Nicely inconsistent edos

Both consistency and Pepper ambiguity are properties of edos rather
than val/breeds (could people kindly register their opinions on
whether "val" or "breed" is better?) and it seems worthwhile to
compare them. In particular, suppose an edo n has a smaller ambiguity
in a particular odd limit than any smaller edo, and yet is not
consistent. Such a system might be called "nicely inconsistent". For
example, 24 is nicely inconsistent in the 13, 15, 17, 19, and 21
limits, and 436 is nicely inconsistent in the 29, 31, 33, 35, 37 and
39 limits. In terms of logflat badness, the rounded val has a max
error badness under 1 for 19, 21, and 23, but *not* for the above
systems. This is because its "5" is relatively bad, and so its 25
twice as bad. On the other hand, *all* primes up to 29 come in flat,
and its inconsistency isn't strong; it is clear the rounded val is the
correct one to use.

It might also be noted that it is possible to be nicely inconsistent,
yet still have an edo on the nice list of edos with smaller ambiguties
than any previous edo which is consistent. An example is 526 in the 27
limit, which is nicely inconsistent, yet falls after the consistent
edo 422 on the nice list.

When the limit gets as high as this, consistent systems on the nice
list are always nanotemperaments. This is largely due to the fact that
consistency becomes increasingly rare in higher limits. For 29, the
only nice consistent system under 10000 is 1578. For 31, it is 6650
and 9934; 33, 6850 and 9934; 35, 4501 and 9934; 37, 4501; 39, 4501;
41, 43, 45, nothing. Moreover, 23 has only 311, 1578 and 2460; 25 311
and 1578; 27 422 and 1578.

>Both consistency and Pepper ambiguity are properties of edos rather
>than val/breeds (could people kindly register their opinions on
>whether "val" or "breed" is better?)

I like val. It sounds nicer, and the endless re-engineering of
terminology around here ticks me off. As for Paul's grandfathering,
the first time I noticed it was in the Middle Path paper, take
from that what you will.

>and it seems worthwhile to
>compare them. In particular, suppose an edo n has a smaller ambiguity
>in a particular odd limit than any smaller edo, and yet is not
>consistent. Such a system might be called "nicely inconsistent".

Oh, blah. I've always viewed consistency as a continuous thing
anyway. Besides, you had me convinced (years ago) that we should
be talking about vals, not ETs.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Oh, blah. I've always viewed consistency as a continuous thing
> anyway.

By "consistency" I mean that the rounded values for each interval of
the diamond lead to a unique val. Another way to say it is that the
rounded val gives all the rounded values, but in fact you could start
by rounding any basis of consonances, not just primes.

Anyway, I don't see it as continuous.

Besides, you had me convinced (years ago) that we should
> be talking about vals, not ETs.

If you have consistency you don't much need to worry about the
difference. However, while consistency is a weak condition in the
lower limits, by the 9-limit it has become strong enough that it is
pretty much filtering out all the crap, and after a while you are
forced to admit it is filtering out too much.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I like val. It sounds nicer, and the endless re-engineering of
> terminology around here ticks me off.

Thanks for responding. I've pretty much concluded I must re-engineer
"standard val" however, as the term is not being accepted. You didn't
like "patent val". Why not? It has half the syllables of your
proposal, is more clearly a term of art, and corresponds more
precisely in meaning.

>> Oh, blah. I've always viewed consistency as a continuous thing
>> anyway.
>
>By "consistency" I mean that the rounded values for each interval of
>the diamond lead to a unique val. Another way to say it is that the
>rounded val gives all the rounded values, but in fact you could start
>by rounding any basis of consonances, not just primes.
>
>Anyway, I don't see it as continuous.

I just don't round it.

> Besides, you had me convinced (years ago) that we should
>> be talking about vals, not ETs.
>
>If you have consistency you don't much need to worry about the
>difference.

But I do worry about the difference.

>and after a while you are
>forced to admit it is filtering out too much.

Exactly.

-Carl

At 03:23 PM 2/11/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
>> I like val. It sounds nicer, and the endless re-engineering of
>> terminology around here ticks me off.
>
>Thanks for responding. I've pretty much concluded I must re-engineer
>"standard val" however, as the term is not being accepted. You didn't
>like "patent val". Why not? It has half the syllables of your
>proposal, is more clearly a term of art, and corresponds more
>precisely in meaning.

Patent is fine by me. I just prefer immediate because I have a
vague idea I've seen it used for this kind of thing before.
Standard should go.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Both consistency and Pepper ambiguity are properties of edos rather
> than val/breeds (could people kindly register their opinions on
> whether "val" or "breed" is better?)

I don't like either of them. I'd prefer our theory to be accessible to
outsiders and so I'd prefer to keep the term "prime mapping" (or
just "mapping" for short), which has been in use for longer than
either val or breed.

But I'm starting to feel like a stuck record, as it seems no one else
much cares about accessibility.

-- Dave Keenan

Dave Keenan wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" > <genewardsmith@...> wrote:
> >>Both consistency and Pepper ambiguity are properties of edos rather
>>than val/breeds (could people kindly register their opinions on
>>whether "val" or "breed" is better?)
> > > I don't like either of them. I'd prefer our theory to be accessible to > outsiders and so I'd prefer to keep the term "prime mapping" (or > just "mapping" for short), which has been in use for longer than > either val or breed.
> > But I'm starting to feel like a stuck record, as it seems no one else > much cares about accessibility.

I'm with you! And I've been feeling the same way.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> wrote:
>
> Both consistency and Pepper ambiguity are properties of edos rather
> than val/breeds (could people kindly register their opinions on
> whether "val" or "breed" is better?)

If we must use "monzo", then let's use "breed" too -- graham is a genius and actually contributed a great deal to the develpoment of this field.

> and it seems worthwhile to
> compare them. In particular, suppose an edo n has a smaller ambiguity
> in a particular odd limit than any smaller edo, and yet is not
> consistent. Such a system might be called "nicely inconsistent". For
> example, 24 is nicely inconsistent in the 13, 15, 17, 19, and 21
> limits, and 436 is nicely inconsistent in the 29, 31, 33, 35, 37 and
> 39 limits. In terms of logflat badness, the rounded val has a max
> error badness

What is "max error badness"?

> under 1 for 19, >21, and 23,

Are these limits? For which edo? Or are they edos? In which limit?

> but *not* for the above
> systems.

You've completely lost me. Can you restate the above more clearly please? My IQ was higher than Aaron's when I was in the first grade, but may be substantially declining . . . :)

> This is because its "5" is relatively bad, and so its 25
> twice as bad. On the other hand, *all* primes up to 29 come in flat,
> and its inconsistency isn't strong; it is clear the rounded val is the
> correct one to use.

Are you talking about 24 or 436?

> It might also be noted that it is possible to be nicely inconsistent,
> yet still have an edo on the nice list of edos with smaller ambiguties
> than any previous edo which is consistent. An example is 526 in the 27
> limit, which is nicely inconsistent, yet falls after the consistent
> edo 422 on the nice list.
>
> When the limit gets as high as this, consistent systems on the nice
> list are always nanotemperaments. This is largely due to the fact that
> consistency becomes increasingly rare in higher limits. For 29, the
> only nice consistent system under 10000 is 1578. For 31, it is 6650
> and 9934; 33, 6850 and 9934; 35, 4501 and 9934; 37, 4501; 39, 4501;
> 41, 43, 45, nothing. Moreover, 23 has only 311, 1578 and 2460; 25 311
> and 1578; 27 422 and 1578.
>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >Both consistency and Pepper ambiguity are properties of edos rather
> >than val/breeds (could people kindly register their opinions on
> >whether "val" or "breed" is better?)
>
> I like val. It sounds nicer, and the endless re-engineering of
> terminology around here ticks me off.

How about the endless engineering of terminology without any attempt at consensus or even feedback?

> As for Paul's grandfathering,
> the first time I noticed it was in the Middle Path paper, take
> from that what you will.

I don't know what to take from that. Everything there was posted here first, and I avaoided both "monzo" and "breed" there. But Graham really is "cross-breeding" temperaments.

> >and it seems worthwhile to
> >compare them. In particular, suppose an edo n has a smaller ambiguity
> >in a particular odd limit than any smaller edo, and yet is not
> >consistent. Such a system might be called "nicely inconsistent".
>
> Oh, blah. I've always viewed consistency as a continuous thing
> anyway. Besides, you had me convinced (years ago) that we should
> be talking about vals, not ETs.
>
> -Carl
>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> as it seems no one else
> much cares about accessibility.
>
> -- Dave Keenan

I sure do, which is why I used neither "monzo" nor "breed" in my paper, despite the pleas I received.