Re: BASIC: What was temperament again?

In-Reply-To: <3B1ECFB7.81667562@z.zgs.de>
Klaus wrote:

> What are the language regulations for "5-limit harmony", explained
> for a pedant? Is it, as I believe, shorthand for "some scale that
> happens to be consistent up to and including the 5th harmonic"? Or
> is the presence of the 5th harmonic or a 5-limit interval in the
> scale (while everything else may have been fiddled with) reason
> enough to call the whole thing "5-limit"? (An answer to this, and
> I'll stop nagging...)

Oh no! This one's been left for me to trip over!

"5-limit harmony" means the intervals 3:2, 5:4, 6:5 and inversions and
complements or approximations thereof are considered to be consonances.
This is roughly the way Common Practice harmony sees it, so it's not
such an advanced concept. If they're tuned just, you call it "5-limit
JI". 5-limit scales are a fuzzier concept. Some say any scale with
5-limit intervals is a 5-limit scale, others that all notes have to be
connected by 5-limit intervals.

Graham

Paul Erlich schrieb:
>
> --- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:
> >
> > What are the language regulations for "5-limit harmony", explained
> > for a pedant?
>
> The term "limit" comes from Harry Partch. He primarily meant, "all
> intervals using ratios with odd numbers up to and including 5 are
> considered consonant; other ratios are considered dissonant". Others
> after Harry Partch have tended to use "limit" to refer to the highest
> _prime_ number in a tuning system, rather than the highest _odd_
> number in a consonant chord. But this is a departure from Partch's
> primary usage. When the question is one of verticality, I think the
> odd specification makes a lot more sense than the prime one (since
> you can approximate _any_ interval with arbitrary accuracy using 5-
> prime-limit ratios -- thus the prime limit doesn't really tell you
> anything about the kind of harmonic sounds a composer may wish to
> use). Hope that answers your question.
>
> >(An answer to this, and
> > I'll stop nagging...)
>
> Please, keep nagging . . . I love it!

Well, you asked for it...

I was really wondering about the application of the "n-limit"
terminology to temperaments.

A meantone scale will have a just 5/4 or 6/5, but not a just 3/2 of
4/3. So Partch (I think) would object to the fifths und probably not
call such a scale (let alone 12tET) "5-limit".

Graham, in his reply, actually made a distinction (not very
Partchian presumably) of 5-limit JI (which I took for granted: hence
my misunderstanding and question) and plain 5-limit. Which I do take
to refer to some kind of consistency.

So unless you (or Graham or anybody) need to correct me about the
above, please learn me this slightly finer point:

12tET is consistent for the 5-limit while dividing the 3rd
arithmetically :p. 34tET is also consistent at the 5-limit (I
suppose), but it makes a difference (however inexact in a JI sense,
but this have figured out by the sweat of mine own brain) between
9/8 and 10/9 (a fact a like). There surely must be a fine difference
in the names of these two kinds of consistency?

Klaus

--- In tuning@y..., klaus schmirler <KSchmir@z...> wrote:

> I was really wondering about the application of the "n-limit"
> terminology to temperaments.
>
> A meantone scale will have a just 5/4 or 6/5, but not a just 3/2 of
> 4/3. So Partch (I think) would object to the fifths und probably not
> call such a scale (let alone 12tET) "5-limit".

Well, Partch did claim that Western common-practice music already
used 5-limit harmony . . . albeit out-of-tune. Putting aside Partch,
let's just say that there will always be errors in tuning, and
meantone errors are not large enough to significantly impair the
interpretation of 5-limit consonances as such.

> Graham, in his reply, actually made a distinction (not very
> Partchian presumably) of 5-limit JI (which I took for granted: hence
> my misunderstanding and question) and plain 5-limit. Which I do take
> to refer to some kind of consistency.

Not necessarily.
>
> So unless you (or Graham or anybody) need to correct me about the
> above, please learn me this slightly finer point:
>
> 12tET is consistent for the 5-limit while dividing the 3rd
> arithmetically :p.

?

> 34tET is also consistent at the 5-limit (I
> suppose), but it makes a difference (however inexact in a JI sense,
> but this have figured out by the sweat of mine own brain) between
> 9/8 and 10/9 (a fact a like). There surely must be a fine difference
> in the names of these two kinds of consistency?
>
The difference is not one of consistency, but of unique articulation.
34-tET uniquely articulates the primary _and_ secondary 5-limit
ratios, while 12-tET only uniquely articulates the primary 5-limit
ratios. Secondary 5-limit ratios are ones which are not 5-odd-limit
but can be formed by combining two 5-odd-limit ratios. You might also
say that 34-tET uniquely articulates the primary intervals in (1 3 5
9) -- that is, the intervals where the highest odd factor in the
numerator is either 1, 3, 5, or 9, and the highest odd factor in the
denominator is either 1, 3, 5, or 9 -- while 12-tET does not.