TGs, Unison Vectors, and Octaves

When calculating tonal generators from the interval ratios of a given scale (organized in an order of ascending pitch), followed by determining the unison vectors (or the geometric steps of the geometric steps, organized in a non-redundant order of ascending pitch, themselves), is it INVALID to consider a (beginning, or starting) set of interval ratios which span multiple octaves (as opposed to "octave-reducing" all such interval ratios prior to performing the above described analysis)?

Or, do such calculations remain (in some way) meaningful where "non-octave reduced" interval ratios are utilized as the original "arguments" or "independent variables" of the above described algorithm?

Sincerely, J Gill

--- In tuning-math@y..., J Gill <JGill99@i...> wrote:
> When calculating tonal generators from the interval ratios of a
given scale
> (organized in an order of ascending pitch), followed by determining
the
> unison vectors (or the geometric steps of the geometric steps,
organized in
> a non-redundant order of ascending pitch, themselves), is it INVALID
to
> consider a (beginning, or starting) set of interval ratios which
span
> multiple octaves (as opposed to "octave-reducing" all such interval
ratios
> prior to performing the above described analysis)?
>
> Or, do such calculations remain (in some way) meaningful where
"non-octave
> reduced" interval ratios are utilized as the original "arguments" or
> "independent variables" of the above described algorithm?

I'm not sure exactly what you are asking here, but I think the
answer is that its's perfectly valid. I assume we are talking
only about scales that do repeat at the octave. In that case, factors
of two are irrelevant to most calculations. You can leave them in or
take them out. The two common conventions are:
1. All ratios are octave-reduced to between 1/1 and 2/1,
2. All factors of two are eliminated so that ratios only consist of
odd numbers.

I hope this helps.
-- Dave Keenan

Dave,

Thank you for your response to my post. It did help me determine that the my question, as it was framed, did not accurately reflect what I have been wondering about here.

While multiplying or dividing all of the interval ratios of a scale by a common factor does not alter that scale's resultant "unison vectors", movement (or "modulation") of a (physical) "pattern" of scale steps (on a keyboard) certainly appears to do so.

For instance (as I am certain that you must allready know):

The 3-note scale made up of 4/5, 1/1, 5/4 where the (lowest) pitch of 4/5 is the reference note, is equivalent to the scale 1/1, 5/4, 25/16 (beginning at its reference pitch of 4/5 Hz [cps]), having a single tonal generator of 5/4, and no unison vector.

Yet raising the 4/5 Hz pitch by one octave to become 8/5, where the new set of 3-notes then becomes 1/1, 5/4, and 8/5, *and* referencing the scale to the 1/1 Hz pitch (amounting to a "modulation") yields a scale which has the tonal generators 5/4, 32/25, and a single unison vector of 128/125.

So, in thinking along the lines of the lowest pitch in a set of notes being thought of as the "reference" tonic, while also thinking about the ordering of interval ratios in an order of ascending pitch (from left to right in the set) when setting forth to calculate "tonal generators" and on to the "unison vectors", it appeared to me that the mere act of raising the pitch of 4/5 Hz up an octave to 8/5 Hz was causing an alteration of the scale's "unison vectors". Hence my (somewhat misguided) question...

I believe that I was, in asking the above question, failing to recognize that a "modulation" occurs in the above operation as well, since my approach was to automatically consider the lowest pitch tone in a set to be the "reference tonic". Therein lies my folly in framing my question as I did.

Thank you, nevertheless, for pointing out the parameters which exist, and the conventions which are followed, in the derivation of "tonal generators" and "unison vectors".

Best Regards, J Gill

--- In tuning-math@y..., J Gill <JGill99@i...> wrote:
> Dave,
>
> Thank you for your response to my post. It did help me determine that the
> my question, as it was framed, did not accurately reflect what I have been
> wondering about here.
>
> While multiplying or dividing all of the interval ratios of a scale by a
> common factor does not alter that scale's resultant "unison vectors",
> movement (or "modulation") of a (physical) "pattern" of scale steps (on a
> keyboard) certainly appears to do so.
>
> For instance (as I am certain that you must allready know):
>
> The 3-note scale made up of 4/5, 1/1, 5/4 where the (lowest) pitch of 4/5
> is the reference note, is equivalent to the scale 1/1, 5/4,
> 25/16 (beginning at its reference pitch of 4/5 Hz [cps]), having a single
> tonal generator of 5/4, and no unison vector.

Assuming "tonal generator" means "step size", and assuming the interval of equivalence is the
octave (2:1), it seems you are missing something. There is an additional step size of 32:25, in
addition to two of 5:4 (we like to use "/" for pitches and ":" for intervals).
>
> Yet raising the 4/5 Hz pitch by one octave to become 8/5, where the new set
> of 3-notes then becomes 1/1, 5/4, and 8/5, *and* referencing the scale to
> the 1/1 Hz pitch (amounting to a "modulation") yields a scale which has the
> tonal generators 5/4, 32/25, and a single unison vector of 128/125.

The single unison vector of 128/125 was there just the same in the first scale.
>
> So, in thinking along the lines of the lowest pitch in a set of notes being
> thought of as the "reference" tonic,

By thoughts on tonics are in a much different realm. I see periodicity blocks, etc. as
fundamentally "pre-tonal", as the appearance of diatonic and chromatic scales in Pythagorean
and meantone tuning preceded the appearance ot tonality in Western music.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, July 28, 2001 10:35 AM
> Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves
>
>
>
> [My?] thoughts on tonics are in a much different realm.
> I see periodicity blocks, etc. as fundamentally "pre-tonal",
> as the appearance of diatonic and chromatic scales in Pythagorean
> and meantone tuning preceded the appearance [of] tonality
> in Western music.

Hey Paul,

This is a cool idea, and I agree with it.

It seems to me like you're alluding to my idea of "finity",
in that the composers make use of unison-vectors and the
listeners pick that up, without anyone really being very
conscious of it all. Am on I the right track?

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: Paul Erlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, July 28, 2001 10:35 AM
> > Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves
> >
> >
> >
> > [My?] thoughts on tonics are in a much different realm.
> > I see periodicity blocks, etc. as fundamentally "pre-tonal",
> > as the appearance of diatonic and chromatic scales in Pythagorean
> > and meantone tuning preceded the appearance [of] tonality
> > in Western music.
>
>
> Hey Paul,
>
> This is a cool idea, and I agree with it.
>
> It seems to me like you're alluding to my idea of "finity",
> in that the composers make use of unison-vectors and the
> listeners pick that up, without anyone really being very
> conscious of it all. Am on I the right track?

Sure -- I've been implying something to that effect specifically on
the Tuning List, rather than here. And of course, your inquiries into
finity are what led me to study PBs in the first place.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, July 29, 2001 5:46 PM
> Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves
>
> > [monz:]
> > It seems to me like you're alluding to my idea of "finity",
> > in that the composers make use of unison-vectors and the
> > listeners pick that up, without anyone really being very
> > conscious of it all. Am on I the right track?
>
> Sure -- I've been implying something to that effect specifically on
> the Tuning List, rather than here. And of course, your inquiries into
> finity are what led me to study PBs in the first place.

Right... duh! on my part... of course I remember that --
I was *with* you when it all started!

But as far as the *unconscious* aspect of it... you haven't
really made that explicit, and I suppose that's what I was
doing here. That's probably the thing I find most interesting
about your avenue of exploration. It seems that there are
mathematical ways of modeling our unconscious tonal habits
after all.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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