triprime comma

hi Gene,

regarding

http://tonalsoft.com/enc/triprime-comma.htm

can you please write a short one-sentence description
of what a triprime comma is? for use as the first sentence
of the definition. thanks.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
> hi Gene,
>
> regarding
>
> http://tonalsoft.com/enc/triprime-comma.htm
>
> can you please write a short one-sentence description
> of what a triprime comma is? for use as the first sentence
> of the definition. thanks.

The triprime commas of a rank two temperament are the reduced commas
of the temperament (meaning they are greater than one and not a power)
which factor into three primes or less. The coefficients of the monzos
of the triprime commas can be taken from the coefficients of the
wedgie for the temperament up to determination of the sign. Hence,
they may be computed very quickly and easily from the wedgie. They
have various uses; for instance they can be used to find a comma basis
for the temperament.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The triprime commas of a rank two temperament are the
> reduced commas of the temperament (meaning they are greater
> than one and not a power) which factor into three primes or
> less. The coefficients of the monzos of the triprime commas
> can be taken from the coefficients of the wedgie for the
> temperament up to determination of the sign. Hence,
> they may be computed very quickly and easily from the
> wedgie. They have various uses; for instance they can be
> used to find a comma basis for the temperament.

Thanks ... i'm using this now for the lead definition on
the new version of the webpage.

Are triprime commas only applicable to rank-2 temperaments?
Can this be generalized any further?

-monz

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

> Are triprime commas only applicable to rank-2 temperaments?
> Can this be generalized any further?

Rank n temperaments always come equipped with n+1 prime factor commas.
For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and 3^28/5^19 as
two-prime commas, which you can see directly derive from the val,
<12 19 28|. A rank three temperament likewise has four prime commas, etc.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Are triprime commas only applicable to rank-2 temperaments?
> > Can this be generalized any further?
>
> Rank n temperaments always come equipped with n+1 prime factor commas.
> For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and 3^28/5^19 as
> two-prime commas, which you can see directly derive from the val,
> <12 19 28|. A rank three temperament likewise has four prime commas,
etc.

We might use "n-prime comma" as the generic name, with specific
values of n leading to 2-prime commas, 3-prime commas, etc.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Are triprime commas only applicable to rank-2 temperaments?
> > Can this be generalized any further?
>
> Rank n temperaments always come equipped with n+1
> prime factor commas.

Good! Thanks.

> For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and
> 3^28/5^19 as two-prime commas, which you can see directly
> derive from the val, <12 19 28|. A rank three temperament
> likewise has four prime commas, etc.

Are you sure that 3^28/5^19 is correct? Its cents value
is ~314.7794608, which seems rather large for a "comma".

I'm quite intrigued by this ... i've always viewed the
just intonation lattice without the identity prime 2,
so that a rank-2 temperament grid-lattice like 5-limit 12-et
only needs 2 two-prime commas to close itself into a torus.
It's different when you toss the identity-vector into
the mix, eh?

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Rank n temperaments always come equipped with n+1
> > prime factor commas. For instance, 5-limit 12-et has
> > 3^12/2^19, 2^7/5^3 and 3^28/5^19 as two-prime commas,
> > which you can see directly derive from the val,
> > <12 19 28|. A rank three temperament likewise has
> > four prime commas, etc.
>
> We might use "n-prime comma" as the generic name, with
> specific values of n leading to 2-prime commas, 3-prime
> commas, etc.

I was thinking exactly the same thing when i read your
previous post. But can we get a nicer name than "n-prime
comma"? That will work if it seems like the best
appellation, but it seems kind of clunky to me.

-monz

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

> Are you sure that 3^28/5^19 is correct? Its cents value
> is ~314.7794608, which seems rather large for a "comma".

It's big all right, but consider the |0 28 -19>. Clearly

<12 19 28|0 28 -19> = 19*28 - 28*19 = 0.

> I'm quite intrigued by this ... i've always viewed the
> just intonation lattice without the identity prime 2,
> so that a rank-2 temperament grid-lattice like 5-limit 12-et
> only needs 2 two-prime commas to close itself into a torus.
> It's different when you toss the identity-vector into
> the mix, eh?

It's important that you treat 2 like any other prime for some purposes.

The lattice of pitch classes is also important; a fact not really
deriving from the mathematics, but from the fact that octave
equivalence reflects a real, audible reality. Also, lattices of pitch
classes can involve tempering, and generators which are not primes,
and this is significant when dealing with Tonescape style
representations. It's best not to get locked into too rigid a picture.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > Are you sure that 3^28/5^19 is correct? Its cents value
> > is ~314.7794608, which seems rather large for a "comma".
>
> It's big all right, but consider the |0 28 -19>. Clearly
>
> <12 19 28|0 28 -19> = 19*28 - 28*19 = 0.

Somehow i knew it had to make sense ... and that you'd easily
be able to show how. :)

> > I'm quite intrigued by this ... i've always viewed the
> > just intonation lattice without the identity prime 2,
> > so that a rank-2 temperament grid-lattice like 5-limit 12-et
> > only needs 2 two-prime commas to close itself into a torus.
> > It's different when you toss the identity-vector into
> > the mix, eh?
>
> It's important that you treat 2 like any other prime for
> some purposes.
>
> The lattice of pitch classes is also important; a fact
> not really deriving from the mathematics, but from the
> fact that octave equivalence reflects a real, audible
> reality.

This is really interesting to me, as it involves my
ideas about finity ... namely that each of the lowest
few prime-factors has its own unique affect (musical
"flavor" etc.), and that the "audible reality" of
identity is simply the affect of prime-factor 2.

Extrapolating from this, the powerful resonance which
i think is the primary affective quality of prime-factor 3
results in its being used so often as a generator of
linear chain-of-5ths tunings.

This procedure has been applied by analogy to many
other sizes of generators ... but what you say here,
and the ideas i've just mentioned, makes me wonder how
valid that application really is ... perhaps the other,
different affective qualities of the other low primes
indicates that they should be treated in different
manners. I don't know exactly what i mean by this ...
just kind of thinking "out loud" here on the list.

-monz

> Also, lattices of pitch
> classes can involve tempering, and generators which are not primes,
> and this is significant when dealing with Tonescape style
> representations. It's best not to get locked into too rigid a
picture.

Hi,

Would it make any sense to call them "prime-free commas",
"prime-deleted commas" or even "maximal commas"?

Based on Gene's examples, it _seems_ that the factorisation
of each comma lacks exactly one of the primes that does not
exceed the limit. If this is a _necessary_ consequence of the
lattice structure (Gene?), isn't each comma maximal in terms
of the natural ordering by division of the (reduced) products
of powers of its prime factors? Sorry, I'm rusty on lattice
theory, but I suspect I need the concept of (lattice) "ideal"
to explain myself clearly here ...

Regards,
Yahya

-----Original Message-----
[monz]
Are triprime commas only applicable to rank-2 temperaments?
Can this be generalized any further?

[Gene]
Rank n temperaments always come equipped with n+1 prime factor commas.
For instance, 5-limit 12-et has 3^12/2^19, 2^7/5^3 and 3^28/5^19 as
two-prime commas, which you can see directly derive from the val,
<12 19 28|. A rank three temperament likewise has four prime commas, etc.

...

[Gene]
We might use "n-prime comma" as the generic name, with specific
values of n leading to 2-prime commas, 3-prime commas, etc.

...

[monz]
I was thinking exactly the same thing when i read your
previous post. But can we get a nicer name than "n-prime
comma"? That will work if it seems like the best
appellation, but it seems kind of clunky to me.

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

> I was thinking exactly the same thing when i read your
> previous post. But can we get a nicer name than "n-prime
> comma"? That will work if it seems like the best
> appellation, but it seems kind of clunky to me.

Syzycomma, from syzygy comma?

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

> Based on Gene's examples, it _seems_ that the factorisation
> of each comma lacks exactly one of the primes that does not
> exceed the limit.

It's more specific than that: for 11-limit miracle 225/224, 1375/1372
and 2401/2400 are all commas lacking exactly one prime, but none is a
triprime comma since they have *four* primes in the factorization. The
triprime commas include 243/242, 1029/1024, 16875/16803, 823543/819200
and there are ten of them in total.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > I was thinking exactly the same thing when i read your
> > previous post. But can we get a nicer name than "n-prime
> > comma"? That will work if it seems like the best
> > appellation, but it seems kind of clunky to me.
>
> Syzycomma, from syzygy comma?

I guess that works a little better. Comments from anyone?
Should we transfer this discussion to the tuning-jargon list?

-monz