SUBharmonics, Relatedness and Euler

I see that I have been guilty of trying to fit everything to the scheme
where frequencies are in proportion a:b:c:d so that all frequencies are
DIVISIBLE by a common factor whereas it is just as feasible to have them
have a common MULTIPLE.

Of course that means that a minor chord becomes 1/6:1/5:1/4 (a sort of
inverted major) rather than 10:12:15 and the numbers become much
smaller. This is particularly applicable to some of the 4 note chords
that were raised recently (I can't remember which post) and it was these
that got me thinking about this question again. Of course I have always
known that the minor can be expressed this way, but have previously
regarded it as less satisfactory than the major.

It surprises me that I have thought like that (sounds weird I know)
because in the harmonics theory both situations are equally common.

It seems that a major chord such as 4:5:6 somehow evokes the note at 1
which is the key two octaves lower, whereas the minor 1/6:1/5:1/4 evokes
the note at 1 which is two octaves above the fifth. Does anyone else
think of it in this way?

Over the last few days I have been trying to develop an understanding of
the relationship between two notes as being the sum of all possible
relationships that they have will any and all other notes. This is a
bit like Feynman diagrams turned into music. As an example, I had
previously considered the relationship between 1 and 12 as being like
this:
1--2--4
| | |
3--6--12

but also recognising that there are paths from 1 to 12 via all of these
methods 1-12, 1-2-12, 1-3-12, 1-4-12, 1-6-12, 1-2-4-12, 1-2-6-12 and
1-3-6-12. It is these 8 paths that make the relationship 1 to 12 a very
strong one. This is the calculation method that I use in the harmonics
theory.

However two days ago I began to think about the relationship of 2 to 3
and realised that you cannot do it as above and so I started thinking
like 2-6-3, 2-12-3, 2-18-3 and 2-1-3, 2-1/2-3, 2-1/3-3 and so on with
many more steps. Of course I would have to allow the 1-12 relationship
to also include 1-24-12 and 1-1/2-12 and so on to be fair.

My hope is that I will find a way that adds up all these possibilities
with due allowance for the length of the path and the size of the
numbers (without getting infinity - which is a definite possibility) to
get an answer that says definitively what the strength of the
relationship is.

Then in the middle of all this thinking Monzo posts a link to his
translation of Euler's writing on music at
http://www.ixpres.com/interval/monzo/euler/euler-en.htm - POOOWWWEEEEEE!
Here is another man addressing a very similar question and getting a
slightly different answer, but providing great clues. I have never seen
anyone else attempt to do this entirely from first principles before,
something that I have been fiddling with on and off for over 30 years.

Recognising the similarity of this method to Feynman diagrams I see that
it may be a way to link the Harmonics theory to Quantum Physics. The
idea dawns that in all the vibrations in the universe the relatedness
(as measured in some way hinted at above) is the fundamental thing about
how much they will interact and therefore the resulting physics.

Dream on ...

[still trying to flush that next digest out :]
-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
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Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

Ray Tomes wrote:

>I see that I have been guilty of trying to fit everything to the scheme
>where frequencies are in proportion a:b:c:d so that all frequencies are
>DIVISIBLE by a common factor whereas it is just as feasible to have them
>have a common MULTIPLE.

Yes. Of course any Just chord has both a GCD and an LCM, but what we mean
is that for some chords the numbers are a lot smaller when expressed as
divisions of the LCM (LCM=1 instead of GCD=1). These are called utonal
chords, as opposed to otonal. In the utonal form of ratio 1/a:1/b:1/c:1/d
the number 1 represents the so called "guide tone" which is the lowest
harmonic that they all have in common (an actual frequency), while in the
otonal form it represents the virtual fundamental.

There are also chords in which all the intervals are simple but you don't
see this in either the otonal or utonal form of expression (and they are
_not_ contradictory). This was first pointed out by Paul Erlich, see
"Anomalous Saturated Suspensions" (ASS) on Graham Breed's web site or via
Monzo's dictionary. Here's the simplest one:

Minor 7th
C EbG Bb
5:6
4:5
5:6
2 : 3
2 : 3
5 : 9

otonal form: 10 :12 :15 :18
utonal form: 1/18 :1/15 :1/12 :1/10

>Of course that means that a minor chord becomes 1/6:1/5:1/4 (a sort of
>inverted major)

Yes. A shame the word "inverted" already has a quite different meaning in
music. I think I have heard people call it the "dual". Note that the minor
7th above is its own dual.

>rather than 10:12:15 and the numbers become much
>smaller. ... Of course I have always
>known that the minor can be expressed this way, but have previously
>regarded it as less satisfactory than the major.

It _is_ less satisfactory! It's just not quite as bad as 10:12:15 would
make it seem. The maths may be symmetrical but the psychoacoustics isn't.
While the minor (or more generally otonal) chords get their consonance from
the consonance of the individual intervals that make them up, the major or
otonal dual, although it contains exactly the same intervals, gets an extra
boost because all intervals have the same relatively high-pitched virtual
fundamental. The virtual fundamental of the minor triad is so low (due to
10:12:15) that the ear-brain says (to some degree) forget that, lets just
go for the 3 separate but relatively high-pitched virtual fundamentals
(from the three separate intervals).

This is where Sethares theory is limited, it predicts essentially the same
dissonance for a major and a minor (otonal and utonal) having the same mean
frequency. Paul Erlich is still trying to figure out how to extend harmonic
entropy to triads while avoiding this limitation.

>It seems that a major chord such as 4:5:6 somehow evokes the note at 1
>which is the key two octaves lower, whereas the minor 1/6:1/5:1/4 evokes
>the note at 1 which is two octaves above the fifth. Does anyone else
>think of it in this way?

No. I think the minor evokes the virtual fundamental of the 2:3 as the
strongest of the three. Subharmonics are a mathematical construct, not a
psychoacoustic one. Of course the guide tone is actually there and doesn't
have to be "evoked", but it's just a harmonic among others. Its amplitude
may be either increased or reduced depending on relative phases.

>Over the last few days I have been trying to develop an understanding of
>the relationship between two notes as being the sum of all possible
>relationships that they have will any and all other notes. This is a
>bit like Feynman diagrams turned into music. As an example, I had
>previously considered the relationship between 1 and 12 as being like
>this:
> 1--2--4
> | | |
> 3--6--12
...
>My hope is that I will find a way that adds up all these possibilities
>with due allowance for the length of the path and the size of the
>numbers (without getting infinity - which is a definite possibility) to
>get an answer that says definitively what the strength of the
>relationship is.

Let us know if you get some results. Doesn't look too promising to me.

>Recognising the similarity of this method to Feynman diagrams I see that
>it may be a way to link the Harmonics theory to Quantum Physics. The
>idea dawns that in all the vibrations in the universe the relatedness
>(as measured in some way hinted at above) is the fundamental thing about
>how much they will interact and therefore the resulting physics.
>
>Dream on ...

I'm glad you said that last. In music, a fundamental thing is how the
ear-brain has evolved (and been trained) according to whatever could be
most easily cobbled together from whatever went before, in order to give
the greatest reproductive survival to its user, by some incredibly
circuitous causal pathways.

Of course the physics determines that most sounds of interest will have
partials which are harmonic or nearly so, diminishing in amplitude with
frequency. But all the shortcuts taken by the ear-brain system; what
information it ignores or throws away etc. are highly relevant. If you
ignore that, you're just doing numerology.

I love this one that Paul Erlich (who sometimes calls himself Brett Barbaro
on this list) hit me with recently:

If you listen to a noise signal along with a copy that is delayed by 20ms
you will hear a 50Hz tone, where none exists.

Regards,
-- Dave Keenan
http://dkeenan.com

> From: rtomes@kcbbs.gen.nz (Ray Tomes)
>
> I see that I have been guilty of trying to fit everything to the scheme
> where frequencies are in proportion a:b:c:d so that all frequencies are
> DIVISIBLE by a common factor whereas it is just as feasible to have them
> have a common MULTIPLE.

Let's look at it in an other way.
(Excuse me while talking in statements and
using other expressions than used in the tuninglist)
Different tones sound well together when wavelengths fit:
A soundwave never comes alone,
it carries always some overtones and some undertones with it.
Different soundwaves fit when they got common overtones or
common undertones relative closely to the chord.
Talking about theoretical overtones and undertones,
we are talking about harmonics and subharmonics.
Every chord has somewhere a smallest common harmonic and
a largest common subharmonic.
The smallest common harmonic is the fundamental of
an other harmonic range made out of all common harmonics.
The distance between the smallest common harmonic and
the largest common subharmonic is something equivalent with
weighting d*n. (for what it's worth)
Mirror-symmetrical chords sounds tensionless to me.

Peter

[Ray Tomes, TD 185.13]
>
> I see that I have been guilty of trying to fit everything to
> the scheme where frequencies are in proportion a:b:c:d so that
> all frequencies are DIVISIBLE by a common factor whereas it is
> just as feasible to have them have a common MULTIPLE.
>
> Of course that means that a minor chord becomes 1/6:1/5:1/4
> (a sort of inverted major) rather than 10:12:15 and the numbers
> become much smaller.
> <snip>
> Of course I have always known that the minor can be expressed
> this way, but have previously regarded it as less satisfactory
> than the major.

Paul Erlich has posted (with particular reference to Terhardt)
on the fact that utonal [= minor chords expressed with common
multiple: see Partch, _Genesis of a Music_] chords *are* less consonant
than their corresponding 'inverted' otonal [= major].

This is primarily due to the phenomenon of the 'virtual
fundamental', and to summation and difference tones.

(If I've garbled the information, Paul will straighten it out.)

So, in a musically significant way, the 'common multiple' notation
*is* less satisfactory than the 'common divisor' version, Partch's
use of it notwithstanding.

[Ray]
>
> Over the last few days I have been trying to develop an
> understanding of the relationship between two notes as being the
> sum of all possible relationships that they have will any and all
> other notes. This is a bit like Feynman diagrams turned into
> music. As an example, I had previously considered the
> relationship between 1 and 12 as being like this:
>
> 1--2--4
> | | |
> 3--6--12
>
> <snip: all the rest of this post could have been quoted>
>
> However two days ago I began to think about the relationship of
> 2 to 3 and realised that you cannot do it as above
> <snip>
> My hope is that I will find a way that adds up all these
> possibilities with due allowance for the length of the path and
> the size of the numbers (without getting infinity - which is a
> definite possibility) to get an answer that says definitively
> what the strength of the relationship is.

Ray, that's exactly what I've been attempting to do with my
tonal lattice diagrams, ever since I first digested Partch's
theories and realized what a vast playing field we're on now
(in extended JI composition).

I am very mathematically-challenged, but recently I've found
out about 'lattice theory', and began investigating. The
lattice diagrams we use do not meet the formal definition of
the lattices known by mathematicians.

But in a book on lattices recently [sorry, don't have the
reference], I saw the following 'Hasse diagram' in the description
of a poset (= partially-ordered set):

> The following diagram shows clearly the relationship of
> Divisibility [exhibiting the three postulates 'reflexivity',
> 'anti-symmetry', and 'transitivity'] for the set of integers
> consisting of 1, 2, 3, 4, 6, 12:
>
> 12
> / \
> / \
> / \
> 6 4
> / \ /
> / \ /
> / \ /
> 3 2
> \ /
> \ /
> \ /
> 1
>
> Here if a number x a divisor of a number y (x|y)
> then x es BELOW y in the diagram (unless of course x=y)
> and is joined to it by a RISING line -- either a
> line-segment or a broken line ...

Don't know if that helps, but it's awfully similar to your
diagram.

[Ray]
> The idea dawns that in all the vibrations in the universe the
> relatedness (as measured in some way hinted at above) is the
> fundamental thing about how much they will interact and therefore
> the resulting physics.

Exactly the perception I had when I first 'invented' lattices.

For me, the important thing was to pare all ratios down to
their prime-factorizations, because a diagram of that shows the
*patterns of relationship* inherent in the musical system at
large, in its simplest possible representation.

I searched for years to find significant information on Euler's
musical theories, and when I finally got hold of Bailhache's
paper, I too was surprised to see how much Euler's work has in
common with my own.

Joseph L. Monzo monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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[Dave Keenan TD186.4]

>This is where Sethares theory is limited, it predicts essentially the same
>dissonance for a major and a minor (otonal and utonal) having the same mean
>frequency. Paul Erlich is still trying to figure out how to extend harmonic
>entropy to triads while avoiding this limitation.

Ah fun!

>If you listen to a noise signal along with a copy that is delayed by 20ms
>you will hear a 50Hz tone, where none exists.

Fascinating. One of the ways that I analyse time-series for cycles is
to do a lagged autocorrelation and this shows this type of thing well.
By lagged autocorrelation I mean take a correlation between a series and
itself at lags 1, 2, 3, etc up to n and then plot the result. This will
show major peaks at lags which are equal to any frequency present plus
multiples thereof. One reason for doing this is that many series (I am
typical using natural series such as geomagnetic or economic etc) have
strong autocorrelations at the first few multiples but lesser ones at
higher multiples. I believe that this is true because of modulations by
larger cycles. The autocorrelation peaks usually die out exponentially
and the rate is a measure of the stability of the frequency. Your
example hits the ultimate extreme in one direction!

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

[Peter Mulkers TD186.6]

>Let's look at it in an other way.
>(Excuse me while talking in statements and
> using other expressions than used in the tuninglist)
>Different tones sound well together when wavelengths fit:
>A soundwave never comes alone,
>it carries always some overtones and some undertones with it.

That is the logical explanation, but I have not seen the presence of
many undertones (sometimes 1/2) but then again I haven't looked all that
hard. Can anyone comment about the commonness of undertones in real
instruments?

Perhaps this is the explanation for the fact that the common multiple is
more important than the common factor, because overtones are more
powerful and common than undertones. That answers the question
regrading Paul's formula in the previous post.

>Different soundwaves fit when they got common overtones or
>common undertones relative closely to the chord.
>Talking about theoretical overtones and undertones,
>we are talking about harmonics and subharmonics.
>Every chord has somewhere a smallest common harmonic and
>a largest common subharmonic.
>The smallest common harmonic is the fundamental of
>an other harmonic range made out of all common harmonics.
>The distance between the smallest common harmonic and
>the largest common subharmonic is something equivalent with
>weighting d*n. (for what it's worth)
>Mirror-symmetrical chords sounds tensionless to me.

And of course the ratio of the two when factorised gives the range of
each prime index for all the original chords.

[Monzo TD 186.7]
>This is primarily due to the phenomenon of the 'virtual
>fundamental', and to summation and difference tones.

>(If I've garbled the information, Paul will straighten it out.)

>So, in a musically significant way, the 'common multiple' notation
>*is* less satisfactory than the 'common divisor' version, Partch's
>use of it notwithstanding.

Just finished replying to the above and then read this. Are you saying
the same thing as I am in my second paragraph above?

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
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