re: consistency

>In something as common as the standard JI major scale (1/1, 9/8, 5/4, 4/3,
>3/2, 5/3, 15/8, 2/1), 26e, which is consistent through the 13-limit, would
>seem to be inconsistent not only because of the 15/8, but also because
>of the inconsistent (i.e. not the best ((LOG(N)-LOG(D))*(n/LOG(2))
>approximation if N/D=32/27 and n=26) 7/26 32/27 between the 4/3 and
>the 9/8...

Should that be surprising?

>while on the other hand you have say the Pythagorean 1/1, 9/8, 81/64, 4/3,
>3/2, 27/16, 243/128, 2/1, which would seem to be entirely consistent in 5e
>despite the fact that 81/64 and 4/3 are both approximated by 2/5, and that
>243/128 and 2/1 are both approximated by 5/5...

This touches on another issue called uniqueness. Consistency means that no
JI interval has more than one ET representation. Uniqueness means that no
ET interval has more than one JI representation. You need both for 1:1
correspondence between ET and JI.

>So I guess some of the things I'm wondering are these: Is consistency more
>of a theoretically useful concept, or is it a really musically detectable
>(or useful) phenomena...

Well, consistency is mostly an improved way to measure the accuracy of a
temperament. In the old days, people just measured the deviations from the
intervals they liked, and wound up getting a much greater error when they
tried to play chords. Likewise, lack of uniqueness can cause effective
errors (try modulating from a utonal tetrad to a 5:6:7:9 chord in 12tET!)
not apparent from plain deviation.

In the above examples, you have to ask yourself what property of the just
scales you want to keep. If you aren't playing them as chords, consistency
doesn't make much difference.

-Carl

[Carl Lumma:]
> Should that be surprising?

Well, unless I'm missing something (which is certainly possible...), I
do think it is surprising (or at least that it's probably not going to
be too readily apparent) that in 22e, which is consistent through the
11-limit, a 1/1, 81/64, 3/2 Pythagorean triad is inconsistent, or that
when one says that 26e is consistent through the 13-limit, that a 1/1,
5/4, 3/2, 16/9 V chord is going to be inconsistent... (etc.)

Dan

>> Should that be surprising?

>Well, unless I'm missing something (which is certainly possible...), I
>do think it is surprising (or at least that it's probably not going to
>be too readily apparent) that in 22e, which is consistent through the
>11-limit, a 1/1, 81/64, 3/2 Pythagorean triad is inconsistent, or that
>when one says that 26e is consistent through the 13-limit, that a 1/1,
>5/4, 3/2, 16/9 V chord is going to be inconsistent... (etc.)

Dan, 81:64 is an 81-limit interval, and the interval between 16/9 and 5/4,
64:45, is a 45-limit interval. If you think limits can only be prime, see
the three references to "9-limit" in Partch's _Genesis of a Music_, where
the term is introduced (unfortunately with multiple meanings). In terms of
consistency, only an odd-limit definition makes sense; a prime-limit
definition would be impossible to formulate.

[Paul H. Erlich:]
> 81:64 is an 81-limit interval, and the interval between 16/9 and
5/4, 64:45, is a 45-limit interval. In terms of
> consistency, only an odd-limit definition makes sense; a prime-limit
definition would be impossible to formulate.

Right, but there are plenty of cases where the 81-limit interval (say
7 & 20e) and the 45-limit interval (say 16 & 18e) are consistent
despite the fact that the temperament is not... (And perhaps what I'm
interested in here is something related to consistency but not
consistency?) I think that your comments from the other related post
("Consistency does not apply to scales within an ET, it applies to
consonant harmonies within an ET." etc.) are perhaps more to the
point... and, as I really doubt that this is all 'old hat' to most
here (though I may be wrong and maybe I'm the only old hat!), perhaps
me trying to publicly figure it out will help others get a handle on
it (or inquire if they don't) as well... so (Paul, Carl, or anyone
else who might contribute) keep it coming, I appreciate it.

Dan

>Right, but there are plenty of cases where the 81-limit interval (say
>7 & 20e) and the 45-limit interval (say 16 & 18e) are consistent
>despite the fact that the temperament is not...

Absolutely. Two intervals from the 45-limit are not every interval from
the 45-limit.

>("Consistency does not apply to scales within an ET, it applies to
>consonant harmonies within an ET." etc.) are perhaps more to the
>point...

Well, I don't know about that. The kind of improvement in accuracy that
consistency gives is usually only noticed in harmonies, but there's nothing
about consistency that says it isn't useful for scales per se. It's just
an evaluation of the mapping between one set and another. Paul is
injecting some other stuff here (about the accuracy of dissonant intervals
being irrelevant, and so forth). The limit thing is just a blanket to
evaluate ET's. If you're basic consonance is the 7:9:11 triad, you might
not care if the ET is fully 11-limit consistent.

>and, as I really doubt that this is all 'old hat' to most here (though I may
>be wrong and maybe I'm the only old hat!), perhaps me trying to publicly
>figure it out will help others get a handle on it (or inquire if they don't)
>as well... so (Paul, Carl, or anyone else who might contribute) keep it
>coming, I appreciate it.

It wasn't long ago that I made that same mistake with the prime vs. odd
limit thing.

-Carl

Dan Stearns wrote,

>Right, but there are plenty of cases where the 81-limit interval (say
>7 & 20e) and the 45-limit interval (say 16 & 18e) are consistent
>despite the fact that the temperament is not...

A single interval has no opportunity for inconsistency. A triad does, and
yes, in general even if n-ET is not totally consistent in the m-limit, there
will be _some_ m-limit triads that will be consistent in n-ET.

>(And perhaps what I'm
>interested in here is something related to consistency but not
>consistency?)

Patrick Ozzard-Low (author of _21st Century Orchestral Instruments) came up
with a notion of fractional consistency, which addresses the "_some_" above.
Perhaps that's what you're interested in?

[Paul H. Erlich:]
>A single interval has no opportunity for inconsistency.

But all those were examples of either the Pythagorean major triad or
the syntonic V tetrad...

>Patrick Ozzard-Low (author of _21st Century Orchestral Instruments)
came up with a notion of fractional consistency, which addresses the
"_some_" above. Perhaps that's what you're interested in?

Well, unfortunately I'm not really sure... is any of this on the net,
or is it only in the book that you cited?

Dan

I wrote,

>>Patrick Ozzard-Low (author of _21st Century Orchestral Instruments)
>>came up with a notion of fractional consistency, which addresses the
>>"_some_" above. Perhaps that's what you're interested in?

Dan Stearns wrote,

>Well, unfortunately I'm not really sure... is any of this on the net,
>or is it only in the book that you cited?

That book is downloadable in its entirety from the net. Let me see if I can
find the URL (anyone else have it)?

Found it! http://www.c21-orch-instrs.demon.co.uk/