Yahoo Tuning Groups Ultimate Backup tuning The lie of the land

In trying to design the best possible AJI system, there is a situation
that I think exists but which I am not sure whether anyone else has any
awareness of at all. It took me some years to work this out, but I
think it is possible to explain it and people may have an intuitive
grasp of it already. In essence the idea is that octaves (in quantities
of over two anyway) are not the intervals that most nearly make notes
match up as we move up the scale. I know that this sounds like
blasphemy, but let me put the arguments.

If all "C"s are equivalent and all "F"s are equivalent then it doesn't
matter what chord inversions we do they should be equally harmonic.
However I would suggest that extended 5ths sounds better than extended
4ths (say F-C-G vs G-C-F). The reason that this is so (if it is agreed
that it is) is that the land is not flat, but for every so many octaves
the harmony actually wants a 5th thrown in.

So I would agree that a single octave is the most consonant interval,
followed by a fifth and a fourth. This is accepted by all I think.
Also, a double octave is the next larger consonant interval. However I
don't think that 4 octaves is as consonant as 3 or 4 octaves plus a
fifth (I guess that is technically a 26th or a 33rd :)

As further evidence for this I want to look at two notes at a ratio of
16 in comparison to another two at ratio 24. The 1:16 pair have dual
consonance with other in between notes at 2, 4 and 8. However a 1:24
ratio has consonance with 2, 3, 4, 6, 8, 12 which is much richer. In
part this is due to the greater interval, but even so, a 1:12 ratio has
consonance with 2, 3, 4, 6 which is better than for 1:16.

I suggest in fact that for about every 4.5 octaves a shift of a fifth is
required for the greatest consonance - F0 F1 F2 F3 C4 C5 C6 C7 G7 G8 G9.

The full formula for this is that ratios of 2, 3, 5, 7 etc should occur
in inverse proportion to p*ln(p) which gives relative proportions of
100, 42, 17, 10 etc. and creates the pattern of relationships shown in
http://www.kcbbs.gen.nz/users/rtomes/rt-ha-nm.gif where the taller lines
are at the harmonic numbers that have the greatest consonance with 1.

The reason that I raise all of this is that when the best ratios are
being sought (based on the previous chords etc) for any given note
played, we cannot ignore which octave it is played in. If a given note
is in a higher octave then it is more acceptable to have an additional
ratio of a higher prime (in the numerator) than it is if the same note
is in a lower octave.

-- 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

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

Ray Tomes wrote:

>In essence the idea is that octaves (in quantities
>of over two anyway) are not the intervals that most nearly make notes
>match up as we move up the scale. I know that this sounds like
>blasphemy, but let me put the arguments.

Not blasphemy in this list.

Most people on this list accept that octave equivalence is not total, but
it is often convenient to assume that it is. And some would agree that
there is a weak tritave equivalence (or fifth equivalence) too.

>If all "C"s are equivalent and all "F"s are equivalent then it doesn't
>matter what chord inversions we do they should be equally harmonic.
>However I would suggest that extended 5ths sounds better than extended
>4ths (say F-C-G vs G-C-F). The reason that this is so (if it is agreed
>that it is) is that the land is not flat, but for every so many octaves
>the harmony actually wants a 5th thrown in.

There are several possible explanations for this, some of which are
equivalent but at different levels of description. But of course they must
all deny strict octave equivalence. Mine would be that their dissonance is
dominated by the outer intervals and 9:16 is way more complex than 4:9, its
proximity to 5:9 brings it down a little, but not enough to be better than
4:9. And of course the inner intervals are more consonant to start with.

>So I would agree that a single octave is the most consonant interval,
>followed by a fifth and a fourth. This is accepted by all I think.

Yes. Provided we are talking about JI with typical harmonic timbres, and
the intervals all overlap significantly in their frequency range (e.g. same
upper note or same lower note or something in between. Since consonance for
a given interval increases with frequency it is possible to have a high
fifth which is more consonant than a low octave.

>Also, a double octave is the next larger consonant interval. However I
>don't think that 4 octaves is as consonant as 3 or 4 octaves plus a
>fifth (I guess that is technically a 26th or a 33rd :)

A 28th and a 36th I believe, but it's much easier to understand "<n>
octaves plus <whatever>" particularly once one gets past a 13th. Or just
give the ratios, 1:12 and 1:24. But I even think that one or two octaves
plus a fifth (1:3, 1:6) and two octaves plus a major third (1:5) are about
as consonant as any number of octaves (if the lower note is the same in
every case).

>As further evidence for this I want to look at two notes at a ratio of
>16 in comparison to another two at ratio 24. The 1:16 pair have dual
>consonance with other in between notes at 2, 4 and 8. However a 1:24
>ratio has consonance with 2, 3, 4, 6, 8, 12 which is much richer. In
>part this is due to the greater interval, but even so, a 1:12 ratio has
>consonance with 2, 3, 4, 6 which is better than for 1:16.

I think these reasons are spurious, but we have to be careful here. Are you
talking about the dissonance of the bare dyad (interval) or are you talking
about the _contribution_ of this dyad to the dissonance of various chords
of which it might be a subset. There are definitely synergies occuring in
some chords and not others which I think renders such a project fairly
meaningless.

It's a pity you didn't join the list a few months back. We had a heavy
session on formulae for dyadic dissonance (dissonance of bare intervals).
I'll refer to various people below. You can find their stuff by going to
<http://www-math.cudenver.edu/~jstarret/microtone.html> and then to the
"microtonalists" page and then find the persons name alphabetically.

Now dyadic dissonance has been modelled very well by Bill Sethares using
calculations which take the Fourier transform of the tones involved and sum
the contribution of every pair of frequencies to the overall dissonance,
based on an experimentally determined dissonance curve for a pair of sines.
This even works for non-harmonic timbres and partly works for more than two
notes.

But this is not the sort of calculation you can do in your head. What we
wanted was a simple formula that gave a reasonable approximation for
typical harmonic timbres.

We broke the dyadic dissonance down into three components called
COMPLEXITY, TOLERANCE and SPAN. COMPLEXITY relates to the size of the
numbers in the ratio and gives the dissonance for dyads that are both
simple and narrow, under certain conditions, TOLERANCE refers to the
consonance of very complex dyads due to their nearness to simpler ones, and
SPAN refers to the consonance of very wide dyads despite their complexity.

We compared our experience (and some deliberate experiments) with about 18
different COMPLEXITY formulae. We came to a reasonably strong consensus
that prime factorisation was a red herring in this regard. At least two of
us started out supporting a weighted prime exponent scheme but changed our
tune by the end. What we came up with was that for a typical harmonic
timbre, dyads that can be represented as a ratio x:y in lowest terms, where
x+y < 17, have relative dissonances given by the following formulae in the
following cases:
When the dyads all have their
lowest note the same Min(x,y)
highest note the same Max(x,y)
mean frequency the same (x+y)/2
mean pitch the same sqrt(x*y)

The first two have also been less accurately called "numerator" and
"denominator" respectively, based on a convention that considers the ratio
to be always a fraction greater than one. But note that musical interval
ratios are really unordered, so x:y = y:x.

It is tempting to conclude from the above formulae that the dissonance is
given by the period of the virtual fundamental, but this only works when
the dyads being compared are in the same frequency range. Consider a fifth
versus an octave where the high note of the octave is the same as the low
note of the fifth. I don't think anyone finds these to have the same
dissonance, although they have the same virtual fundamental.

Note that the above conditions ensure that one dyad fits inside the span of
the other. Unfortunately we haven't yet considered how to compare the
dissonance of dyads not having a significant part of their span in common.

Obviously the x+y < 17 is not a sharp cutoff. It might be pushed to 19 for
a timbre rich in higher harmonics or lowered to 13 for one weak in higher
harmonics.

The reason for having any cutoff at all, is that the ear-brain system has a
pitch error TOLERANCE. Clearly a ratio of 1000:2001 sounds pretty much as
consonant as an octave. But even a ratio like 8:13 is slightly less
dissonant than expected from its COMPLEXITY, due to its proximity to 3:5
and 5:8. Experiments have shown this tolerance to be a bell curve with a
standard deviation of about 1% (17 cents) in the midrange. Here's a rough
justification of the x+y < 17 cuttoff. The difference between the major and
minor whole tones 8:9 and 9:10 is 21.5c, the familiar syntonic comma, not
much more than one standard deviation, and these two intervals don't seem
to have a dissonance hump between them. Now 8+9 = 17 and 9+10 = 19.

Paul Erlich has found a way to combine some simple COMPLEXITY measures with
TOLERANCE to give curves very similar to those of Sethares, although he
tends to assume octave equivalence. He calls this Harmonic Entropy.

With regard to SPAN, some say that very wide intervals are neither
consonant nor dissonant. I prefer to stick to talking about degrees of
dissonance (rather than consonance) since we can all agree that wide
intervals are not dissonant. Maybe we need a term "asonant".

The really hard part is dealing with more than two notes. More about this
later.

>I suggest in fact that for about every 4.5 octaves a shift of a fifth is
>required for the greatest consonance - F0 F1 F2 F3 C4 C5 C6 C7 G7 G8 G9.

I would say that with the lowest note kept constant, the following
intervals are all roughly equally consonant 1:2, 1:3, 1:4, 1:5, 1:6, etc.

>The full formula for this is that ratios of 2, 3, 5, 7 etc should occur
>in inverse proportion to p*ln(p) which gives relative proportions of
>100, 42, 17, 10 etc. and creates the pattern of relationships shown in
>http://www.kcbbs.gen.nz/users/rtomes/rt-ha-nm.gif where the taller lines
>are at the harmonic numbers that have the greatest consonance with 1.

I don't understand the relevance of 4 and 5 digit harmonic numbers to
music. Where does the p*ln(p) prime weighting come from? How does it relate
to human psycho-acoustics? I have to admit to being highly skeptical of any
scheme for modelling the subjective human experience of dissonance that
also claims to explain the structure of the universe!

Can you give the complete formula for your proposed complexity (or
dissonance) metric for a ratio x:y, assuming that x:y is in lowest terms
and that x = 2^a * 3^b * 5^c * 7^d * ... and y = 2^A * 3^B * 5^C * 7^D *
... or some such. I'll include it with the other 18 formulae in my Harmonic
Complexity metrics spreadsheet (on my website).

>The reason that I raise all of this is that when the best ratios are
>being sought (based on the previous chords etc) for any given note
>played, we cannot ignore which octave it is played in. If a given note
>is in a higher octave then it is more acceptable to have an additional
>ratio of a higher prime (in the numerator) than it is if the same note
>is in a lower octave.

I would agree with this if "prime" was changed to simply "whole number",
but not for the reasons you give. Please read Sethares and Erlich's stuff.
For Harmonic Entropy you can look it up in Monzo's dictionary.

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

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

[Dave Keenan TD 186.3]
... snip

>Ray Tomes wrote:
>>Also, a double octave is the next larger consonant interval. However I
>>don't think that 4 octaves is as consonant as 3 or 4 octaves plus a
>>fifth (I guess that is technically a 26th or a 33rd :)

>A 28th and a 36th I believe, but it's much easier to understand "<n>
>octaves plus <whatever>" particularly once one gets past a 13th.

One of us ran out of fingers and toes but I think it was you :)
You have to take 7*(3 or 4)+5 I think.

> Or just give the ratios, 1:12 and 1:24.

Yeah lets do that and save taking our socks off (joke about using toes).

>>As further evidence for this I want to look at two notes at a ratio of
>>16 in comparison to another two at ratio 24. The 1:16 pair have dual
>>consonance with other in between notes at 2, 4 and 8. However a 1:24
>>ratio has consonance with 2, 3, 4, 6, 8, 12 which is much richer. In
>>part this is due to the greater interval, but even so, a 1:12 ratio has
>>consonance with 2, 3, 4, 6 which is better than for 1:16.

>I think these reasons are spurious, but we have to be careful here. Are you
>talking about the dissonance of the bare dyad (interval) or are you talking
>about the _contribution_ of this dyad to the dissonance of various chords
>of which it might be a subset. There are definitely synergies occuring in
>some chords and not others which I think renders such a project fairly
>meaningless.

Well I guess that the argument does slightly beg the question of
possible other notes and the possibility of them relating to both.
I am happy for now if that much is accepted, although I think that there
is a smaller factor in relation to just the two notes themselves.

>Now dyadic dissonance has been modelled very well by Bill Sethares using
>calculations which take the Fourier transform of the tones involved and sum
>the contribution of every pair of frequencies to the overall dissonance,
>based on an experimentally determined dissonance curve for a pair of sines.
>This even works for non-harmonic timbres and partly works for more than two
>notes.

I have looked at Bill's graphs and they look good to me, but I haven't
seen any that go outside the ratios 1:1 to 1:2. Are there any?

>We broke the dyadic dissonance down into three components called
>COMPLEXITY, TOLERANCE and SPAN.

The following discussion is very interesting and I will need to digest
it and try to translate it to compare to my own views. Chances are that
will take a couple of days. I will be away from home from tomorrow for
2 weeks and so will get onto this after that. (The tuning list posts
will briefly shrink by about 50% :)

... lots of interesting stuff snipped which I hope to come back to ...

>>I suggest in fact that for about every 4.5 octaves a shift of a fifth is
>>required for the greatest consonance - F0 F1 F2 F3 C4 C5 C6 C7 G7 G8 G9.

>I would say that with the lowest note kept constant, the following
>intervals are all roughly equally consonant 1:2, 1:3, 1:4, 1:5, 1:6, etc.

In that case my argument depends on the ability to combine with as yet
unspecified third and fourth notes etc.

>>The full formula for this is that ratios of 2, 3, 5, 7 etc should occur
>>in inverse proportion to p*ln(p) which gives relative proportions of
>>100, 42, 17, 10 etc. and creates the pattern of relationships shown in
>>http://www.kcbbs.gen.nz/users/rtomes/rt-ha-nm.gif where the taller lines
>>are at the harmonic numbers that have the greatest consonance with 1.

>I don't understand the relevance of 4 and 5 digit harmonic numbers to
>music.

Clearly if the fundamental is in our hearing range then they will be at
most of interest to dogs (joke). However, if the fundamental is
something much lower, perhaps 1 oscillation per 3 seconds or less, then
these harmonics will be what occupies the interesting part of the
hearing range.

Sidetrack for a moment. I put the 1 per 3 seconds figure because that
is the typical rate of breathing for a walking person. If all the most
closely related harmonics of 1/3 Hz are played then it might well be
that it has some power to entrain a person's breath (or heart or brain
or whatever). I am plotting to make music do more than just give the
emotions a little prod. :-)

Additional sidetrack. One of the ways that music works on the emotions
is through the implied fundamentals of chords hitting resonances within
the human body and mind. We have these things which the Indians call
chakras which are located at close to 1/12s of the way up the human body
and therefore one falls between the legs (1/2 way) and one in the heart
(3/4 way) and so on. These chakra are supposed to be associated with
various emotions and attributes of humans and so hitting chords and
sequences that activate these can set off emotions. The fundamental
waves of the human body are ~1/3 Hz breathing but varies with bodily
activity, ~1 Hz heart likewise, ~8 Hz brain which varies with type of
mental activity. There are interconnections between these and the
chakras also. It seems that the frequencies which activate chakras may
be in these ELF (extra Low Frequency) range rather than in the hearing
range. They are therefore mainly activated by rhythm and implied
fundamentals.

So to return to the 4 and 5 digit harmonics. If we want all the
relationships to many notes starting from the fundamental then the
initial pattern of numbers that you see at the low end is probably
pretty much what you would expect. What happens then is that each
higher harmonic is considered in relationship to all the existing ones,
taking account of things something like your span etc above. Through
this process the whole pattern is developed.

>Where does the p*ln(p) prime weighting come from?

When the calculations are done for the harmonics the relationship of
every note to every other is taken as follows:

note relates to notes
1 --> 2 3 4 5 6 7 8 9 10 11 12 13 etc
2 --> 4 6 8 10 12 etc
3 --> 6 9 12
4 --> 8 12
5 --> 10
6 --> 12 etc etc etc

and that is how the pattern develops. On this basis the strongest
harmonics in the 48 to 96 range are 48,60,72, 96 or a major chord and
the next strongest 48,54,56,60,64,72,80,84,90,96 which is the just
intonation scale plus Eb and Bb if we are in C. On this basis 34560 is
the strongest relationship.

When the above pattern is calculated out to high numbers then the ones
with the most ways of being produced (for the size of the harmonic -
since the numbers go up rapidly anyway) turn out to be 1 2 4 12 24 48
144 288 1440 2880 5760 17280 34560 69120 etc. On this basis 34560 is
the strongest relationship. These numbers have ratios between adjacent
numbers of 2 2 3 2 2 3 2 5 2 2 3 2 2 etc and in this series the numbers
appear with frequency 1/(p*ln(p)).

>How does it relate to human psycho-acoustics?

I assume that it must because human music has so many features that are
totally consistent with the results from my harmonics calculations. I
mentioned the major chord and just intonation scale above. Minor, Blues
and Indian music scales are also found. Additions of the predicted
strong harmonics also develop all the main rhythms. As well as that it
seems to fit the stucture of the universe of which we are children.

>I have to admit to being highly skeptical of any
>scheme for modelling the subjective human experience of dissonance that
>also claims to explain the structure of the universe!

Fine, be skeptical, but check it out.

>Can you give the complete formula for your proposed complexity (or
>dissonance) metric for a ratio x:y, assuming that x:y is in lowest terms
>and that x = 2^a * 3^b * 5^c * 7^d * ... and y = 2^A * 3^B * 5^C * 7^D *
>... or some such. I'll include it with the other 18 formulae in my Harmonic
>Complexity metrics spreadsheet (on my website).

Well it depends on where the x:y is within the structure (unless we
interpret it as absolute x and y which I think would be a mistake).
I recently posted that I was trying to solve this one with a slightly
new perspective - as yet incomplete.

I can give an average value based on different a set of octaves picked
from various parts of the harmonic pattern. I hope to soon produce a
set of graphics with these selections so that you can see how they vary
slightly as we move along the pattern. Briefly though, at the low end
2 and 3 ratios are more important (pythagorean) but quickly the 5s make
a showing (galilean) and higher up the 7s (blues). Eventually 11, 13
and the others come in. From the size of the universe it takes all the
way to the size of an electron and proton for 17 to appear in the e/p
mass ratio of 1:1836 (2*2*3*3*17) and 19 and 23 appear in accelerator
created particles mass ratios. Note that masses are the same as
frequencies according to Planck's and Einstein's laws E=hf and E=mc^2.

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

[Ray Tomes TD 187.4]
>One of us ran out of fingers and toes but I think it was you :)
>You have to take 7*(3 or 4)+5 I think.

Oh yeah. It was me. Sorry. I guess that helps make the point tho. :-)

>Well I guess that the argument does slightly beg the question of
>possible other notes and the possibility of them relating to both.
>I am happy for now if that much is accepted, although I think that there
>is a smaller factor in relation to just the two notes themselves.

Ok. So you are not really talking about the consonance/dissonance of the
pair of notes but rather something I will call their harmonic fecundity or
gregariousness. i.e. their potential for forming relatively consonant
groups with other notes. This is a very useful thing to quantify but you
seem to have been confusing it with how the two notes sound together. The
two are not related. Prime factorisation is definitely relevant to the
former but irrelevant to the latter (except in so far as it aids in
reducing a ratio to lowest terms). The latter depends only on the _size_ of
the numbers involved in the lowest-terms ratio, not their further
factorisations. This is demonstrated by the fact that 8:9 is not more
consonant than 7:8, nor is 15:8 more consonant than 13:7.

>In that case my argument depends on the ability to combine with as yet
>unspecified third and fourth notes etc.

Yes.

...
>Sidetrack for a moment. I put the 1 per 3 seconds figure because that
>is the typical rate of breathing for a walking person. If all the most
>closely related harmonics of 1/3 Hz are played then it might well be
>that it has some power to entrain a person's breath (or heart or brain
>or whatever). I am plotting to make music do more than just give the
>emotions a little prod. :-)

It _may_ be. But I'd wan't some evidence before I bothered trying to
musically use virtual fundamentals significantly below the range of human
hearing.

>Additional sidetrack. One of the ways that music works on the emotions
>is through the implied fundamentals of chords hitting resonances within
>the human body and mind.

Mind yes, body no. The ear-brain responds in some ways the same as it would
if the virtual/implied fundamental were really there, but it does not
constitute air vibrations at that frequency and so is not capable of
exciting mechanical resonances in body parts.

>We have these things which the Indians call
>chakras which are located at close to 1/12s of the way up the human body
>and therefore one falls between the legs (1/2 way) and one in the heart
>(3/4 way) and so on. These chakra are supposed to be associated with
>various emotions and attributes of humans and so hitting chords and
>sequences that activate these can set off emotions. The fundamental
>waves of the human body are ~1/3 Hz breathing but varies with bodily
>activity, ~1 Hz heart likewise, ~8 Hz brain which varies with type of
>mental activity. There are interconnections between these and the
>chakras also. It seems that the frequencies which activate chakras may
>be in these ELF (extra Low Frequency) range rather than in the hearing
>range. They are therefore mainly activated by rhythm and implied
>fundamentals.

I haven't personally seen any scientifically respectable evidence of the
validity of chakra theory, and I think you've got it somewhat garbled
anyway, but we don't need to invoke chakras to establish that various
aspects of music trigger various emotions. As for body resonances, rhythm
yes, low audio frequencies yes, subsonics yes, implied fundamentals no
(irrespective of their frequency).

You could very easily perform an experiment for yourself that would prove
this. Take a high Q mechanically resonant system, say an A440 tuning fork,
and play a chord having 440Hz as its virtual fundamental but not containing
440Hz. To avoid affects of startup and stopping transients ramp the
amplitude up gradually then down gradually and see iff the tuning fork is
ringing. Of course a small amount of actual 440Hz will be generated by
non-linearities (primarily in the speakers), so compare it to the case when
you actually play 440Hz.

Alternatively you could use a 1Hz pendulum and play a pair of audio tones
1Hz apart. There's very little chance that the speakers can generate any
actual 1Hz through nonlinearities, and the result (or lack thereof) will be
visible even while you are playing the tones.

...
>>Where does the p*ln(p) prime weighting come from?
>
>When the calculations are done for the harmonics the relationship of
>every note to every other is taken as follows:
>
>note relates to notes
>1 --> 2 3 4 5 6 7 8 9 10 11 12 13 etc
>2 --> 4 6 8 10 12 etc
>3 --> 6 9 12
>4 --> 8 12
>5 --> 10
>6 --> 12 etc etc etc
>
>and that is how the pattern develops. On this basis the strongest
>harmonics in the 48 to 96 range are 48,60,72, 96 or a major chord and
>the next strongest 48,54,56,60,64,72,80,84,90,96 which is the just
>intonation scale plus Eb and Bb if we are in C. On this basis 34560 is
>the strongest relationship.
>
>When the above pattern is calculated out to high numbers then the ones
>with the most ways of being produced (for the size of the harmonic -
>since the numbers go up rapidly anyway) turn out to be 1 2 4 12 24 48
>144 288 1440 2880 5760 17280 34560 69120 etc. On this basis 34560 is
>the strongest relationship. These numbers have ratios between adjacent
>numbers of 2 2 3 2 2 3 2 5 2 2 3 2 2 etc and in this series the numbers
>appear with frequency 1/(p*ln(p)).

Aha! This makes sense now. This is not a way of determining relative
consonance of intervals or chords, but it may well be a very interesting
and useful way of finding Just scales that maximise the number of _otonal_
chords for a limited number of notes. But I'm not sure it's clear to
everyone yet (me included) how to use it for that purpose or why it works.

The way that many on this list have been looking for such scales
previously, is by drawing graphs (or networks) of the consonant
relationships (that we've been calling lattices) in such a way that the
most consonant chords form highly symmetrical polyhedra and so stand out
for the human visual system.

As you have now recognised, your method ignores utonal chords, mixed
o/u-tonal (ASSs) and inconsistent (contradictory/necesarily tempered)
chords. Whereas our lattices make the opposite mistake of according utonal
equal status to otonal. But lattices merely give less importance to the
last two types rather than ignoring them completely.

>>How does it relate to human psycho-acoustics?
>
>I assume that it must because human music has so many features that are
>totally consistent with the results from my harmonics calculations. I
>mentioned the major chord and just intonation scale above. Minor, Blues
>and Indian music scales are also found.
...

It's gotta be because your method maximises the number of small whole
number ratios available, weighted towards the lowest numbers in a very good
way. And for humans lower numbers = greater consonance.

...
>>Can you give the complete formula for your proposed complexity (or
>>dissonance) metric ...
>
>Well it depends on where the x:y is within the structure (unless we
>interpret it as absolute x and y which I think would be a mistake).
>I recently posted that I was trying to solve this one with a slightly
>new perspective - as yet incomplete.

Yes but how x:y sounds doesn't depend on what other notes are available in
the scale. So again, this establishes that you are not talking about a
consonance/dissonance metric for intervals or chords. But a harmonic
usefulness (fecundity?) metric for scales.

...
> Briefly though, at the low end
>2 and 3 ratios are more important (pythagorean) but quickly the 5s make
>a showing (galilean) and higher up the 7s (blues). Eventually 11, 13
>and the others come in.

Fine. Most people accept that. Except I don't know what Galileo had to do
with 5-limit.

> From the size of the universe it takes all the
>way to the size of an electron and proton for 17 to appear in the e/p
>mass ratio of 1:1836 (2*2*3*3*17) and 19 and 23 appear in accelerator
>created particles mass ratios. Note that masses are the same as
>frequencies according to Planck's and Einstein's laws E=hf and E=mc^2.

Aw c'mon Ray. Surely you know that these constants have been measured to
extreme accuracy and none of them are integers. Proton-electron mass ratio
is given at
http://physlab.nist.gov/cgi-bin/cuu/Value?mesmp|search_for=atomic!
as 1836.152701 with a standard uncertainty of .000037. Does the universe
use temperaments then?

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

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

Dave Keenan [188.13]

>Ok. So you are not really talking about the consonance/dissonance of the
>pair of notes but rather something I will call their harmonic fecundity or
>gregariousness. i.e. their potential for forming relatively consonant
>groups with other notes. This is a very useful thing to quantify but you
>seem to have been confusing it with how the two notes sound together.

I plead guilty. It is as you say above. Thanks for helping me clarify
my thinking on this one.

>It _may_ be. But I'd wan't some evidence before I bothered trying to
>musically use virtual fundamentals significantly below the range of human
>hearing.

There is a fellow called Ed Wagner who measures waves in trees which
travel at just under 1 m/s and similar waves I believe exist in humans
(as well as faster nerve waves). Interestingly, a 1 m/s wave takes
about 3.5 seconds to bounce between our head and feet and back again -
an amount of time which is close to our normal breathing rate.

Rhythms which are near multiples of this rate (of about 17 bpm, but
varying with height) will establish a standing wave pattern in a person.
If energy concentrations occur at certain locations in the body then
various organs can be affected and can affect moods.

>Mind yes, body no. The ear-brain responds in some ways the same as it would
>if the virtual/implied fundamental were really there, but it does not
>constitute air vibrations at that frequency and so is not capable of
>exciting mechanical resonances in body parts.

It is not mechanical, it is electrical and chemical or nerve waves.

>I haven't personally seen any scientifically respectable evidence of the
>validity of chakra theory, and I think you've got it somewhat garbled
>anyway,

Well I probably caused confusion by changing from chakra theory to my
explanation of it half way through. It is not standard theory but my
suggestion that standing waves are involved. Let me suggest that you do
the measurements for this. Take your height and divide by two and
measure where that is in your body (it is where the base of the spine
and sex chakras are). The divide the top half in two again and you get
the heart. Divide the top again and you get the throat and the next
half is between the eyes.

>but we don't need to invoke chakras to establish that various
>aspects of music trigger various emotions. As for body resonances, rhythm
>yes, low audio frequencies yes, subsonics yes, implied fundamentals no
>(irrespective of their frequency).

If you look at any chord then the implied fundamental is the frequency
at which a single regular location in the wave is not moving.

>You could very easily perform an experiment for yourself that would prove
>this. Take a high Q mechanically resonant system, say an A440 tuning fork,
>and play a chord having 440Hz as its virtual fundamental but not containing
>440Hz. To avoid affects of startup and stopping transients ramp the
>amplitude up gradually then down gradually and see iff the tuning fork is
>ringing. Of course a small amount of actual 440Hz will be generated by
>non-linearities (primarily in the speakers), so compare it to the case when
>you actually play 440Hz.

But the human body is not a high Q system. It is probably highly
non-linear.

>Aha! This makes sense now. This is not a way of determining relative
>consonance of intervals or chords, but it may well be a very interesting
>and useful way of finding Just scales that maximise the number of _otonal_
>chords for a limited number of notes. But I'm not sure it's clear to
>everyone yet (me included) how to use it for that purpose or why it works.

Yes, you are right. It does give a set of values which have the maximum
possible number of relationships.

>The way that many on this list have been looking for such scales
>previously, is by drawing graphs (or networks) of the consonant
>relationships (that we've been calling lattices) in such a way that the
>most consonant chords form highly symmetrical polyhedra and so stand out
>for the human visual system.

However there is no reason in these networks to stop after some number
of powers of each prime. There is no reason for the higher number of
powers of 2 and 3 allowed and so on. The harmonics theory makes clear
exactly how important (relatively) each harmonic is. There are no
arbitrary assumptions and it all comes out.

>As you have now recognised, your method ignores utonal chords, mixed
>o/u-tonal (ASSs) and inconsistent (contradictory/necesarily tempered)
>chords. Whereas our lattices make the opposite mistake of according utonal
>equal status to otonal. But lattices merely give less importance to the
>last two types rather than ignoring them completely.

The harmonics theory produces parts where there are more utonal or
otonal relationships amoung the strongest harmonics. I have referred to
these parts as emphasizing major or minor which I think that we agreed
was equivalent.

>It's gotta be because your method maximises the number of small whole
>number ratios available, weighted towards the lowest numbers in a very good
>way. And for humans lower numbers = greater consonance.

Right.

>Aw c'mon Ray. Surely you know that these constants have been measured to
>extreme accuracy and none of them are integers. Proton-electron mass ratio
>is ... 1836.152701 with a standard uncertainty of .000037. Does the universe
>use temperaments then?

Yes, you have got it (OK, I know that you were being cheeky but you did
get it right). I gave the same answer to another question. Look at the
masses of H, He etc and you find 1.008, 4.004 which is not exactly 4
times as much. The reason is that there are always small other energies
involved which are called binding energies or other terms. If you doubt
that the 1836 ratio is real then consider this. Take a mass of 68 times
and electron. This divides 27 times (almost exactly) into a proton or
neutron mass. It so happens that it also divides accurately 3 times
into the muon mass, 4 times into the pion mass and very close to an
integer times into many other particles. The small variations from
integers are no more significant than the same variations for isotopes
which are accepted as being whole numbers of nucleons. What is more I
used this plus other information in 1994 to publicly predict that there
must exist a particle of mass 68 times an electron. Such a particle was
discovered in 1995. That is the true test.