Need help finding unusual linear temperaments

Hi guys,

George Secor and I could use some help with the sagittal notation project.

We're approaching the notation of ratios from a new direction. We expect it will give the same results as before at the level typically used by mere mortals, but we're now going totally off the deep end and attempting a JI notation suitable for use by the gods on Mount Olympus. ;-)

It will have a maximum error of around half a cent (say 0.4 to 0.9 c). It will not be based on an equal division of the octave.

What we're looking at are planar temperaments. Specifically planar temperaments where one generator is fixed at the just fifth (and of course the octave is fixed too). It seems to me that this is equivalent to a _linear_ temperament where the "interval of equivalence" is the apotome 3^7/2^11 (~ 113.685 cents). In any case, what we really want to know is how best to divide up the apotome in a relatively even (Rothenberg proper?) manner, from minus half an apotome to plus half an apotome, so as to closely approximate many commas for many popular ratios.

We've found 3 such temperaments by ad hoc means, but we'd really like to know that we haven't missed any really good ones. What we need is a list of (minimax) generators and periods to examine (where the period is either the apotome or an integer fraction of the apotome, and the generator is smaller than half the period. For example, one such generator is, perhaps unsurprisingly, an approximate syntonic comma (of about 21.57 cents) with a whole apotome period. Another is an approximate minor diesis of about 41.1 cents in a whole apotome, and another is an approximate syntonic comma in a 1/3-apotome period.

It is reasonably safe to assume a 23 prime-limit, although 31 would probably be better. And you can assume an odd (actually factor-of-2-and-3-free) limit of 625, although 1225 would be better. If 625 is too hard I'd settle for 385.

Ideally I'd supply you with a list of commas, giving each a popularity rating but you can probably come up with a reasonable surrogate mathematical function for this yourself. For example a weighted sum of prime exponents such that these ratios are approximately equally popular
175/1
19/1
245/1
13/7
625/1
23/1
49/25

The problem is, while we have stats on the popularity of ratios (from the Scala archive), what we really need is the popularity of their commas.

Take any 2,3-free ratio and add factors of 2 and 3 in such a way as to produce a ratio in the range -apotome/2 to +apotome/2 and you have a comma for the original ratio. There are an infinite number of possible commas for any ratio, however the relative popularity of the different commas will be strongly inversely related to their exponent of 3. Most commas with an absolute 3-exponent greater than 12 can be ignored. All commas with a 3-exponent greater than 17 can be ignored. There are rarely more than 3 commas of significance for any ratio.

A good temperament for our purpose then is one that gets close to many popular commas without having to iterate the generator too many times.

We're not interested in chains of more than 160 generators, (meaning N chains of 160/N generators). And there is unlikely to be anything useful with fewer than 45 generators. Minimax errors should be less than 0.9 cents for the X most popular commas where X is greater than 20, and less than 0.5 cents for the Y most popular commas where Y is greater than 10, and we are interested in what these values of X and Y are for each temperament.

Can anyone help? Do you need more info?

Regards,

--- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@u...> wrote:

> What we're looking at are planar temperaments. Specifically planar
> temperaments where one generator is fixed at the just fifth (and of
course
> the octave is fixed too).

Those are pretty common; you just need a planar temperament where the
fifth works as a generator, and then use a tuning where it is pure.\

It seems to me that this is equivalent to a
> _linear_ temperament where the "interval of equivalence" is the apotome
> 3^7/2^11 (~ 113.685 cents).

This makes zero sense to me. Should I give examples of planar
temperaments?

> We've found 3 such temperaments by ad hoc means, but we'd really
like to
> know that we haven't missed any really good ones. What we need is a
list of
> (minimax) generators and periods to examine (where the period is
either the
> apotome or an integer fraction of the apotome, and the generator is
smaller
> than half the period.

If you are using exact octaves, the period can only be an approximate
apitome. However, it is certainly possible to look for such beasts.

> The problem is, while we have stats on the popularity of ratios
(from the
> Scala archive), what we really need is the popularity of their commas.

Why not simply take lists of commas which pass a goodness test instead?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, David C Keenan <d.keenan@u...>
wrote:
> It seems to me that this is equivalent to a
> > _linear_ temperament where the "interval of equivalence" is the
apotome
> > 3^7/2^11 (~ 113.685 cents).
>
> This makes zero sense to me. Should I give examples of planar
> temperaments?

OK. Forget the planar temperament stuff. The things I'm interested in
are mathematically like a _linear_ temperament, but instead of
repeating at the octave they repeat at the apotome, and instead of
approximating simple-ratio JI harmonies they approximate popular
notational commas.

This is for the purpose of notating JI by means of comma-inflection
symbols applied to a chain of Pythagorean fifths. We want maximum
accuracy for the maximum number of popular commas with the minimum
number of symbols.

We are just concentrating on the region within +-1/2-apotome of a
single note in the chain of fifths.

> > We've found 3 such temperaments by ad hoc means, but we'd really
> like to
> > know that we haven't missed any really good ones. What we need is a
> list of
> > (minimax) generators and periods to examine (where the period is
> either the
> > apotome or an integer fraction of the apotome, and the generator is
> smaller
> > than half the period.
>
> If you are using exact octaves, the period can only be an approximate
> apitome. However, it is certainly possible to look for such beasts.

You have misunderstood me. I'm sorry I wasn't clearer. I hope the
above helps. Octaves are not important. Exact apotomes are. We could
divide the apotome into equal pieces and call the resulting object an
EDA (equal division of the apotome), but equal divisions are not
necessary, all we need is an approximately even (proper) division, and
we can probably get greater accuracy that way.

> > The problem is, while we have stats on the popularity of ratios
> (from the
> > Scala archive), what we really need is the popularity of their commas.
>
> Why not simply take lists of commas which pass a goodness test instead?

I'm certainly open to suggestions. Goodness for what? Actually I don't
need to know what. Just post a list and I'll tell you if I think they
are roughly in popularity order.

The single most popular comma in the entire universe (for notational
purposes such as ours) is of course the syntonic comma 81/80. Then the
septimal comma 64/63. Then the double syntonic comma 6561/6400. After
that it starts to get a little hazy with regard to the commas, but
decreasing popularity of the ratios which the commas are meant to
notate start off like this (with factors of 2 and 3 removed).

1/1
5/1
7/1
25/1
7/5
11/1
35/1
125/1
49/1
13/1
11/5
11/7
17/1
25/7
49/5
13/5
175/1
19/1
245/1
13/7
625/1
23/1
49/25
55/1
77/1
17/5
19/5
35/11
13/11
31/1
343/1
29/1
125/7
55/7
17/11
77/5
19/7
385/1
55/49
17/7
1225/1
37/1
121/1
23/5
19/13
17/13
23/7
25/11

Another reason for looking at these linear temperaments of the apotome
is that they may provide us with a series of proper divisions of the
apotome that increase in resolution but belong to a common "family".
This is so that we can provide a number of notations that trade off
accuracy for number-of-symbols, in different ways, while making it
easy to move between them. Each notation would correspond to a proper
MOS of the same LTA (linear temperament of the apotome).

I note that the first 7 ratios above account for 71% of all ocurrences
of ratios in the Scala archive (when factors of 2 and 3 are removed).
So ideally their commas would all be represented in a reasonably short
chain of generators (say 20 or less).

--- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
wrote:

> You have misunderstood me. I'm sorry I wasn't clearer. I hope the
> above helps. Octaves are not important.

this is going to be a bizarre notation indeed! without octave
equivalence???

> The single most popular comma in the entire universe (for notational
> purposes such as ours) is of course the syntonic comma 81/80. Then
the
> septimal comma 64/63. Then the double syntonic comma 6561/6400.

how did that leapfrog over the diaschisma 2048/2025??

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
> wrote:
>
> > You have misunderstood me. I'm sorry I wasn't clearer. I hope the
> > above helps. Octaves are not important.
>
> this is going to be a bizarre notation indeed! without octave
> equivalence???

Oh dear. I almost despair of communicating my intentions here if _you_
are not getting it Paul. But this certainly is not the kind of "linear
temperament" any of us have ever considered before. So I shouldn't be
surprised.

Let me reassure you that the notation has octave equivalence. It is
the same sagittal notation whose essential details were worked out on
this list over a year ago and have not changed since. The letter-names
and sharps and flats are based on octaves and fifths as you would expect.

What I am investigating here is what happens _after_ you have chosen a
letter and sharps or flats for a ratio, and deals only with the comma
inflections.

For this purpose, I am looking only at the region within half an
apotome either side of any 3-limit pitch and looking at dividing up
this space by considering the proper moments-of-symmetry of a
comma-sized generator iterated modulo this apotome.

> > The single most popular comma in the entire universe (for notational
> > purposes such as ours) is of course the syntonic comma 81/80. Then
> the
> > septimal comma 64/63. Then the double syntonic comma 6561/6400.
>
> how did that leapfrog over the diaschisma 2048/2025??

It seems to me that folks prefer to notate ratios of 25 as stacked
ratios of 5, e.g. 1:5:25 as C E\ G#\\ rather than C E\ Ab'\ where \ is
syntonic comma down and '\ is diaschisma down. (By the way, the
apostrophe ' can also be read, when on its own, as schisma up).

But lets not get bogged down in details like this. I don't mind if you
want to rank the diaschisma as more popular (for comma-inflected
notation) than the double-syntonic comma. It is still likely to
benefit from the same kind of apotome-MOS or
linear-temperament-of-the-apotome.

Regards,
-- Dave Keenan