Yahoo Tuning Groups Ultimate Backup tuning Monz Question: Prime Limit

Monz,

Hello.

I'm wondering if you would kindly elaborate on something I've read in
your book and seen on your site too (if memory serves)? That of: You
feel that there is a certain Prime Limit beyond which we would tend
to interpret the intervals as lower number integer ratios. I believe
it was the 13 Prime Limit (correct me if wrong). Please explain what
led you to this realization about this perceptual limit. And perhaps
why it may be important to establish a limit that is not exceeded in
the construction of a scale system.

I should admit here that I frequently cross this line. Perhaps even
this would be a good topic for research: To "A&B" some
chords/progressions from higher and lower limits (for which the
higher may perceived as lower), than the limit you have said, just to
hear if there is any significantly difference between them. Could be
very interesting as well.

Respectfully,

Jacky

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Jacky Ligon" <jacky_ekstasis@y...> wrote:
> http://www.egroups.com/message/tuning/12456
>
> Monz,
>
> ... You feel that there is a certain Prime Limit beyond which
> we would tend to interpret the intervals as lower number integer
> ratios. I believe it was the 13 Prime Limit (correct me if wrong).
> Please explain what led you to this realization about this
> perceptual limit. And perhaps why it may be important to
> establish a limit that is not exceeded in the construction
> of a scale system.
>
> I should admit here that I frequently cross this line.
> ...

Hi Jacky.

I don't have any definite strong beliefs on exactly what this
limit is, and in the second MIDI example of 'Hendrix Chord'

http://interval/monzo/hendrix/hndrx-ex.mid
http://interval/monzo/hendrix/hendrix.htm

I used primes as high as 271 - up there in La Monte Young
territory - in my chord-'root' movements.

Thru the mid-1990s, while working on the original version of
my book, I began to form a pretty clear idea that somewhere
between 11 and 23 there was a kind of perceptual prime-limit
within which we place what we hear, regardless of how it's
actually tuned.

Of course, as was noted here again recently (in fact I think
it was you), 'context is everything'. So different listeners,
different composers, different performers, different audio
equipment, etc. etc. etc., all play a part in what these limits
are. So I can't define it more precisely than I have above.

I have tended in my own music to see both 13 and 19 (maybe 23)
as limits beyond which I don't often feel the need to go.
I pretty much hear a lot of my 'head music' in 13-limit, but
I really love the 4:6:10:14:19 version of the 'Hendrix Chord',
and in this piece:

http://www.ixpres.com/interval/monzo/altrock/altrock.htm

I like the way I used 43/32.

A lot of my ideas on this came together 2 years ago while I
was making tons of lattice-diagrams of ancient Greek tuning
systems, and developing my concepts of 'finity' and 'bridging'.
Follow the links from here:

http://www.ixpres.com/interval/dict/finity.htm

to follow my train of thought on that.

Many Greek theorists were firm believers in the need for all
ratios in a tetrachord to be superparticular. (Look those up
in my Dictionary too if you need to.) In some cases, this
led to ridiculously high prime-factors for a note that would
almost certainly be perceived as a ratio with a much lower
prime-limit. This is how I first developed the idea of 'bridging'
(altho Margo Schulter later supplied the term itself).

I myself would never say that 'it may be important to establish
a limit that is not exceeded in the construction of a scale system'
in a general sense. Indeed, my great fascination with tuning
history is the wonderful variety of tuning systems that I've
found everywhere and everytime. And I emphasize that these are
*systems*.

Just the very fact that humankind has found so many various ways
to carve up and systematize the virtual pitch-continuum was
interesting in itself to me.

Nature presents us with absolute finite limits in hearing at the
upper and lower ends of the continuum (and they vary for every
individual, but fall within the range of roughly 16 to 20000 Hz).

So I began to wonder what the limits are in terms of perceiving
harmonic relationships, or in terms of hearing a particular
interval as a particular interval _gestalt_. (I think Erlich's
harmonic entropy concept is important here.)

Unfortunately, the contexts are so varied that it's been very
difficult for me to quantize it any better than 'somewhere between
11- and 23-prime-limit', but I hold out the hope that with the
growing interest in microtonality, someday we'll figure it out.

And then of course, as with all great scientific discoveries,
it will open doors into myriad other worlds of inquiry...

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Jacky Ligon <jacky_ekstasis@yahoo.com>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> I don't have any definite strong beliefs on exactly what this
> limit is,
> I used primes as high as 271 - up there in La Monte Young
> territory - in my chord-'root' movements.
>

Joe,

I was remembering something from your book about this, but was at my
day job, and couldn't access the information. What I was referring to
was from the chapter "Using Higher Primes/Beyond 19", in which you
say in relation to 17 and 19 identities: "indicates to me that
already a kind of identity limit has been reached at 13". I can see
that you do consider a much broader palette from your post. Thanks
for the clarification. I found this chapter of your book particularly
interesting.

>
> Thru the mid-1990s, while working on the original version of
> my book, I began to form a pretty clear idea that somewhere
> between 11 and 23 there was a kind of perceptual prime-limit
> within which we place what we hear, regardless of how it's
> actually tuned.

Ok, 11-23. Quite allot of choices contained therein!!!

>
> Of course, as was noted here again recently (in fact I think
> it was you), 'context is everything'. So different listeners,
> different composers, different performers, different audio
> equipment, etc. etc. etc., all play a part in what these limits
> are. So I can't define it more precisely than I have above.

You have cleared away the clouds like the sun on a rainy day.

>
> A lot of my ideas on this came together 2 years ago while I
> was making tons of lattice-diagrams of ancient Greek tuning
> systems, and developing my concepts of 'finity' and 'bridging'.

And every time I encounter this in tuning, I think about our
discussion of this during our visits - which is very common to my
harmonic scales. It would seem a very valid argument too. By how many
cents must two ratios that lie near unison for the phenomenon to be
considered "bridging"? Perhaps this is "context dependant" as well?
>
> Many Greek theorists were firm believers in the need for all
> ratios in a tetrachord to be superparticular.

I'm very aware of Ptolemy.

(Look those up
> in my Dictionary too if you need to.) In some cases, this
> led to ridiculously high prime-factors for a note that would
> almost certainly be perceived as a ratio with a much lower
> prime-limit. This is how I first developed the idea of 'bridging'
> (altho Margo Schulter later supplied the term itself).

Even though "a note that would almost certainly be perceived as a
ratio with a much lower prime-limit", I still find myself compelled
to explore these regions, if out of nothing but pure curiosity (note
my Lab contribution).
>
>
> I myself would never say that 'it may be important to establish
> a limit that is not exceeded in the construction of a scale system'
> in a general sense. Indeed, my great fascination with tuning
> history is the wonderful variety of tuning systems that I've
> found everywhere and every time. And I emphasize that these are
> *systems*.

Forgive the poor wording of my question. I was not my intent at all
to suggest that you would advocate an arbitrary limit.

>
> So I began to wonder what the limits are in terms of perceiving
> harmonic relationships, or in terms of hearing a particular
> interval as a particular interval _gestalt_. (I think Erlich's
> harmonic entropy concept is important here.)

I also am very interested in these findings.

>
> Unfortunately, the contexts are so varied that it's been very
> difficult for me to quantize it any better than 'somewhere between
> 11- and 23-prime-limit',

This is a huge resource Monz. Thanks for all of the clarification.

Respectfully,

Jacky Ligon

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Jacky Ligon" <jacky_ekstasis@y...> wrote:
> http://www.egroups.com/message/tuning/12475
>
> I was remembering something from your book about this, but was
> at my day job, and couldn't access the information. What I was
> referring to was from the chapter "Using Higher Primes/Beyond 19",
> in which you say in relation to 17 and 19 identities: "indicates
> to me that already a kind of identity limit has been reached at
> 13". I can see that you do consider a much broader palette from
> your post. Thanks for the clarification. I found this chapter of
> your book particularly interesting.

Oh....OK, Jacky. Thanks for pinpointing the info in my book.
Now I can answer you specifically on that point.

What I developed there was the idea that in traditional 'common-
practice' harmonic theory, chord-members are stacked in '3rds',
but that after the '13th', the cycle returns back to the origin,
i.e., the '15th' = the 'double 8ve'.

As you (and probably everyone else on this List) well know, this
is not necessarily the case with just-intonation. A 5-limit
system can build chords out of just-tuned 'major-' and 'minor-3rds'
(5/4s and 6/5s) in the same way.

But if one's JI system follows the harmonic series strictly into
higher prime- or odd-limits, the '3rds' get progessively smaller,
so that, in an otonality for instance, after the 13th harmonic,
the 15th harmonic is *not* a return to the origin, but rather
an entirely new form of '7th' (the 'major 7th', whereas the 7th
harmonic is closer to a 'minor 7th').

Similarly, if one builds a JI system up to the 31st harmonic,
again the 29th harmonic will be a kind of 'minor 7th' and the
31st will sound like either a wide 'major 7th' or a narrowed
'8ve'.

And what I noticed about this was the traditional harmonic
prescription that 'major' and 'minor' intervals are not used
simultaneously, which seemed to have an echo in JI in that
a nice typically tonal chord would have a '7th' that can be
any one out of the set of 7th, 15th, 29th, or 31st harmonics,
but not more than one of them together.

Now of course, lots of music has been written that doesn't
follow this 'rule' at all - some of it by me :) There's no
reason why any set of tones can't be used, if it sounds right
or suits the compositional purpose.

But I did think that there was a very interesting parallel
with traditional theory there, and so I felt this observation
was a sort of vindication of my prior idea that 13 is a kind
of limit.

So, that fleshes out a bit what I wrote in my book.

> > Many Greek theorists were firm believers in the need for all
> > ratios in a tetrachord to be superparticular.
>
> I'm very aware of Ptolemy.

But there were *lots* of others! Many of them are in my book too:
Eratosthenes, Didymus, etc.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> http://www.egroups.com/message/tuning/12464
>
> Hi Joe... I'm getting "error 404" for practically all the links
> on this post... the ones in the directory "interval," anyway.
> You might want to check those and/or post the files to egroups.
> Thanks!

Wow, Joe and everyone else, I'm terribly sorry! I guess my
fingers just couldn't keep up with my mind today. Here's a
repost of that message with all the correct links. Again,
my apologies... thanks Joe!

------------------------

From: Monz <MONZ@J...>
Date: Thu Sep 7, 2000 2:41pm
Subject: Re: Monz Question: Prime Limit

--- In tuning@egroups.com, "Jacky Ligon" jacky_ekstasis@y... wrote:
> http://www.egroups.com/message/tuning/12456
>
> Monz,
>
> ... You feel that there is a certain Prime Limit beyond which
> we would tend to interpret the intervals as lower number integer
> ratios. I believe it was the 13 Prime Limit (correct me if wrong).
> Please explain what led you to this realization about this
> perceptual limit. And perhaps why it may be important to
> establish a limit that is not exceeded in the construction
> of a scale system.
>
> I should admit here that I frequently cross this line.
> ...

Hi Jacky.

I don't have any definite strong beliefs on exactly what this
limit is, and in the second MIDI example of 'Hendrix Chord'

http://www.ixpres.com/interval/monzo/hendrix/hndrx-ex.mid
http://www.ixpres.com/interval/monzo/hendrix/hendrix.htm

I used primes as high as 271 - up there in La Monte Young
territory - in my chord-'root' movements.

http://www.ixpres.com/interval/monzo/altrock/altrock.mid

I like the way I used 43/32.

http://www.ixpres.com/interval/dict/finity.htm

to follow my train of thought on that.

Just the very fact that humankind has found so many various ways
to carve up and systematize the virtual pitch-continuum was
interesting in itself to me.

Nature presents us with absolute finite limits in hearing at the
upper and lower ends of the continuum (and they vary for every
individual, but fall within the range of roughly 16 to 20000 Hz).

And then of course, as with all great scientific discoveries,
it will open doors into myriad other worlds of inquiry...

-monz
http://www.ixpres.com/interval/monzo/homepage.html