re: Consistency parts?

All-

Think I've finally turned up some new ground with this consistency thing.
According to my earlier post in this thread, an ET is consistent with
respect to a saturated chord if and only if it is consistent with respect
to every triad contained within that chord. That's pretty much exactly how
Paul Hahn's site has _defined_ consistency for over a year now. I just
never understood it properly before. Or maybe I did, and then forgot.
Anyway, that's not the new part.

The new part is the program found at...

ftp://lumma.org/pub/consist_triad.ss (SWL run for Chez Scheme)
ftp://lumma.org/pub/triad_run.txt (Chez Scheme code)

Like the other two programs at my ftp site, this one checks a range of ET's
for consistency. You call it by typing...

(chart min-ET max-ET)

...where min-ET and max-ET are whole numbers representing the ET's at the
ends of your range. So if you want to check all ET's between 5 and 41,
inclusive, you'd type (chart 5 41). If you want to check just 41tET, type
(chart 41 41).

The difference between this program and the others is the way it checks for
consistency. Instead of using the nifty shortcut outlined in Paul
pseudocode, it actually checks all the triads at each limit. Importantly,
it returns the triads that fail. So, even if the ET is not consistent at
the limit, you can see where. And you can construct chords with which the
ET is consistent by simiply making sure they don't contain any of the
failing triads.

Below, I've run the program on some of my favorite ET's. The limit is
backed off, starting at 19, until full consistency is achieved. To find
the triads that fail at a given limit, look at everything listed from the
bottom up to and including the limit in question. So if I want to find
13-limit chords consistent with 5tET, I just make sure they don't contain
any (13 7), (13 11), or (11 5) triads. Wait a minute! Those are pairs,
not triads! The "1" is implied. So the best 11/1 in 5tET is not the sum
of the best 5/1 and 11/5.

If you see a blank next to a limit, it means you can add that identity to
any set without changing the consistency value of the set. For example,
12tET has a blank next to 19. This means that you can add the 19-identity
to any set with which 12tET is consistent, and the new set will be
consistent too!

If anybody spots an error in my reasoning here, please let me know. Any
questions are welcome. Enjoy!

(5
19 (19 5) (19 13) (19 15)
17 (17 3) (17 5) (17 9) (17 13) (17 15)
15 (15 7) (15 11)
13 (13 7) (13 11)
11 (11 5)

(7
19 (19 5) (19 15)
17 (17 5) (17 9) (17 11) (17 15)
15 (15 7)
13
11 (11 7)
9 (9 7)
7 (7 5)

(10
19 (19 3) (19 9) (19 11) (19 17)
17
15
13
11 (11 5)
9 (9 5)

(12
19
17 (17 11)
15 (15 13)
13 (13 5) (13 7) (13 11)
11 (11 3) (11 9)

(15
19 (19 17)
17 (17 3) (17 9) (17 13) (17 15)
15 (15 7)
13 (13 7)
11
9 (9 7)

(17
19
17 (17 3) (17 7) (17 9) (17 11) (17 13)
15 (15 7) (15 9) (15 11) (15 13)
13 (13 5)
11 (11 5)
9 (9 5)
7 (7 5)
5 (5 3)

(19
19 (19 7) (19 9) (19 13) (19 15)
17 (17 7) (17 9) (17 13) (17 15)
15 (15 11)
13 (13 11)
11 (11 7)

(22
19 (19 3) (19 7) (19 9) (19 15) (19 17)
17
15
13 (13 3) (13 7) (13 9)

(25
19 (19 3) (19 13) (19 15)
17 (17 3) (17 13) (17 15)
15 (15 7) (15 9) (15 11)
13 (13 5) (13 7) (13 9) (13 11)
11 (11 3)
9 (9 3)
7 (7 3)

(26
19 (19 11) (19 15)
17 (17 15)
15 (15 3) (15 5) (15 9) (15 13)

(27
19 (19 11) (19 15) (19 17)
17 (17 3) (17 5) (17 7) (17 9)
15 (15 3) (15 5) (15 7) (15 9) (15 13)
13
11 (11 3) (11 5) (11 7) (11 9)

(29
19 (19 17)
17 (17 5) (17 7) (17 11) (17 13) (17 15)

(31
19 (19 9) (19 11)
17 (17 9) (17 11)
15
13 (13 9) (13 11)

(34
19 (19 3) (19 9) (19 11) (19 13) (19 15)
17
15 (15 7)
13 (13 7)
11 (11 7)
9 (9 7)
7 (7 3) (7 5)

(35
19 (19 3) (19 5) (19 7)
17 (17 13)
15 (15 3) (15 5) (15 7)
13 (13 3) (13 5) (13 7) (13 11)
11
9 (9 3)

(37
19 (19 3) (19 15)
17 (17 3) (17 15)
15 (15 9)
13
11
9 (9 3)

(41
19 (19 17)
17 (17 5) (17 7) (17 15)

(53
19 (19 17)
17 (17 11)
15
13
11 (11 7)

(58
19 (19 5) (19 7) (19 9) (19 11) (19 13) (19 15)

(72
19 (19 13)

-Carl

>According to my earlier post in this thread, an ET is consistent with
>respect to a saturated chord if and only if it is consistent with respect
>to every triad contained within that chord. That's pretty much exactly how
>Paul Hahn's site has _defined_ consistency for over a year now.

And how I've defined it for over four years.

>Importantly,
>it returns the triads that fail. So, even if the ET is not consistent at
>the limit, you can see where. And you can construct chords with which the
>ET is consistent by simiply making sure they don't contain any of the
>failing triads.

Cool! Maybe you should print out Patrick Ozzard Low's "fractional
consistency" measure; that is, what percentage of the triads succeed.

>Wait a minute! Those are pairs,
>not triads! The "1" is implied. So the best 11/1 in 5tET is not the sum
>of the best 5/1 and 11/5.

You also need to check triads that don't include the 1.

This is something I've been wondering for awhile now about
consistency - in short does consistency apply to slightly stretched or
compressed octaves? I mean if you look at an ET like 17 as something
like ~17.1-tET (or a 2:3 / 10, or whatnot), a ~6.7� error at the 17/17
might not be what some would like to call naturally occurring or
tolerable, but this slight change -- if I'm understanding this
correctly -- would make 17-tET consistent through the 15 odd limit (as
opposed to the 3), and that seems like quite a difference for what I
would think could perhaps still be seen as 17-tET with 'tuning errors'
that fall within the realm of a naturally occurring possibility.

Dan

Dan Stearns wrote,

>This is something I've been wondering for awhile now about
>consistency - in short does consistency apply to slightly stretched or
>compressed octaves?

Yes -- if you look at the definition of "consistency" on Monz's site, you
will find:

"Consistency may be defined for non-octave equal temperaments but then even
as well as odd numbers must be considered, and the consistency will be
through an "integer limit" rather than an odd limit.

The data for case are tabulated here. The ordinary, odd-limit consistency of
an integer ET can be read from this table as well: it is the largest odd
number not exceeding the integer limit."

The word "here" links to
ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt.

>I mean if you look at an ET like 17 as something
>like ~17.1-tET (or a 2:3 / 10, or whatnot), a ~6.7� error at the 17/17
>might not be what some would like to call naturally occurring or
>tolerable, but this slight change -- if I'm understanding this
>correctly -- would make 17-tET consistent through the 15 odd limit (as
>opposed to the 3),

You must have done something wrong -- the integer limit of 17.0-tET is 4,
but the integer limit of 17.1-tET is only 7.

[Paul H. Erlich:]
> You must have done something wrong -- the integer limit of 17.0-tET
is 4, but the integer limit of 17.1-tET is only 7.

Yep. Got it now though.

thanks,
Dan

[Paul H. Erlich:]
>if you look at the definition of "consistency" on Monz's site, you
will find:

"Consistency may be defined for non-octave equal temperaments but then
even as well as odd numbers must be considered, and the consistency
will be through an "integer limit" rather than an odd limit."

Now that I think I've got the non-octave consistency math straitened
out, one of the things I was wondering (along the lines of the 17-tET
example I gave) was that if slight errors (+ / - differences of a
couple of cents across an ET) can (sometimes drastically) change an
ETs consistency level, say 14-tET as ~13.95 (0 86 172 258 344 430 516
602 688 774 860 946 1032 1118 1204) which is consistent through the 7
integer limit as opposed to the hardly different 14-tET which is only
consistent through the 3 odd limit... is the straight odd limit
consistency measure not somewhat 'misleading' without somehow taking
these slight deviations into consideration?

Dan

>Now that I think I've got the non-octave consistency math straitened
>out, one of the things I was wondering (along the lines of the 17-tET
>example I gave) was that if slight errors . . . can (sometimes drastically)
change an
>ETs consistency level . . . is the straight odd limit
>consistency measure not somewhat 'misleading' without somehow taking
>these slight deviations into consideration?

Odd-limit consistency or integer-limit consistency -- either one can
suddenly change with a small change in the tuning. As for being misleading,
consistency is meant to prevent the misleading effects of straight
errors-vs.-JI computations, which often use inconsistent sets of intervals
as approximations to JI. Basically, if you find a low ET that seems to
approximate JI intervals in some limit well, you'd better make sure the ET
is consistent in that limit to be sure you're not fooling yourself.

[Paul H. Erlich:]
>As for being misleading, consistency is meant to prevent the
misleading effects of straight errors-vs.-JI computations, which often
use inconsistent sets of intervals as approximations to JI.

Right, and as such it certainly is a useful tool or measure, but I
don't think that obviates the point I'm trying to make, or rather the
question I'm trying to ask: If 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, shouldn't that
be somehow taken into account (as actual ET music will inevitably
include some notable degrees of tuning inaccuracy, unless everything
is tuning table synth based)?

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

Dan

[Paul H. Erlich:]
>As for being misleading, consistency is meant to prevent the
misleading effects of straight errors-vs.-JI computations, which often
use inconsistent sets of intervals as approximations to JI.

Right, and as such it certainly is a useful tool or measure, but I
don't think that obviates the point I'm trying to make (or question
I'm trying to ask): 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)? BTW, is this form of ET
consistency your own original conception Paul?

Dan

[I wrote:]
>the question I'm trying to ask: If 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,
shouldn't that be somehow taken into account

An even better example -- though working in the opposite direction
(i.e., from a higher consistency to lower consistency) -- would have
been something like ~26.03-tET (26.026-tET would be 0 46 92 138 184
231 277 323 369 415 461 507 553 599 646 692 738 784 830 876 922 968
1014 1060 1107 1153 1199), as the largest cents deviation from 26-tET
is a whopping ~1�, and yet the consistency tumbles from an integer
limit of 14 all the way down to one of 8... is this not somehow
'important,' or am I really missing something here?

Dan