understanding ennealimmal

hi Gene,

i'm posting this here as i think it's more on-topic for this list.
when i understand it better, and can answer your concerns better,
we'll return it to the Tonescape list.

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> Date: Wed Jun 8, 2005 12:27 am
> Subject: Re: : wrapping ennealimmal
>
> --- In tonescape@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Are those generator values correct for the tuning you wanted?
> > "temper the 3 and 5, for instance by tuning them to 612-et".
> > ... ?
>
>
> The tuning is fine. The generators are not a very good
> choice. {3,5} gives you triads, which is nice, but it
> doesn't work well with ennealimmal. {6/5,12/7} will give
> you another incomplete tetrad which works much better
> with ennealimmal. Breed style generators, namely 49/40
> and 10/7, would be possible, or 49/40 and 7/6 would work.

All the work i've ever done always used either 2/1 as
the identity interval, or no identity interval at all.

I've been confused in the past about the exact distinction
between "identity interval" and "period", and i'm completely
at a loss here.

On my Encyclopedia definition page for "ennealimmal",
the data you give is:

period
~1/9-octave = ~133.3...
ratio 27/25 = ~133.2375749 cents.

generator
ratio 36/35 = ~48.7703814 cents
ratio 21/20 = ~84.46719347 cents

as i understand it, you generate the scale by incrementing
the powers of the ratio generators, and the resulting
cents values are the addition of the generator cents
values modulo the period ... correct?

but why is it important that the period is ~1/9-octave?
what does that have to do with what we hear? or is it
simply a mathematical property of the scale that doesn't
necessarily have anything to do with any audible properties?
please help me get this.

what exactly is ennealimmal supposed to be good at?

IOW, i know that meantone is useful because it gets rid
of the problems associated with the syntonic-comma difference
between many 5-limit JI pitches, while preserving other
important aspect of 5-limit JI harmony.

i understand that tunings of the augmented family let you
circulate thru the augmented triad and come back to your
starting point, and ditto for diminished tetrads in the
diminished family. And i also understand that the period
of augmented is a ~major-3rd and that of diminished
a ~minor-3rd. But since i haven't really played around
with those tunings (outside of 12-edo, which belongs
to both), i really don't get it. i'm still hearing the
scales in terms of octave-equivalence.

anyway, if you can see what i'm getting at with those
last two paragraphs, try to explain ennealimmal to me.
if you have links to old posts, i'll read them too.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> All the work i've ever done always used either 2/1 as
> the identity interval, or no identity interval at all.

That's fine, but then you need to distinguish that from the period.

> I've been confused in the past about the exact distinction
> between "identity interval" and "period", and i'm completely
> at a loss here.

If you try to make one of your generators an octave, you find it
doesn't always work, and you need to make the nth part of an octave a
generator instead. This is then called the "period"; one reason for
that name is that DE type scales repeat at the period.

That ennealimmal has a 1/9 octave period does not mean there are not
other choices where it is a "linear" temperament in a strict sense. If
you use "tritave equivalence", at 3, then 3 can be used as the period
for constructing scales, and 5/3 as the "generator". This is not a
BP type scale, as -18 generators nets you a 2. Since +18 generators
then gives a 3/2 (tritave inversion of 2), you have 2s and 3s; since
+19 generators gives a 5/2, 5s also, and since +4 generators is
already 18/7, 7s. So the big deal with ennealimmal and 1/9 octave is
strictly an octave thing, but since you probably want to use octave
equivalence (the only kind which really works very well) it's
important in practice.

> On my Encyclopedia definition page for "ennealimmal",
> the data you give is:
>
> period
> ~1/9-octave = ~133.3...
> ratio 27/25 = ~133.2375749 cents.
>
> generator
> ratio 36/35 = ~48.7703814 cents
> ratio 21/20 = ~84.46719347 cents
>
>
> as i understand it, you generate the scale by incrementing
> the powers of the ratio generators, and the resulting
> cents values are the addition of the generator cents
> values modulo the period ... correct?

You can create a DE scale for ennealimmal by taking multiples of
a generator within a period and reducing to the period, which is
exactly what you'd do for a scale with an octave period. The
difference now is you take nine copies of this and stack them, making
an octave in total.

> but why is it important that the period is ~1/9-octave?
> what does that have to do with what we hear?

As you see, it is important for scale construction. For something like
Tonescape, which assumes octave equivalence, it is important for how
the tuning is represented; nine 27/25s add up to an octave, and so are
octave equivalent to a 1/1. Also, 9 7/6s, which is a 7-limit
consonance, come to 2 octaves. Instead of considering the note classes
to be arranged in a line defined by a generator ("linear temperament")
you get a band, which wraps around to a cylinder. You might call it a
cylindrical temperament; anyway, in the case of ennealimmal there are
nine lines of generators running along the cylinder. You can actually
get cylinder pictures of linear temperaments, such as miracle or
meantone, but with ennealimmal you are stuck with them, so that if the
octave is the interval of equivalence it is innately cylindrical.

> what exactly is ennealimmal supposed to be good at?

If you take Dave Keenan's definition of "just intonation", and ask
what is the simplest rank two tuning which gives you just intonation,
the answer is ennealimmal. What it is good at, therefore, is getting
to "just" 7-limit intervals by means of just two generators rather
than four, and it is remarkably how very good it is at that.

> IOW, i know that meantone is useful because it gets rid
> of the problems associated with the syntonic-comma difference
> between many 5-limit JI pitches, while preserving other
> important aspect of 5-limit JI harmony.

A basis for ennealimmal is {2401/2400, 4375/4374}. It gets rid of
certain very fine distinctions deriving from these.