2 types of consistency (was: Patrick Ozzard Low)

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, February 05, 2002 4:26 PM
> Subject: [tuning] Re: Patrick Ozzard Low
>
>
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_33601.html#33708
>
> > > By the way, the "(which I have)" system that I referred to
> > > above is the one that I intend to start composing in. It's
> > > a near-just 15-limit system of only 17 tones/octave, and
> > > it maps consistently (meaning: like intervals span the same
> > > number of degrees within the system)
> >
> > this alternate meaning of consistency is one that is often
> > found as well, especially in graham breed's writings
> > (monz take note).
> >
>
> ****This isn't so great, is it? These *two* ideas of "consistency"
> are *totally* different, if I'm understanding them correctly.
> Shouldn't there be another term for *one* of them??

yeah, as most of you can imagine, my basket is in a bunch over
this now.

my short-term solution is to label them with their authors's
names, as Manuel did with stability and impropriety. so we'll
have Erlich consistency and Breed consistency?

hopefully we can do better than that... any ideas for a new name?

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:

> my short-term solution is to label them with their authors's
> names, as Manuel did with stability and impropriety. so we'll
> have Erlich consistency and Breed consistency?

i hope not; as george secor was apparently using 'consistency' in
both senses before either graham or i were born, perhaps we should
let him think of names.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > my short-term solution is to label them with their authors's
> > names, as Manuel did with stability and impropriety. so we'll
> > have Erlich consistency and Breed consistency?
>
> i hope not; as george secor was apparently using 'consistency' in
> both senses before either graham or i were born, perhaps we should
> let him think of names.

You don't need two names, because you are really talking about the
same thing occurring in two different types of situations. In the
first place, we wouldn't have any reason to discuss consistency if
there weren't instances in which it were lacking, so I would prefer
to frame a tentative definition in terms of its opposite:

inconsistency - the property of a scale or tonal system such that, in
its employment for the portrayal or approximation of consonant
harmonies based on small-number ratios, one or more of its
constitutent intervals spans an inappropriate or unexpected number of
degrees relative to what exists for the majority of the other
intervals in that scale or system

To illustrate:

1) If a Partch 11-limit hexad (4:5:6:7:9:11) is reduced to a single
octave to form a scale:

1/1, 9/8, 5/4, 11/8, 3/2, 7/4, 2/1

an inconsistency is found in that the 3:4 (or perfect fourth) from
9/8 to 3/2 is 3 degrees, whereas the 3:4 from 3/2 to 2/1 (also a
perfect fourth) is only 2 degrees.

2) If in 24-EDO one attempts to represent a 7-limit tetrad (4:5:6:7)
by the following:

C, 0 degrees
E, 8 degrees
G, 14 degrees
B-sesquiflat, 19 degrees

the triad, 4:5:6, consisting of C, E, and G will be consistent, but
one of the three ratios of 7 will not, inasmuch as the nearest
approximations of these are:

for 4:7, 19 degrees; consistent with C to B-sesquiflat
for 6:7, 5 degrees; consistent with G to B-sesquiflat
for 5:7, 12 degrees; inconsistent with E to B-sesquiflat, which is
11 degrees

In each case an interval was found to span an inappropriate or
unexpected number of degrees, hence the inconsistency of the scale or
tonal system.

Monz, you're free to use or improve upon any of the above as you see
fit, or as any others in the Tuning List may suggest.

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> To illustrate:
>
> 1) If a Partch 11-limit hexad (4:5:6:7:9:11) is reduced to a single
> octave to form a scale:
>
> 1/1, 9/8, 5/4, 11/8, 3/2, 7/4, 2/1
>
> an inconsistency is found in that the 3:4 (or perfect fourth) from
> 9/8 to 3/2 is 3 degrees, whereas the 3:4 from 3/2 to 2/1 (also a
> perfect fourth) is only 2 degrees.

i wouldn't call that inconsistency, and i don't think graham would either. that's simply a lack of the CS property, as Kraig Grady communicated it to us.

> 2) If in 24-EDO one attempts to represent a 7-limit tetrad (4:5:6:7)
> by the following:
>
> C, 0 degrees
> E, 8 degrees
> G, 14 degrees
> B-sesquiflat, 19 degrees
>
> the triad, 4:5:6, consisting of C, E, and G will be consistent, but
> one of the three ratios of 7 will not, inasmuch as the nearest
> approximations of these are:
>
> for 4:7, 19 degrees; consistent with C to B-sesquiflat
> for 6:7, 5 degrees; consistent with G to B-sesquiflat
> for 5:7, 12 degrees; inconsistent with E to B-sesquiflat, which is
> 11 degrees
>
> In each case an interval was found to span an inappropriate or
> unexpected number of degrees, hence the inconsistency of the scale or
> tonal system.

right, this is the definition of consistency as it currently stand in monz' dictionary. but i think graham nicely pointed out other used of the term.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> right, this is the definition of consistency as it currently stand in monz' dictionary. but i think graham nicely pointed out other used of the term.

As far as I can see they all could be subsumed under the same mathematical treatment, which could define consistency in terms of the existence or non existence of certain mappings, but I don't know that anyone but me would like the resulting definitions.

>>>>By the way, the "(which I have)" system that I referred to
>>>>above is the one that I intend to start composing in. It's
>>>>a near-just 15-limit system of only 17 tones/octave, and
>>>>it maps consistently (meaning: like intervals span the same
>>>>number of degrees within the system)
>>>
>>>this alternate meaning of consistency is one that is often
>>>found as well, especially in graham breed's writings
>>>(monz take note).
>>
>>****This isn't so great, is it? These *two* ideas
>>of "consistency" are *totally* different, if I'm
>>understanding them correctly. Shouldn't there be
>>another term for *one* of them??
>
>yeah, as most of you can imagine, my basket is in a bunch over
>this now.

I'm not familiar with Graham's usage here, but this sounds like
constant structures.

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> I'm not familiar with Graham's usage here, but this sounds like
> constant structures.

you're right, CS as Kraig defined it _is_ George's other meaning of
consistency, but Graham brought up some others -- are you in digest
mode, or just 'digesting'?

>>I'm not familiar with Graham's usage here, but this sounds like
>>constant structures.
>
>you're right, CS as Kraig defined it _is_ George's other meaning
>of consistency, but Graham brought up some others -- are you in
>digest mode, or just 'digesting'?

Digesting, I'm afraid. Internet has been down at work for the
last few... feels like days, but it's only been a day. Found
out that not only am I mentally and physically addicted, I depend
on the 'net for virtually all of my financial, legal, etc. etc.
stuff. Anyway, I've powered up the old dial-up line.

Thing is, "consistency" is a handy term with a useful logical
meaning, that's bound to come up in discourse in any topic,
including tuning. It seems silly to put all these in a
dictionary of terminology. I'd say the def. in your paper should
be the only 'special' one; let the others come naturally in the
context of individual conversations, at least for now.

-Carl

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > To illustrate:
> >
> > 1) If a Partch 11-limit hexad (4:5:6:7:9:11) is reduced to a
single
> > octave to form a scale:
> >
> > 1/1, 9/8, 5/4, 11/8, 3/2, 7/4, 2/1
> >
> > an inconsistency is found in that the 3:4 (or perfect fourth)
from
> > 9/8 to 3/2 is 3 degrees, whereas the 3:4 from 3/2 to 2/1 (also a
> > perfect fourth) is only 2 degrees.
>
> i wouldn't call that inconsistency, and i don't think graham would
either. that's simply a lack of the CS property, as Kraig Grady
communicated it to us.

Now suppose that I had started with modulo 6 and attempted to map
that 6-tone chord onto it. Would I then have an inconsistent mapping
of that chord? (In other words, do you find anything wrong with my
definition of inconsistency, and if it's okay, then could the CS
property be considered a particular type of consistency?)

>
> > 2) If in 24-EDO one attempts to represent a 7-limit tetrad
(4:5:6:7) ...
> >
> > In each case an interval was found to span an inappropriate or
> > unexpected number of degrees, hence the inconsistency of the
scale or
> > tonal system.
>
> right, this is the definition of consistency as it currently stand
in monz' dictionary. but i think graham nicely pointed out other used
of the term.

And I agree with Graham that you don't need a need a new term for all
of the cases for which he applies the term "consistency".

--George