Consistency

For anyone interested in the topic of "consistency" (and who might not
have seen these already), I just happened upon these (other) charts
and brief bits of text at Paul Hahn's site (Paul Erlich had given the
html to one of these charts not too long ago):

<http://library.wustl.edu/~manynote/music.html>

Dan

Dan Stearns wrote,

>Can the slight altering of ETs with low
>consistency levels (the 14 to 13.95-tET example for instance - which
>taken as cents in the first octave is for all intents and purposes
>14-tET) result in a very different consistency measure, and if so,
>shouldn't that be taken into account (as actual ET music will
>inevitably include some notable degrees of tuning inaccuracy, unless
>everything is tuning table synth based)?

OK, Dan, you could take this into account by assigning (sorry to add more
measures, Carl) a worst degree_ of worst inconsistency to each ET in each limit.
For example, since slight alteratins in 14-tET increase its consistency limit
from 4 to 7, we know that whatever inconsistencies occur in triads containing
ratios of 5 or 7 in 14-tET are very slight; that is, the inconsistency amounts to
a tiny fraction of a scale step. Since I'm away from my office at the moment,
let's look at an easy example, 24-tET. I think the "worst" 7-limit triad from a
consistency standpoint in 24-tET is 4:5:7, the example with which we all should
be familiar by now. The degree of inconsistency could be measured by the
difference between the deviation of the best approximation of an interval and the
deviation of the approximation required to make the chord consistent. In this
case, the worst degree of inconsistency is in the 4:5, since 400 cents is only 14
cents sharp, while 350 cents, which would make the chord consistent, is 36 cents
flat. 36-14=22 cents, which represents .44 of a step of 24-tET.

So, I would say that for 24-tET in the 7-limit, the worst degree of worst
inconsistency is .44. You'll find it's much lower for 14-tET.

>BTW, is this form of ET
>consistency your own original conception Paul?

Yes. I posted it to the list back when it was on the Mills server in digest 6??
or 7??, and its definition appears in a footnote of my paper, TTTTTT. Paul Hahn
then generalized this to higher-level consistency, where mine is level 1. I then
showed that consistency of level N>1.5 is simply equivalent to a measure of the
whether the maximum error in the given limit is less than 1/2N of a step.
the maximum error in the given limi

>OK, Dan, you could take this into account by assigning (sorry to add more
>measures, Carl) a worst degree_ of worst inconsistency to each ET in each
>limit.

Actually, this is just fractional Hahn levels, no?

>>BTW, is this form of ET consistency your own original conception Paul?
>
>Yes. I posted it to the list back when it was on the Mills server in digest
>6?? or 7??, and its definition appears in a footnote of my paper, TTTTTT.

Hate to say it, but Erv Wilson claims to have been using consistency since
the 60's.

-Carl

In-Reply-To: <9s8a8n+u3p0@eGroups.com>
Gene wrote:

> I denote by h46 the map sending primes to the nearest 48-et value,
> with everything else following from that, and similarly for h34. In
> that case, in the 11-limit we don't have h46=h34+h12, but if we
> instead send 11 to 41 steps of the 12-et, we have a map g12(2)=12,
> g12(3)=19, g12(5)=28, g12(7)=34 g12(11)=41 which tells us how many
> scale steps we use for anything in the 11-limit, and then we do have
> h46=h34+g12, as well as consistency.

That all makes sense, assuming you meant 48 to be 46. But we're still
getting different results, aren't we?

Me:
> > > > Taking nearest prime approximations, I get
> > > > [1, -2, -8, -6].

Gene:
> > > This is what I get for 34, not 46.

Me:
> > Well, you can get anything you like from 34 because, again, it
> isn't
> > consistent.

Gene:
> I can get anything I like from anything, consistent or not, but why
> should I want to? The above is derived from the nearest prime
> approximations for 34, it seems to me.

It can be assumed that an equal temperament uses the closest
approximations to the relevant consonance limit. Any more specific rule,
and you have to say what it is. Nearest primes is fine, but you didn't
say you were using it before.

I still get this mapping as [1, -2, -8, 11].

h46 is [46, 73, 107, 129, 159]. h34 is [34, 54, 79, 95, 118].

Me:
> > 46+58 is the linear temperament consistent with both 46- and 58-
> equal.

Gene:
> This is definately something we want a notation for, but so is the et
> and generator defined by 46 and 58, which is what I've been using it
> for. What would you say to 46&58 or 46^58 to mean the temperament
> obtained by intersecting the kernels of 46 and 58?

What's the difference?

I got into special symbols before. After a while, I couldn't remember
what they meant, and I don't think anyone else ever worked it out. It's
best to call the temperament m+n that with m intervals of one size and n
of another. That means the approximation is only defined where m and n
are consistent[1]. Anything more specific and you could say so in
English.

It may be you're defining the smaller ET as step sizes of its closest
approximation in the larger one. I don't like that written as a sum,
because one argument depends on the other in a non-obvious way.
h46.subscale(34)+h46 maybe.

Graham

[1] You could work it out if only one of m and n is consistent, but m+n
is. The trouble is, that would mean using the result of the sum to
determine one of the arguments, which I don't like.

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <9s8a8n+u3p0@e...>

> Nearest primes is fine, but you didn't
> say you were using it before.

It's what I mean by hn, as I've explained a few times.

> I still get this mapping as [1, -2, -8, 11].

I thought you said [1,-2,-8,-6], which is the same mod 17 as
[1,-2,-8,11], but *not* the same mod 23.

> h46 is [46, 73, 107, 129, 159]. h34 is [34, 54, 79, 95, 118].

And 159/4 = 11 (mod 23), 118/3 = -6 = 11 (mod 17.)

> Me:
> > > 46+58 is the linear temperament consistent with both 46- and 58-
> > equal.

> Gene:
> > This is definately something we want a notation for, but so is
the et
> > and generator defined by 46 and 58, which is what I've been using
it
> > for. What would you say to 46&58 or 46^58 to mean the temperament
> > obtained by intersecting the kernels of 46 and 58?

> What's the difference?

By 46+58 I would mean very specifically the 7/52 (=mediant(4/29,3/23))
generator in the 104-et.

> I got into special symbols before. After a while, I couldn't
remember
> what they meant, and I don't think anyone else ever worked it out.
It's
> best to call the temperament m+n that with m intervals of one size
and n
> of another.

The 104 et would have MOS of size 58 and 46, so we could regard the
58 as the white keys and the 46 as the black keys, so to speak.

That means the approximation is only defined where m and n
> are consistent[1]. Anything more specific and you could say so in
> English.

My approach gives a generator to any two n and m; the mapping to
primes is not required to define it, so good mathematical style would
be not to require it.