Yahoo Tuning Groups Ultimate Backup tuning Re: meditation on a one-string chord

At 10:59 AM 7/29/01 -0400, you wrote:

>Hi,
>
>You wrote:
>
> > Consider a keyboard (for computational convenience) with a bass-note
> > fundamental pitch of 1 Hz (cycle per second), 72 keys per octave, and
> > plenty of keys to span higher octaves, as well:
> >
> > (1) Here I see that the trend of greater and greater sub-
> > divisions of the octave ultimately tends toward a limit
> > where, for instance, on the keyboard described above,
> > there exists an individual key for *each and every*
> > possible harmonic, or "partial", of the 1 Hz bass-note.
> > Thus, the sound spectrum approaches one that could *also*
> > be characterized as exclusively "harmonic", where,
> > relative to the 1 Hz "tonic", ALL of its natural harmonics are
> > represented, and ALL of the harmonics of any of the keys
> > also represent integer multiples of that 1 Hz fundamental.

Jon: Note that when I use the language, "trend of greater and greater sub-
divisions of the octave ultimately tends toward a limit....", I am speaking in
terms of the dominant trend in the extreme in a mathematical perspective,
as opposed to the perspective of "how might I construct or tune practically
play-able instrument". I am not trying to deem a set number as adequate,
but simply pointing out a characteristic through which more sub-divisions
of the octave with "keys" (keys in terms of actual physical keyboard keys)
available to potentially play them will eventually *approach* a *limit* (in the
mathematical sense of limits) where the "keys" played will allow the player
(at least) an *opportunity* to generate such frequencies (which are those
frequencies, or pitches, which are integer multiples of lower pitched "keys").

I am not trying to state that any particular ET or EDO of some given number
of subdivisions of the octave is *the* system by which *all* of this works
ideally, or even adequately. It was intended as statement about the trends
that emerge as one compares the implications of smaller (ie 12) scales
relative to larger (ie 72, or what have you), and not as a tuning system
proposal, per se.

>I don't get this: why do you say there is a key for every harmonic on such
>an instrument? Having the fundamental be 1 Hz or 100 kHz doesn't change
>any of this. Each *low* harmonic of 1 Hz will be represented quite well by
>a degree of the 72-tet scale, but already in the 8th octave there will be
>more than 72 partials to map onto notes, and you won't have enough.

Agreed. The general trend I described remains, nevertheless.

>Similar problem for harmonics of the 72-tet keys: in fact *none* of them,
>except for harmonics of keys a whole number of octaves above 1 Hz, will
>also be harmonics of 1 Hz. Consider the 1st overtone of the 15th key in
>the lowest octave - which harmonic of 1 Hz does this represent? It falls
>between 2 and 3 Hz.

When you say "1st overtone", I assume you mean the 2nd harmonic.
When you say "15th key on the keyboard", if I assume you mean the
key tuned to 15 Hz (cycles per second), with a 2nd harmonic of 30 Hz.

It seems that, instead of thinking of the keyboard in terms of referencing
to the "1st key" (tuned to 1 Hz) at the leftmost position on the keyboard,
you may well be considering the viewpoint of the reference pitch existing,
instead, somewhere more in the middle of the hypothetical keyboard,
where the harmonic integer multiples of numbers which are *less* than
a value of one certainly DO exist at rational, as opposed to integer, values,
indeed. Your point is well taken that my perspective is based (in its strictest
sense) on analyzing the multiples of 1 Hz!

If one considers a tetrachord formed from four "tones" (which each themselves
represent a fundamental frequency as well as equal energy at integer multiples
of those individual fundamental frequencies), this represents a "best-case"
theoretical scenario for the possibility of "common overtones" occuring [which
is probably only *approached* by the frequency spectrums of organ pipes,
and stringed instruments which have been (amplitude) "compressed/limited"
to a large extent (where distortion products at non-harmonically related pitches
will occur due to the non-linear nature of such signal processing techniques)].

If we set the frequencies of this above described "tetrachord" of "tones"at 1 Hz,
3 Hz, 5 Hz, and 7 Hz, and analyze the number of frequencies between 1 Hz and
45 Hz at which "common overtones" can possibly exist, these frequencies (in Hz) are:

3 5 6 7 9 10 12 14 15 18 20 21 24 25 27 28 30 33 35 36 39 40 42 45

which comprise a total of only 24 out of the 44 possible frequencies of where "commonality"
could possibly exist (within the integer-multiple frequencies of 2 Hz to 45 Hz). Overtones
from the 5 Hz "tone" in this "tetrachord" will contribute to the potential number of "common
frequencies" a factor of 3/5 less than the 3 Hz "tone" will, and, similarly, the 7 Hz "tone"
will contribute a factor or 3/7 less than the 3 Hz "tone", etc., ... in terms of any given unit
of *bandwidth* (or arithmetic ranges of frequency). If the reference pitch is increased,
for instance to 2 Hz instead, and the analysis limit is retained at 45 Hz, potential
"commonality" is further divided by a comensurate amount within that 45 Hz bandwidth.

My thoughts on a continuum existing between scales which sub-divide the octave by
amounts small and large does (I think) retain merit in the face of the realities above,
since the number of *choices* of possible chords played in a large scale with many
individual choices (accomplished by up to 10 fingers, anyway), and the possibility
of the resultant frequencies present (when considering the "tones" as described above)
being equal to an integer multiple value of *other* frequencies which are also present
is increased (relative to a smaller scale), due to the number of scale intervals from which
a musician may choose.

>Thanks for explaining this statement -- Jon

Please let me know if I have succeeded in accomplishing your above request!
I am posting this to the "tuning" group as well as replying to you, and I would be
just as interested in other comments which folks may have regarding this stuff...

Best Regards, J Gill

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗J Gill <JGill99@imajis.com>

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗J Gill <JGill99@imajis.com>

--- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:

> Using some such logic, but eliminating the keyboard and the chord,
> can we attempt to draw inferences as to how the Rishi-musicologists
> would have arrived at the number 22 for microtones?

Haresh,

I read your following post clarifying your question quoted above.

While the subject matter you speak of is very interesting to me, and
I have attempted to absorb the content of some of the articles and
postings which exist (including your posts), I consider myself more
of a dazed neophyte than an authority on these interesting subjects.

I find that there appear to be a total of 24 harmonics of a
fundamental tone (within the first 44 harmonics, not counting the
fundamental), the frequency of which can potentially "coincide"
with the (fundamental and/or harmonics) of tones whose fundamental
frequencies exist at the multiples 3, 5, and 7 (where all tones
discussed above consist of energy at the fundamental as well as -
potentially - all harmonic integer multiples of those fundamentals).

From the same data, one can state that the number would be 22 if the
maximum number of harmonics considered is arbitralily limited to
either 40 or 41 (as the figure of the first 45 harmonics is also an
arbitrary choice for an upper limit to such an analysis).

While these results are interesting (to me), and potentially useful
in analysis, nothing about what I am doing indicates that such
information would shed any light whatsoever on the historical data
which you have inquired about in your post...

While I would bet that you, having demonstrated your strong interest
in these matters (in your posts and communications with several other
knowledgable group members), have *allready* become aware of the
links to information below, I will include them just in case they may
be helpful to you in your research:

Have you read this? Lots of info here which "snows" me:
http://www.its.caltech.edu/~vidya/music/srutis.html

There is Joe Monzo's interesting paper at:
http://www.ixpres.com/interval/monzo/indian/indian.htm

There is a link in Monz's document (above) to Paul Erlich's and Joe
Monzo's:
http://www.ixpres.com/interval/td/erlich/srutipblock.htm
which has some thoughts by Erlich regarding the possible evolution of
22, rather than 24 tones (which is followed by a differing
viewpoint). You will find these statements at the end of the document.

Best Regards, J Gill

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, July 30, 2001 12:09 PM
> Subject: [tuning] Re: meditation on a one-string chord
>
>
> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:
> >
> > There is Joe Monzo's interesting paper at:
> > http://www.ixpres.com/interval/monzo/indian/indian.htm
> >
> > There is a link in Monz's document (above) to Paul Erlich's and Joe
> > Monzo's:
> > http://www.ixpres.com/interval/td/erlich/srutipblock.htm
> > which has some thoughts by Erlich regarding the possible evolution
> > of 22, rather than 24 tones (which is followed by a differing
> > viewpoint). You will find these statements at the end of the
> > document.
>
> Actually, this disturbs me a bit because the thoughts I offered were
> a suggestion to help Monz make his viewpoint (which I myself did not
> share) more viable -- and then the "differing viewpoint" merely takes
> issue with some terminology that I was clearly indicating that I was
> unsure about, rather than concerning the content of my suggestion to
> Monz. My own view is seen higher up in the page.

Hi Paul,

Sorry I never got around to doing more with that page.
Lots of other new ones got in the way. :)

One of these days I'll take a closer look and properly address the
points you made. Meantime, if J. and you and anyone else interested
can sort it out without me, I'll add the summary of your conclusions
to the webpage (so keep that in mind as y'all post).

love / peace / harmony ...

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

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🔗J Gill <JGill99@imajis.com>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "J Gill" <JGill99@i...> wrote:

> > There is Joe Monzo's interesting paper at:
> > http://www.ixpres.com/interval/monzo/indian/indian.htm
> >
> > There is a link in Monz's document (above) to Paul Erlich's and
Joe
> > Monzo's:
> > http://www.ixpres.com/interval/td/erlich/srutipblock.htm
> > which has some thoughts by Erlich regarding the possible
evolution
> of
> > 22, rather than 24 tones (which is followed by a differing
> > viewpoint).
> > You will find these statements at the end of the document.
>
> Actually, this disturbs me a bit because the thoughts I offered
were
> a suggestion to help Monz make his viewpoint (which I myself did
not
> share) more viable -- and then the "differing viewpoint" merely
takes
> issue with some terminology that I was clearly indicating that I
was
> unsure about, rather than concerning the content of my suggestion
to
> Monz. My own view is seen higher up in the page.

Paul,

Thanks for setting me straight on that. I apologize for making those
assumptions about the material in question. It was not my intention
to "speak for you" in describing that portion the document, I was
just trying to indicate that a mention of the "22 note" question
seemed to appear at that location. I realize that I should have been
more cautious as to the potential inaccuracy of the comment which I
did make.

Sincerely, J Gill :)