Re: Consonance, intervals and scales



Tomislav Novak wrote:
>
> Joey Goldstein <nospam@xxxxxxxxxxx> writes:
>
> > Tomislav Novak wrote:
> >>
> >> Hi!
> >>
> >> I'm trying to understand music theory better to improve my playing and
> >> (possibly in the future) songwriting ability. There are few things I'd like
> >> to know but can't figure out by myself... :-/
> >>
> >> I've read that the perception of consonance depends on the number of common
> >> higher harmonics of two tones.
> >
> > I guess you could look at it that way.
> > The idea is really about frequency ratios though. Intervals with the
> > simplest fre ratios are the most "harmonious" to the human ear,
> > providing that the overtones produced by the instruments (i.e. the
> > timbre of the instruments) playing the interval are also in harmony.
> > (Some instruments have overtones that are not in tune compared to the
> > overtone series.)
>
> Well, isn't this the same thing in purely mathematical sense? The first common
> harmonic of the tones with the simplest frequency ratios is closest possible
> to the fundamental frequency (for example, consider a tone with frequency f
> and a tone perf 5th apart with freq 3/2f - their first common harmonic is 3f).
> If my reasoning is correct. :-)

You're looking up top the common harmonics of the tones involved in the interval.
I'm looking down to a the fundamental tone that is the "root" of both tones.
If the harmonics of the two tones are out of tune with each other the
interval will sound sour too.
If the freq ratio of the two tones is offset significantly from a simple
freq ratio then the interval will sound sour too.

> > 1:1 = unison
> > 2:1 = octave
> > 3:1 = perf 12th
> > 3:2 = perf 5th
> > 4:1 = double oct
> > 4:2 = oct
> > 4:3 = perf 4th (Note: In classical style perf 4ths were treated *as if*
> > they were a "dissonance". Ultimately the concept of dissonance, i.e. the
> > necessity for resolution, is related to style.)
> > 5:1 = maj 17th
> > 5:2 = maj 10th
> > 5:3 = maj 6th
> > 5:4 = maj 3rd
>
> A propos intervals, I have another question. If the minor third is the most
> important interval in minor chords, how come those chords are still
> recognizable when played in an inversion which doesn't contain the minor third?
> For example, if A minor chord is played as C E A (so C is the fundamental),

Be careful how you use the word "fundamental" please.

> there is a major third between C and E, a perfect fourth between E and A and a
> major 6th between C and A, but one still perceives the chord as A minor?

This chord voicing, when heard out of a musical context *will* sound
like a chord with root on C.
If heard in a musical context where an Am chord is expected it may be
heard as having a root on A.

Minor triads have much more harmonic ambiguity, as regards to root
feeling, than major triads do. this is because each interval in a minor
triad has a different acoustical root.

A C E
A-E is a perf 5th @ 2:3 with an acoustical root of A.
A-C is a min 3rd @ 5:6 with an acoustical root of F.
C-E is a maj 3rd @ 5:4 with an acoustical root of C.

The voicings of Am that have the strongest/unambiguous feeling that A is
the root will have A as the lowest tone in the voicing. 2:3, the perfect
5th, is the simplest freq ratio of all the freq ratios formed by the
intervals in this chord. When perf 5ths are presnt above the bass note
of a chord the ear is often persuaded to hear the root of the 5th as the
overall root of the chord. Therefore, in a root position minor triad,
the min 3rd is heard as a distorted version of the 5th partial.

A C E = 4:5*:6 ("*" indicates a distorted partial)

I have written about acoustical roots several times in this forum. Try a
search at the Google archives.
In general:
Chords in which the component intervals closely mirror the tones of a
single fundamental tone's overtone series will have the strongest sense
of root.
Chords in which the component intervals do not closely mirror the tones
of a single fundamental tone's overtone series will have a more
ambiguous sense of root, if any at all.
I.e. Some chord's, like major triads, have a strong sense of root and
this root is felt to be active in almost all inversions and/or
permutations of the component tones of the chord.
Some chords do *not* have a clear sense of root.
Some chords will have a different root feeling in the various inversions
possible of the initial chord.

> > [Note: In 12 TET tuning these simple ratios are considerably more
> > complex, but we seem happy to treat the 12 TET intervals *as if* they
> > are suitable replacements for their simpler relatives.]
>
> I guess the ear doesn't distinguish the minor frequency difference of the
> harmonics (because perf. fifth in 12 TET is 2^(7/12):1 which is very very
> close to 3:2, but not the same)?

I think we hear the proportions of the distances between the pitches and
if the proportions are somewhat close to pure intervals we seem to be
willing to accept them as being close enough symbolically for music making.

> >> The second question is why is the (major) scale constructed the way it is (I'm
> >> not refering to the whole-half steps pattern in equal temperament, but the
> >> intervals it contains - major second, major third, perfect fourth etc.)?
> >
> > The perf 12th (3:1) and perf 5th (at 3:2) intervals are the 1st
> > intervals that appear in the overtone series in which a tone appears
> > that is not an octave equivalent of the fundamental. So perf 5ths (and
> > their kin) are the most harmonious intervals possible next to unisons
> > and octaves. The perf 5th became the yardstick with which our ancestors
> > began the search for more tones that went well together.
>
> Thanks for this excellent and detailed explanation. That's what I was looking
> for. :-)
>
> --
> Pardon me for breathing, which I never do anyway so I don't know why I bother
> to say it, oh God I'm so depressed.

--
Joey Goldstein
http://www.joeygoldstein.com
joegold AT sympatico DOT ca
.

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