RE: complexity weighting triads; Human Hearing and consonance

D. Wolf wrote,
>> This difficulty here is that harmonic chords and their subharmonic
>> inversions will have the same complexity quantity.
>> Most listeners would probably find the latter to be more complex.

Dale Scott wrote,
>> I don't think most people would agree that, for example,
>> 3:5:7 is more consonant than 3:5:8, which is why many
>> people on the list aren't satisfied with the use of n*d
>> as a consonance rating.

I wrote,
>> Compare the sound of 4:5:6:7:9:11 with 1/11:1/9:1/7:1/6:1/5:1/4.
>> Are they equally consonant?

Peter Mulkers wrote,

>I'm sorry. I see you are right about consonance. But please wait.
>In fact I was just talking about complexity. Not consonance.
>I thought complexity and consonance were two different things.
>Complexity is just a matter of how complex a range of numbers
>behave to each other, a matter of Maths (and maybe Physics).
>And consonance as a matter of how we hear and feel all this,
>a matter of Human Perception and anatomy of the human ear.
>No ?

The reason we are looking at numerical complexity in the first place is
because we believe it is related to consonance. The three factors we've
identified as important to consonance are complexity, tolerance, and span
(thanks Dave Keenan). For small enough numbers, tolerance is not too
important, and if we compare intervals of similar sizes, span is not
important. So the examples above are fairly valid in equating complexity
with dissonance. If we divorce complexity from the perception of dissonance,
how could we possibly decide on one complexity measure over another? Without
human perception, the numbers representing the musical tones have no
meaning. Physics is relevant but only to the extent that we are dealing with
the physics of the human hearing apparatus.

The three major phenomena which explain the consonance of simple ratios --
roughness, tonalness, and nonlinear combination tones -- may each have a
different complexity formulation that best models their effects. For
example, the roughness of otonal chords (harmonic series chords) and utonal
chords (their mirror reflections) tend to be similar, while tonalness and
combination tones clearly favor otonal over utonal chords. A simple formula
like LCM or what have you is not going to get at the effects of all three.

Glen wrote,

>> [Glen Peterson:]
>> >I have also been wondering if I even consider "fourths and fifths" to
>> be consonant or dissonant. They are almost neither.
>>
>> [Paul Erlich:]
>> > If a fifth is not consonant, what is?
>>
>> [Dan Stearns]
>> "fourths and fifths" occupy some
>> indeterminate zone between (overtly) 'sweet' and 'sour.'

>Well put. They somehow sound too plain to my ears to get that feeling of
>consonance. Maybe consonance is the wrong word?

Here's where the cultural/historical aspect of my theory comes in. In
medieval music, only fourths and fifths were consonances. Ever since thirds
and sixths took over that role in the Renaissance, and triadic texture
became the norm for stability, a bare fourth or fifth would be an
interruption that texture, a sonority with lower tension than that
associated with the point of repose. "Too plain" is an apt description. In
jazz, triads are "too plain." In my decatonic system, 7-limit tetrads become
the norm, the point of lowest tension. But I don't think "plainness" takes
away from "consonance". What could be more consonant that plain old octaves
and unisons?

>Have I got this right?

>Roughness = Proximity to a small whole number ratio creates a tension
called
>roughness. i.e., 301/200 will be heard as a mistuned 3/2.

No. Roughness is the sensation created when partial tones interfere with one
another. This happens if they are too close to be separately resolved by the
cochlea (i.e., less than a minor third), but too far to blend into a single,
slowly beating tone.

For timbres with harmonic partials, roughness is minimized at simple-integer
ratios, because then pairs of partials coincide and do not interfere with
one another.

>Tonalness = How closely the interval fulfils the ear's desire for
completion
>of a harmonic series of tones.

Yes. Or partial completion -- at least enough to determine the fundamental.

>Difference tones = The difference between two pitches creates another
pitch.
>Most noticeable in with high frequency pitches played closely together.

Yes. There are also other combination tones, such a 2*a-b, etc.

>If I'm on the right track, then what does this mean, "For sine waves,
simple
>integer ratios no longer minimize roughness, but they still maximize
>tonalness?" I would think that, "For sine waves, Tonalness is less of a
>factor, because there is no harmonic series in the actual tones. Roughness
>and difference tones still have an effect."

A harmonic series is made up of sine waves, not the other way around.
Therefore putting sine waves in simple-integer ratios fulfills the ear's
desire for completion of a harmonic series of tones. However, since sine
waves don't have upper partials, simple-integer ratios won't minimize
roughness.

>What about, "Simple-integer ratios come into the picture because if the
>heard tones are to be understood as harmonic overtones of some missing
>fundamental or root, they must form a simple-integer ratio with one
>another." If this is true, why are simple integer ratios so powerful with
>sine waves?

Simple-integer ratios can matter for sine waves (a) because overtones _are_
sine waves by definition, and (b) because difference tones are most clearly
defined for sine waves, and of the three phenomena, difference tones impose
the tighest tolerance on the simple-integer ratios. However, I would say
that simple-integer ratios are not very powerful with sine waves because
there is no roughness-minimization at work (except at 1:1).

>Why does Utonality make any sense at all?

It makes sense for tones with harmonic partials. In a Utonality
1/a:1/b:1/c,..., there is a common overtone, which is the a_th harmonic of
the first note, the b_th harmonic of the second note, the c_th harmonic of
the third note, etc.

>I had a 23 note JI glass organ for a while,

>http://www.organicdesign.org/Glen/Instruments/Glass_Organ/glass_organ.html

>the tone of which had a very profound fundamental, and weak partials in
>various patterns. My experience in playing it was that Otonalities and
>Utonalities were equally consonant,

Were you only playing triads or did you try tetrads, pentads, hexads as
well? Did you try chords that were neither otonal or utonal (like
10:12:15:18 or 12:14:18:21)? By the way, your layout has some severe
departures from the Incipient Tonality Diamond -- what were the exact
Otonalities and Utonalities you were trying to use? Also, perhaps the
voicings (i.e., inversions) you used in the utonalities were more favorable
than the ones in the otonalities, counteracting the expected tonalness
effects.

What about intervals -- were the simple ratios clearly different from the
complex ones? It is possible that whatever weak partials you may have had
were not harmonic ones, so there would be no particular preference for
small-integer ratios except:

>except with high notes close together
>where difference tones were audibly harmonizing with the Otonalities, and
>not with the Utonalities.

Right -- difference tones favor otonal chords.

[Paul Erlich:]
>But I don't think "plainness" takes away from "consonance".

Nor do I... I would like to clarify that my gustatory analogy was
offered by way of sympathizing with Glen Peterson's
comments/experience, but that I personally can't seem to muster much
in the way of enthusiasm for the implications of what a clinical
(i.e., tuning isolated, or non-musically applied) fifth or fourth
strikes me as sounding like, as it seems to me that these sorts of
descriptive implications are much more telling in (though that's not
to say entirely dependant on) an actual musical context.

>What could be more consonant that plain old octaves and unisons?

Seldom have I ever played a unison line with another instrument and
had it sound anything less than electric - well certainly not "plain"
anyway... Do the opening octaves (or even just the opening octave) of
say Ives' "CONCORD SONATA" sound particularly consonant (or plain)?

Dan

Alright, let's see if I have it straight now,

Roughness = when the partials of an interval do not line up well, it creates
roughness. Different intervals may be more or less rough with different
timbres.

Tonalness = How closely an interval fulfils the ear's desire for completion
of a harmonic series of tones. At least to the degree where a fundamental
is perceived.

Combination tones = The difference tone or other pitches perceived by the
ear when two pitches are played simultaneously. Difference tones are most
noticeable when high frequency pitches are played closely together.

> Simple-integer ratios can matter for sine waves (a) because
> overtones _are_
> sine waves by definition, and (b) because difference tones
> are most clearly
> defined for sine waves, and of the three phenomena,
> difference tones impose
> the tighest tolerance on the simple-integer ratios. However,
> I would say
> that simple-integer ratios are not very powerful with sine
> waves because
> there is no roughness-minimization at work (except at 1:1).

And I still say, even with sine waves, I hear Utonalities to be just as
consonant as Otonalities, unless there is an audible difference tone.

> >> Compare the sound of 4:5:6:7:9:11 with 1/11:1/9:1/7:1/6:1/5:1/4.
> >> Are they equally consonant?

For the lay people among us, (myself included) here is an analyses of this
system:

1/1 0.00 cents C +0
9/8 203.91 cents D +4
5/4 386.31 cents E -14
11/8 551.32 cents F# -49
3/2 701.96 cents G +2
7/4 968.83 cents A# -31

1/1 0.00 cents C +0
8/7 231.17 cents D +31
4/3 498.04 cents F -2
16/11 648.68 cents F# +49
16/9 996.09 cents A# -4
8/5 813.69 cents G# +14

Attached please find my attempt at making a .midi file of this.
Unfortunately, I haven't figured out how to get this box of silicon wafers
to play a *$&% sine wave yet, except by writing C programs to use the PC
speaker. Ugh! Any help with creating a readable .MIDI file or changing the
timbres would be appreciated. Here's my best shot using Cakewalk. I was
able to play it with RealPlayer.

Might I dare to agree with Mr. PC appeal and suggest that people try to
include MIDI files with their posts more often? It means much more to me
when I can hear what people are talking about.

Anyway, with the organ timbre I chose, the Otonality is MUCH more consonant.
This was not true of my glass organ as I remember it.

---
Glen Peterson
30 Elm Street North Andover, MA 01845
(978) 975-1527
http://www.OrganicDesign.org/Glen/Instruments

Glen Peterson wrote,

>Alright, let's see if I have it straight now,

>Roughness = when the partials of an interval do not line up well, it
creates
>roughness. Different intervals may be more or less rough with different
>timbres.

Right.

>Tonalness = How closely an interval fulfils the ear's desire for completion
>of a harmonic series of tones. At least to the degree where a fundamental
>is perceived.

Right.

>Combination tones = The difference tone or other pitches perceived by the
>ear when two pitches are played simultaneously.

Well, the missing fundamental can be perceived (due to tonalness) even when
it differs from any of the combination tones. Combination tones arise from
mechanical nonlinearities in the ear, while the missing fundamental is the
brain's attempt to fit what is heard into a harmonic series. The two can be
close in frequency and even (I think John Chalmers referenced this) beat
against one another.

>Anyway, with the organ timbre I chose, the Otonality is MUCH more
consonant.
>This was not true of my glass organ as I remember it.

Were you using the same voicings for both? I note that the voicings you give
are not mirror inverses of one another, which may be affecting your results.
Also, what kind of organ timbre did you choose? What about astring or brass
timbre?

--- In tuning@y..., "Paul H. Erlich" <PErlich@xxxxxxxxxxxxx.xxxx wrote:
> D. Wolf wrote,

> PE:Roughness is the sensation created when partial tones interfere >with one
> another. This happens if they are too close to be separately >resolved by the
> cochlea (i.e., less than a minor third), but too far to blend into >a single,
> slowly beating tone.
> For timbres with harmonic partials, roughness is minimized at >simple-integer
> ratios, because then pairs of partials coincide and do not >interfere with
> one another.

> >DW: Difference tones = The difference between two pitches creates >>another
> pitch.
> >Most noticeable in with high frequency pitches played closely >>together.

> PE: Yes. There are also other combination tones, such a 2*a-b, etc.

> >DW: If I'm on the right track, then what does this mean, "For sine >>waves,
>>simple
> >integer ratios no longer minimize roughness, but they still >>maximize
> >tonalness?" I would think that, "For sine waves, Tonalness is >>less of a
> >factor, because there is no harmonic series in the actual tones. >>Roughness
> >and difference tones still have an effect."

>PE: A harmonic series is made up of sine waves, not the other way >around.
> Therefore putting sine waves in simple-integer ratios fulfills the >ear's
> desire for completion of a harmonic series of tones. However, since >sine
> waves don't have upper partials, simple-integer ratios won't >minimize
> roughness.

> >DW: What about, "Simple-integer ratios come into the picture >>because if the
> >heard tones are to be understood as harmonic overtones of some >>missing
> >fundamental or root, they must form a simple-integer ratio with one
> >another." If this is true, why are simple integer ratios so >>powerful with
> >sine waves?

> PE: Simple-integer ratios can matter for sine waves (a) because >overtones _are_
> sine waves by definition, and (b) because difference tones are most >clearly
> defined for sine waves, and of the three phenomena, difference >tones impose
> the tighest tolerance on the simple-integer ratios. However, I >would say
> that simple-integer ratios are not very powerful with sine waves >because
> there is no roughness-minimization at work (except at 1:1).

> >DW: Why does Utonality make any sense at all?

> PE: It makes sense for tones with harmonic partials. In a Utonality
> 1/a:1/b:1/c,..., there is a common overtone, which is the a_th >harmonic of
> the first note, the b_th harmonic of the second note, the c_th >harmonic of
> the third note, etc.

<SNIP>

> PE: "... difference tones favor otonal chords."

J Gill: Paul, it appears that the (1st order)"difference
tones" between a 1/1 pitch and "superparticular" pitch ratios
3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10,... equal:
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10,...

How does that square with your statement (above) that:
"... difference tones favor otonal chords."???

Curiously, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> J Gill: Paul, it appears that the (1st order)"difference
> tones" between a 1/1 pitch and "superparticular" pitch ratios
> 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10,... equal:
> 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10,...

Right, but why this construction? It seems rather arbitrary.

> How does that square with your statement (above) that:
> "... difference tones favor otonal chords."???

Note the word _Chords_!! By which I mean three or more notes played
_at the same time_. Any interval, just two notes, can be understood
equally as utonal or otonal, so saying "otonal interval" or "utonal
interval" is meaningless outside of some larger context.

What I mean by the statement you quoted, and this should be evident
if you calculate some examples, is that otonal chords will sound very
good even if lots of difference tones are audible, while utonal
chords will turn into a mess (a bigger mess the higher the limit) if
lots of difference tones are audible. Try some examples yourself and
then we can discuss the mathematics of it . . . somewhere.