12&7 (was: the two-dimensional tuning)

Hi Gene,

On Mon, 24 Oct 2005, "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > > (12, 7). By 12&7 I mean a system defined by the corresponding vals;
> > > <12 19| and <7 11| for the 3-limit, and <12 19 28| and <7 11 16| for
> > > the 5-limit.
> >
> > So there's no unique interpretation of the name "m&n" for
> > arbitrary m and n; 12&7 is just a convenient shorthand for
> > all the numbers in those vals.
>
> I use m&n to mean something specific (the rank two temperament defined
> by two "standard" vals) so long as the prime limit is specified. Then
> there's m+n, but that's a whole other story.

So there is a whole lot more behind the notation m&n,
for arbitrary m and n, than meets the eye. I'm aware
that the structures you work with are inherently
mathematical, but still, I wonder about the possibility
of a translation into terms of more general knowledge ...

Is it possible, perhaps, without essential violence, to
explain the meanig of m&n in a way that requires no more
mathematics than, say, the concepts of whole numbers,
ratios and primes? ... One that does not require a deep
understanding of vals, groups or ranks?

If not, I suppose that most list members would have to
take your results and methods on faith.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.361 / Virus Database: 267.12.4/146 - Release Date: 21/10/05

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> Is it possible, perhaps, without essential violence, to
> explain the meanig of m&n in a way that requires no more
> mathematics than, say, the concepts of whole numbers,
> ratios and primes? ... One that does not require a deep
> understanding of vals, groups or ranks?

Roughly speaking, consider what temperament you can find which is
suported by both 12 and 7 edo, and therefore by their sum (in this
case 19.) That would be meantone. However, this is roughly speaking.

Gene:
>>I use m&n to mean something specific (the rank two temperament defined
>>by two "standard" vals) so long as the prime limit is specified. Then
>>there's m+n, but that's a whole other story.

Yahya:
> So there is a whole lot more behind the notation m&n,
> for arbitrary m and n, than meets the eye. I'm aware > that the structures you work with are inherently > mathematical, but still, I wonder about the possibility > of a translation into terms of more general knowledge ... It shouldn't be difficult. If meantone were more commonly accepted I'd even dare say it should be obvious.

> Is it possible, perhaps, without essential violence, to > explain the meanig of m&n in a way that requires no more > mathematics than, say, the concepts of whole numbers, > ratios and primes? ... One that does not require a deep > understanding of vals, groups or ranks?

There wasn't anything about groups in Gene's description anyway. It happens that these things are free abelian groups, but you don't need to know that. Vals describe how the prime-number intervals are approximated in an equal temperament. That should be all you need to know about them for now.

The rank is only the dimension of the temperament. You can't really have deep knowledge of that. An equal temperament has a rank of 1, and a linear temperament has a rank of 2. There are some rank 2 temperaments that, with the latest terminology, aren't linear temperaments, so we need to talk about "rank 2" instead. 5-limit JI is rank 3, and so on. You can't really get very far without this concept.

An m&n temperament will have the following properties, where the properties of m- and n-equal are assumed:

The tuning system (or whatever you call it) includes an MOS of m large and n small, or n large and m small, steps to the octave. Therefore it also includes an MOS of n steps and another of m steps. And two particular tunings will be m-equal and n-equal.

Assume you have m large and n small steps to the octave. If an interval approximates with r steps in m-equal and s steps in n-equal, it will have r large and s small steps in the m&n MOS with m+n notes. This is true for all intervals approximated by the two equal temperaments, and if you swap large and small intervals.

For the case of 12&7, it means you have a chromatic scale of 12 notes and a diatonic scale of 7 notes. A major third is two steps on the diatonic scale and four steps on the chromatic scale. In meantone, this can represent 5:4 because 5:4 maps to 4 steps in 12-equal and 2 steps in 7-equal. If you wanted, you could write 12&7 as a 19 note scale with 12 large and 7 small steps, to demonstrate the rule above. Eytan Agmon noticed that the major third is always 2 diatonic and 4 chromatic steps, but didn't seem to accept the validity of the 19 note scale implied by 12&7.

> If not, I suppose that most list members would have to > take your results and methods on faith.

You shouldn't have to take any of this on faith. The more regular musicians understand these concepts the easier it should be for you to explain them to others. Perhaps you could even explain them in such a way that they don't appear to be mathematical, or they most assuredly are.

What you may have to take on faith is that certain systems -- like 12&7, 12&29, 10&31 and 19&22 -- are the most efficient temperaments given a particular set of criteria. You can run searches on my website so it's only a question of trusting the code behind them. A few years ago that code didn't exist, and so nobody that I know of even guessed that 19&22 was unusually efficient. There's a page on the wiki (riters.com/microtonal or something) that tries to explain the method I follow. I don't know what state it's in now. There's also an older explanation on my website.

Graham