike wrote:

> OK, one more. You said you hated father temperament before. I'm going
> to instead use Uncle temperament instead, where 6 ~4/3's gets you to
> 5/4. I'll skip the math this time, but if you solve you end up with
> 467.48 cents. Load it up, put the Ocarina patch on and watch the magic
> unfold. Now make the generator 1 cent sharper, aka at 468.48 cents
> instead, and watch it all fall apart, a broken, beating, battered mess
> of a temperament only half its former self. Well, OK, I actually like
> the warbling, but you get the point.

Sounds dramatic. I'm always on the lookout for triads that are
similar in error but different in isoharmonicity. Anybody?

-Carl

On Sat, Aug 27, 2011 at 1:54 AM, Carl Lumma <carl@lumma.org> wrote:
>
> Mike wrote:
>
> > OK, one more. You said you hated father temperament before. I'm going
> > to instead use Uncle temperament instead, where 6 ~4/3's gets you to
> > 5/4. I'll skip the math this time, but if you solve you end up with
> > 467.48 cents. Load it up, put the Ocarina patch on and watch the magic
> > unfold. Now make the generator 1 cent sharper, aka at 468.48 cents
> > instead, and watch it all fall apart, a broken, beating, battered mess
> > of a temperament only half its former self. Well, OK, I actually like
> > the warbling, but you get the point.
>
> Sounds dramatic. I'm always on the lookout for triads that are
> similar in error but different in isoharmonicity. Anybody?

Alright kids, fun's over, everyone out of the pool...

-Mike