The '9-limit'...a bogus term?

Hello all,

I've seen some references to the '9-limit' lately. Isn't that a bogus term? I
thought 'limit' in tuning was to refer to prime numbers.

Can someone explain?

Best,
Aaron.

--
OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

>I've seen some references to the '9-limit' lately. Isn't that a bogus
term? I
>thought 'limit' in tuning was to refer to prime numbers.

No, you can check the archives for "odd limit", the highest odd number in
the numerator or denominator of any scale's interval.
To distinguish you can call "limit" also "prime limit".
The Scala SHOW DATA command prints both.

Manuel

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
>
> Hello all,
>
> I've seen some references to the '9-limit' lately.

there are a huge number of references to that in the archives of this
list. any that are particularly confusing to you?

> Isn't that a bogus term? I
> thought 'limit' in tuning was to refer to prime numbers.
>
> Can someone explain?
>
> Best,
> Aaron.

aaron, unfortunately you've heard a 'bogus' definition of limit -- or
at least a later one. the term 'limit' was originally introduced by
harry partch, and you can see three references to '9-limit' in the
*index* of _genesis of a music_ alone! here's how partch classed
ratios below an octave (and then assumed octave-equivalence):

ratio of 1 -- 1:1
ratios of 3 -- 3:2, 4:3
ratios of 5 -- 6:5, 5:4, 5:3, 8:5
ratios of 7 -- 8:7, 7:6, 7:5, 10:7, 12:7, 7:4
ratios of 9 -- 10:9, 9:8, 9:7, 14:9, 16:9, 9:5
ratios of 11 -- 12:11, 11:10, 11:9, 14:11, 11:8, 16:11, 11:7, 18:11,
20:11, 11:6
etc.

now the limits are defined as follows:
1-limit -- ratio of 1
3-limit -- ratio of 1 and ratios of 3
5-limit -- ratio of 1, ratios of 3, and ratios of 5
7-limit -- ratio of 1, ratios of 3, ratios of 5, and ratios of 7
9-limit -- ratio of 1, ratios of 3, ratios of 5, ratios of 7, and
ratios of 9
11-limit -- ratio of 1, ratios of 3, ratios of 5, ratios of 7, ratios
of 9, and ratios of 11
etc.

when the ratios are interpreted as intervals, you have a guide to
partch's consonance measure, according to the 'one-footed bride'.
ratio of 1 is most consonant, ratios of 3 are next most consonant,
ratios of 5 are next most consonant, etc. i've been able to reproduce
this pattern with my 'octave-equivalent harmonic entropy model',
which you can learn more about on the harmonic entropy list.

when the ratios are interpreted as pitches, each limit gives you a
tonality diamond. since i named a total 29 ratios above, we see that
the 11-limit tonality diamond contains 29 pitches -- the 'core' of
partch's tuning system -- while the 5-limit tonality diamond contains
7 pitches, etc.

funny you bring this up now -- earlier this morning, monz and i were
working on the definitions of 'ratio of' and 'limit' in his
dictionary. they're still under development -- in particular, i'm
hoping we'll see "those ratios with odd factors no larger than n" as
the initial, brief definition of limit under the "odd" usage -- but
you can look at them now and let us know what you think:

http://sonic-arts.org/dict/ratio-of.htm
http://sonic-arts.org/dict/limit.htm

what's interesting is that, under this definition, partch's
otonalities and utonalities cease to be the only saturated chords:

http://sonic-arts.org/dict/saturat.htm

once one gets beyond the 7-limit . . .