Di Veroli book and old chestnut

C. di Veroli:
UNEQUAL TEMPERAMENTS: THEORY, HISTORY AND PRACTICE (2008)

http://www.lulu.com/content/5179073

- the author, I'm sure, would be happy to have your order.

One hoary old unanswered question I bumped into reading the first few
dozen pages:
Why is 4:5:6 more consonant than 10:12:15?

The author puts it down to beats and (non-)coincidence of partials in
the constituent intervals of each triad, but that obviously doesn't do
the job. But almost in the same breath he mentions the smallness of the
integer frequency ratios, which is actually a property of
the 'periodicity note', i.e. the pitch '1' of which all heard
frequencies are harmonics, - not the coincidence or otherwise of
partials.

I can't immediately find a good basic discussion of the possible
relation between periodicity and chord-consonance online - is there
one? Carl?

Given the still-disputable nature of the question I don't think we can
hold it against him not to have answered it...

~~~T~~~

> I can't immediately find a good basic discussion of the possible
> relation between periodicity and chord-consonance online - is
> there one? Carl?

The idea dates back to Galileo, who first proposed the n*d
consonance ranking (maybe it was Galileo Sr., I forget).

Rameau championed related ideas (guidetone or whatever he
called it).

It's been prominent in psychoacoustics at least since
Carl Seashore in the '20s, if memory serves. Though usually
credited with devising the competing theory, several sections
I've read in Helmholtz seem to be in support of it also.

Terhardt was a big turning point:
http://www.mmk.e-technik.tu-muenchen.de/persons/ter.html

Peter Cariani is a more recent figure:
http://www.cariani.com

If you think there's a contrast with major/minor triads,
just wait until you get to the 7-limit, 11-limit, and beyond.
The roughness models of Plomp, Levelt, Sethares, et al
completely fail to account for it. It is an obvious
demonstration that periodicity must play at least an equal
if not much greater role in the consonance distinctions
relevant in music.

Paul Erlich's harmonic entropy model can be interpreted as
modeling periodicity. It accounts for the major/minor
asymmetry, anyhow.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I can't immediately find a good basic discussion of the possible
> > relation between periodicity and chord-consonance online - is
> > there one? Carl?
>
> The idea dates back to Galileo, who first proposed the n*d
> consonance ranking (maybe it was Galileo Sr., I forget).
>

You'll have to excuse my ignorance, but I can't find any reference
to 'n*d consonance ranking' either - who invented the terminology? Is
it in tonalsoft under a different name? Presumably Galileo (G. or V.)
called it still something else, if he gave any name at all.

> Rameau championed related ideas (guidetone or whatever he
> called it).
>

That ought to be slightly more comprehensible, and Rameau's (rather
extensive) theoretical works are online.

I am slightly surprised if there is no basic modern explanation of
concepts that are (apparently) so intuitively obvious...
~~~T~~~

> > The idea dates back to Galileo, who first proposed the n*d
> > consonance ranking (maybe it was Galileo Sr., I forget).
>
> You'll have to excuse my ignorance, but I can't find any reference
> to 'n*d consonance ranking' either - who invented the terminology?

That's not terminology! It's called Tenney Height around
here, though I see it's not in the tonalsoft encyclopedia.
But you can google it (in quotes) and get stuff.

Looks like it was Jr...

Galileo Galilei (1638). Discorsi e dimostrazioni matematiche interno à
due nuove scienze attenenti alla mecanica ed i movimenti locali.
Leiden: Elsevier, 1638. Translated by H. Crew and A. de Salvio as
Dialogues concerning Two New Sciences. New York: McGraw-Hill Book Co.,
Inc, 1963.

This also comes up:

http://journals.royalsociety.org/content/k5256595202655r0/

If you have access, perhaps you can comment.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I can't immediately find a good basic discussion of the possible
> > relation between periodicity and chord-consonance online - is
> > there one? Carl?
>
> The idea dates back to Galileo, who first proposed the n*d
> consonance ranking (maybe it was Galileo Sr., I forget).
>
> Rameau championed related ideas (guidetone or whatever he
> called it).
>
> It's been prominent in psychoacoustics at least since
> Carl Seashore in the '20s, if memory serves. Though usually
> credited with devising the competing theory, several sections
> I've read in Helmholtz seem to be in support of it also.
>
> Terhardt was a big turning point:
> http://www.mmk.e-technik.tu-muenchen.de/persons/ter.html
>
> Peter Cariani is a more recent figure:
> http://www.cariani.com
>
> If you think there's a contrast with major/minor triads,
> just wait until you get to the 7-limit, 11-limit, and beyond.
> The roughness models of Plomp, Levelt, Sethares, et al
> completely fail to account for it. It is an obvious
> demonstration that periodicity must play at least an equal
> if not much greater role in the consonance distinctions
> relevant in music.

I've long suspected that the contrast is due to the coincidence (in
otonal chords) vs. non-coincidence (in utonal chords) of
combinational tones, particularly first-order difference tones.

--George

> I've long suspected that the contrast is due to the coincidence (in
> otonal chords) vs. non-coincidence (in utonal chords) of
> combinational tones, particularly first-order difference tones.
>
> --George

If that's true, the effect ought to go away when chords
are played with pure tones at low volume...

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > I've long suspected that the contrast is due to the coincidence (in
> > otonal chords) vs. non-coincidence (in utonal chords) of
> > combinational tones, particularly first-order difference tones.
> >
> > --George
>
> If that's true, the effect ought to go away when chords
> are played with pure tones at low volume...
>
> -Carl

Also, the contrast should be easily observed with pure tones at normal
volume.

--George

>>> I've long suspected that the contrast is due to the coincidence
>>> (in otonal chords) vs. non-coincidence (in utonal chords) of
>>> combinational tones, particularly first-order difference tones.
>>
>> If that's true, the effect ought to go away when chords
>> are played with pure tones at low volume...
>
> Also, the contrast should be easily observed with pure tones at
> normal volume.

Combination tones are basically never strong enough to account
for the effect (maybe in heavy metal guitar work) and I don't
hear it going away with sine tones at low volume either
(do you?).

The hypothesis doesn't reduce the simplicity of the explanation
either, since presumably by coincidence you mean coincidence
with the fundamental, which has to come from the virtual pitch
mechanism anyway (but please demonstrate what you mean by
coincidence otherwise), and virtual pitch explains it quite
nicely without anything additional.

-Carl

This is what i have also been saying for quite some time

I've long suspected that the contrast is due to the coincidence (in
otonal chords) vs. non-coincidence (in utonal chords) of
combinational tones, particularly first-order difference tones.

--George
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

as has been pointed out in Roderer's book difference tones are supplied by the brain even if tones are sepArated in different ears.
In which Case volume would have less affect. But volume always does. i know of nothing that doesn't get less disonant when done softer, including pure noise.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Looks like it was Jr...
>
> Galileo Galilei (1638). Discorsi e dimostrazioni matematiche interno à
> due nuove scienze attenenti alla mecanica ed i movimenti locali.
> Leiden: Elsevier, 1638. Translated by H. Crew and A. de Salvio as
> Dialogues concerning Two New Sciences. New York: McGraw-Hill Book Co.,
> Inc, 1963.
>
> This also comes up:
>
> http://journals.royalsociety.org/content/k5256595202655r0/
>
> If you have access, perhaps you can comment.
>
> -Carl
>

I ought to, but the authorization system throws a loop.
Or perhaps I will only get free access if it's over a year old.

Anyway from what we can read already in the abstract, they haven't
done any work with real neurons, they just show that one theory of
neuron synchronization gives very similar results to one theory of
interval consonance. It's matching one lot of mathematics with another.
The fact that their supplementary materials consist of .wav files
intended to illustrate consonances and dissonances - and they chose to
create these using a rather unconvincing synth piano tone playing what
sound like 12et intervals - is probably symptomatic.
I think the 'Additional material' can be heard for free, so listen up!

I found another article

"Interference effects and phase sensitivity in hearing"
By Brian C. J. Moore

in the Royal Society journal collection which looks a lot solider
(though not directly about consonance etc).

Going back to 'n*d', does it generalize to triads and other chords?
Galilei's simple remark about the regularity of 'pulses striking the
ear' might apply equally well to more complex sounds...

~~~T~~~

Hi Tom,

> Going back to 'n*d', does it generalize to triads and other
> chords? Galilei's simple remark about the regularity of
> 'pulses striking the ear' might apply equally well to more
> complex sounds...
>
> ~~~T~~~

In-vivo single-cell recordings in mammals reveal that the
auditory midbrain contains "combination sensitive" neurons
with best frequencies (usually two or three prominent ones)
that fit in a harmonic series (usually involving the
first 3 harmonics).

http://jn.physiology.org/cgi/content/abstract/90390.2008v1
http://www.vancouver.wsu.edu/fac/portfors/Research.html

The n*d rule has been investigated substantially on this
list, tuning-math, and the harmonic_entropy list in many
listening tests. The rule is actually geomean(a*b*c...)
for a chord a:b:c..., which includes sqrt(n*d) for dyads.
The largest listening test we did was actually done on
tetrads. This was a single-blind online test where mp3
files of about two dozen tetrads were ranked by consonance
by several volunteers. I reproduced the Tenney Height
ranking with only two exceptions, and other participants
got similar results, especially near the top of the ranking.

The rule is also comes out of the harmonic entropy model
in the following way: when the calculation is seeded with
a Farey series, the resulting entropy is proportional
to sqrt(n*d) near the simplest rationals, but has a slight
overall trend as interval size goes up. When the calc is
seeded with a Mann series (n+d), the same thing happens.
When the calc is seeded with a Tenney series, the entropy
again agrees with sqrt(n*d) near the simple rationals, but
no overall trend is observed.

-Carl

I would be curious to see if the tetrads come out in the same order as that found here in this list.
http://anaphoria.com/lullaby.html
which is based on taking the pitch and the first order difference tones and omitting the duplicates.

And how does squaring (n*d) differ in final results from just (n*d), if it does.

In-vivo single-cell recordings in mammals reveal that the
auditory midbrain contains "combination sensitive" neurons
with best frequencies (usually two or three prominent ones)
that fit in a harmonic series (usually involving the
first 3 harmonics).

http://jn.physiology.org/cgi/content/abstract/90390.2008v1 <http://jn.physiology.org/cgi/content/abstract/90390.2008v1>
http://www.vancouver.wsu.edu/fac/portfors/Research.html <http://www.vancouver.wsu.edu/fac/portfors/Research.html>

The n*d rule has been investigated substantially on this
list, tuning-math, and the harmonic_entropy list in many
listening tests. The rule is actually geomean(a*b*
c...)
for a chord a:b:c..., which includes sqrt(n*d) for dyads.
The largest listening test we did was actually done on
tetrads. This was a single-blind online test where mp3
files of about two dozen tetrads were ranked by consonance
by several volunteers. I reproduced the Tenney Height
ranking with only two exceptions, and other participants
got similar results, especially near the top of the ranking.

The rule is also comes out of the harmonic entropy model
in the following way: when the calculation is seeded with
a Farey series, the resulting entropy is proportional
to sqrt(n*d) near the simplest rationals, but has a slight
overall trend as interval size goes up. When the calc is
seeded with a Mann series (n+d), the same thing happens.
When the calc is seeded with a Tenney series, the entropy
again agrees with sqrt(n*d) near the simple rationals, but
no overall trend is observed.

-Carl
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> I would be curious to see if the tetrads come out in the same order as
> that found here in this list.
> http://anaphoria.com/lullaby.html
> which is based on taking the pitch and the first order difference tones
> and omitting the duplicates.
>
> And how does squaring (n*d) differ in final results from just (n*d), if
> it does.
>

I can't substitute for Carl in answering this, but I think the point
is to compare chords with different numbers of notes. So the square
*root* of n*d for a dyad, the cube root of a*b*c for triad, etc.,
otherwise larger chords would start off at a disadvantage.

> > The n*d rule has been investigated substantially on this
> > list, tuning-math, and the harmonic_entropy list in many
> > listening tests. The rule is actually geomean(a*b*
> > c...)
> > for a chord a:b:c..., which includes sqrt(n*d) for dyads.
> > The largest listening test we did was actually done on
> > tetrads.

Now if only we knew how to find those past experiments. Searching the
online message archive usually doesn't give helpful results - the
number of hits is either close to zero or exponentially large.

Was there some sort of control for the 'average pitch' of a chord? I
certainly find some chords more dissonant when transposed lower in pitch.
~~~T~~~

> > And how does squaring (n*d) differ in final results from just
> > (n*d), if it does.
>
> I can't substitute for Carl in answering this, but I think the
> point is to compare chords with different numbers of notes. So
> the square *root* of n*d for a dyad, the cube root of a*b*c for
> triad, etc., otherwise larger chords would start off at a
> disadvantage.

That's right. Also, I would answer that n*d and sqrt(n*d) both
produce the same ranking of dyads. These rules of thumb may be
right quantitatively, but -- I don't know about you -- my ears
aren't that good. But my ears can check a ranking. So that's
what the focus has been on.

> > The n*d rule has been investigated substantially on this
> > list, tuning-math, and the harmonic_entropy list in many
> > listening tests. The rule is actually geomean(a*b*
> > c...) for a chord a:b:c..., which includes sqrt(n*d) for
> > dyads. The largest listening test we did was actually
> > done on tetrads.
>
> Now if only we knew how to find those past experiments.

You can download the chords here:
http://www.soundclick.com/bands/page_music.cfm?bandID=105485
WARNING: Spoilers. If you want to take the test I can put
it together for you.

-Carl

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> I would be curious to see if the tetrads come out in the same
> order as that found here in this list.
> http://anaphoria.com/lullaby.html

Hi Kraig- I'm interested to know too, but I can't figure out
how to read this PDF. First off, I'm seeing note names.
If I'm not to assume 12-ET, what tuning should I use for them?

Are the numbers a dissonance score (higher is less consonant)?

Why are single notes given a score... why is G any different
from C?

> which is based on taking the pitch and the first order
> difference tones and omitting the duplicates.

Hmm...

-Carl

Hi Carl~
on the page before the PDF it says there what each note means. Harmonic on C up to 11.
The numbers are exactly as you intuit. the high the more 'dissonant' . I am in the process of writing this out again. I liked that version cause on two pages i could play the whole piece without turning pages and also jump from section to sections for comparison. I found playing through this one often cannot tell much different between chord to chord so i also became interested in how much difference is needed before we hear any difference. a score with a bland sound file in the works
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Hi Carl~
> on the page before the PDF it says there what each note
> means. Harmonic on C up to 11. The numbers are exactly as
> you intuit. the high the more 'dissonant' . I am in the
> process of writing this out again. I liked that version cause
> on two pages i could play the whole piece without turning pages
> and also jump from section to sections for comparison. I found
> playing through this one often cannot tell much different
> between chord to chord so i also became interested in how much
> difference is needed before we hear any difference. a score
> with a bland sound file in the works

I can't wait to hear it. But I still don't understand you,
grasshopper. How are you ascribing a dissonance of 6 to
CG and of 7 to just the single pitch A# !? Is there a C
drone playing the whole time or something, so that A# is
actually the interval CA#?

-Carl

Carl said~I can't wait to hear it. But I still don't understand you,
grasshopper. How are you ascribing a dissonance of 6 to
CG and of 7 to just the single pitch A# !? Is there a C
drone playing the whole time or something, so that A# is
actually the interval CA#?

yes a single pitch. even though it is alone one hears it in relationship to what goes before. I have no problem with objections to this but it serves the whole in the same way 0/1 and 1/0 serve the scale tree. The purpose is not to analyze single pitches but what ever one uses, this can be done. you can leave them out if you wish.

C G is 3 to 2 with one first order difference tone of . that gives us 6 if you add them all together. 7 is just the 7th harmonic by itself.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Carl said~I can't wait to hear it. But I still don't understand
> you, grasshopper. How are you ascribing a dissonance of 6 to
> CG and of 7 to just the single pitch A# !? Is there a C
> drone playing the whole time or something, so that A# is
> actually the interval CA#?
>
> yes a single pitch. even though it is alone one hears it in
> relationship to what goes before. I have no problem with
> objections to this but

I have no objection to it. Surely the single notes do have
this relation, that evokes something like consonance/dissonance.
But in coming up with all-purpose rules of thumb like n*d,
the goal is to take musical context out of the picture, and
just compare whatever is inherent to the dyads. To come up
with something that works regardless of what pitches one roots
them on. So I'm not sure if your analysis here is directly
comparable to an effort like what we were discussing.

> C G is 3 to 2 with one first order difference tone of 1. that
> gives us 6 if you add them all together. 7 is just the 7th
> harmonic by itself.

Gotcha. Let me think. 3*2 = 6 and 3+2+(3-2)=6. So they
agree. But in general, does

n * d = n + d + (n - d)

n * d = 2n

So the values are only the same when the denominator is 2.

But do they give the same ranking? No. Your measure is the
same as twice the numerator, which is the same ranking as just
using the numerator. Not a bad ranking, but in comprehensive
tests done in 1999 (involving me, monz, Dave K., Paul E,
Daniel Wolf, Eduardo Sabat-Garibaldi, and others), we found
that both the numerator and denominator contribute.

-Carl