Basic Acoustic Questions vis-a-vis Just Intonation

[ evidently this got rejected from microtones@yahoogroups.com ... ]

I am reading the Just Intonation Primer and am on Chapter One. I have several very basic questions.

1: sound = periodic pressure variation + ear drum moving in and out + nerve impulse to brain.

Ok, I have that down. However, I have a question about the periodic pressure variation. Points above the line are due to air compression and points below the line are due to air rarefaction. What does this mean physically about the air of a sound wave? Does it mean that air particles are moving towards each other during compression and that they are moving away from each other during rarefaction? If so, then how could this be? When they move towards each other, arent they at the same time moving away from another point, thus creating rarefaction while creating compression? And at point zero on the sound wave what can we say about air particles?

2: what does the word "diatonic" mean? What is a diatonic scale?

3: if two pure tones both have the same frequency lambda, but different amplitudes A1 and A2, is their superposition (sum) a frequency preserving operation, regardless of what degree of phase shift is applied? What I mean is: assume the first frequency is 0 degrees out of phase, if we put the zero point of the 2nd frequency anywhere > 0 and < 360, can we be certain that in all cases, the superposition leads to a complex tone with the same frequency as its component pure tones?

4a: Can interference beats be created via tones which vary in frequency and amplitude or must they only vary in frequency and have the same amplitude?

4b: Regarding 2 pure tones with frequencies A and B (non-equal), I dont agree with the text here. The first part makes sense. It says that when two tones vary in frequency but have the same amplitude, then and only then are interference beats are created, which at some points double in amplitude and at others completely cancel out. My question here is is the cancellation guaranteed to occur regardless of how out-of-phase the two 2 tones are.

Now my second question has to do with the same 2 pure tones with frequencies A and B (non-equal) but *whose amplitudes also vary*. Here the book does state:

4c: let us be concrete for a moment. Assume two tones of equal amplitude. One has frequency 20 and the other 10. It is understood that the interference beats will not be perceived as such but rather as a single tone with frequency 15 (the average of 20 + 10). It is also understood that the amplitude will vary from 2A to 0 with frequency 10, where A = the amplitude of both frequencies. If this is all correct, I would like to move to point 4d...

4d: when the beat frequency (the absolute value of the difference of the two frequencies which determines the amplitude of the tone superposition) is below 15-20Hz, what do we hear - individual beats or a single fused tone? As the beat frequency increases, supposedly we encounter the limit of frequency discrimination and the critical band? What is the nature of interference beats from 20Hz to the limit of frequency discrimination? What is the nature of interference beats from the limit of frequency discrimination to the critical band? What is the nature of interference beats beyond the critical band?

<quote>
When two pure tones of unequal amplitude are sounded together in the appropriate frequency relationship, beating takes the form of a period amplitude variation, but complete cancellations do not occur.
</quote>

I have two issues with this quote. First, "appropriate frequency relationship" is a vague term lacking mathematical precision. Second, is it a fact that when two pure tones vary in both amplitude and frequency that they can never cancel each other out? I find this hard to believe but would be willing to hear proof to the contrary.

<hacker lang="Perl" type="just another" url="http://www.metaperl.com">
Terrence Brannon
</hacker>
just another business objects layer hacker

[ evidently this got rejected from microtones@yahoogroups.com ... ]

I am reading the Just Intonation Primer and am on Chapter One. I have several very basic questions.

1: sound = periodic pressure variation + ear drum moving in and out + nerve impulse to brain.

Ok, I have that down. However, I have a question about the periodic pressure variation. Points above the line are due to air compression and points below the line are due to air rarefaction. What does this mean physically about the air of a sound wave? Does it mean that air particles are moving towards each other during compression and that they are moving away from each other during rarefaction? If so, then how could this be? When they move towards each other, arent they at the same time moving away from another point, thus creating rarefaction while creating compression? And at point zero on the sound wave what can we say about air particles?

2: what does the word "diatonic" mean? What is a diatonic scale?

3: if two pure tones both have the same frequency lambda, but different amplitudes A1 and A2, is their superposition (sum) a frequency preserving operation, regardless of what degree of phase shift is applied? What I mean is: assume the first frequency is 0 degrees out of phase, if we put the zero point of the 2nd frequency anywhere > 0 and < 360, can we be certain that in all cases, the superposition leads to a complex tone with the same frequency as its component pure tones?

4a: Can interference beats be created via tones which vary in frequency and amplitude or must they only vary in frequency and have the same amplitude?

4b: Regarding 2 pure tones with frequencies A and B (non-equal), I dont agree with the text here. The first part makes sense. It says that when two tones vary in frequency but have the same amplitude, then and only then are interference beats are created, which at some points double in amplitude and at others completely cancel out. My question here is is the cancellation guaranteed to occur regardless of how out-of-phase the two 2 tones are.

Now my second question has to do with the same 2 pure tones with frequencies A and B (non-equal) but *whose amplitudes also vary*. Here the book does state:

4c: let us be concrete for a moment. Assume two tones of equal amplitude. One has frequency 20 and the other 10. It is understood that the interference beats will not be perceived as such but rather as a single tone with frequency 15 (the average of 20 + 10). It is also understood that the amplitude will vary from 2A to 0 with frequency 10, where A = the amplitude of both frequencies. If this is all correct, I would like to move to point 4d...

4d: when the beat frequency (the absolute value of the difference of the two frequencies which determines the amplitude of the tone superposition) is below 15-20Hz, what do we hear - individual beats or a single fused tone? As the beat frequency increases, supposedly we encounter the limit of frequency discrimination and the critical band? What is the nature of interference beats from 20Hz to the limit of frequency discrimination? What is the nature of interference beats from the limit of frequency discrimination to the critical band? What is the nature of interference beats beyond the critical band?

<quote>
When two pure tones of unequal amplitude are sounded together in the appropriate frequency relationship, beating takes the form of a period amplitude variation, but complete cancellations do not occur.
</quote>

I have two issues with this quote. First, "appropriate frequency relationship" is a vague term lacking mathematical precision. Second, is it a fact that when two pure tones vary in both amplitude and frequency that they can never cancel each other out? I find this hard to believe but would be willing to hear proof to the contrary.

<hacker lang="Perl" type="just another" url="http://www.metaperl.com">
Terrence Brannon
</hacker>
just another business objects layer hacker

In-Reply-To: <1F9212F7-2B84-11D6-851F-003065C2A10C@mac.com>
Terrence Brannon wrote:

> [ evidently this got rejected from microtones@yahoogroups.com ... ]

I did get two copies of this at tuning.

> Ok, I have that down. However, I have a question about the
> periodic pressure variation. Points above the line are due to air
> compression and points below the line are due to air rarefaction.
> What does this mean physically about the air of a sound wave?
> Does it mean that air particles are moving towards each other
> during compression and that they are moving away from each other
> during rarefaction? If so, then how could this be? When they move
> towards each other, arent they at the same time moving away from
> another point, thus creating rarefaction while creating
> compression? And at point zero on the sound wave what can we say
> about air particles?

Air particles are moving very fast in random directions all the time.
During compression, they're packed in more tightly than normal, and during
rarefaction they're more spread out. The speed the wave propagates is
much slower than the speed the particles are moving. You can think about
pressure and ignore the particles completely.

> 2: what does the word "diatonic" mean? What is a diatonic scale?

Historically, "diatonic" referred to the Greek genus made up of tones and
semitones. A diatonic scale in modern usage refers to something like the
white notes of a piano keyboard, or the nominals of staff notation.
Diatonic music keeps to one diatonic scale.

Definitions of tuning terms are at <http://www.ixpres.com/interval/dict/>.

> 3: if two pure tones both have the same frequency lambda, but
> different amplitudes A1 and A2, is their superposition (sum) a
> frequency preserving operation, regardless of what degree of
> phase shift is applied? What I mean is: assume the first
> frequency is 0 degrees out of phase, if we put the zero point of
> the 2nd frequency anywhere > 0 and < 360, can we be certain that
> in all cases, the superposition leads to a complex tone with the
> same frequency as its component pure tones?

Yes, the frequency will be preserved. "Frequency" is the number of things
that happen in a given time. If you combine two repeating patterns of the
same length, you'll always get another repeating pattern of that length.
If both tones are pure, the resultant will also be pure.

There are some caveats, like you may find the frequency doubles if you
combine, say

. /\ with /\ to get /\ /\
. ____/ \ / \____ / \/ \

and two pure tones can cancel out, leaving no defined frequency.

> 4a: Can interference beats be created via tones which vary in
> frequency and amplitude or must they only vary in frequency and
> have the same amplitude?

You do get beats in this case, but the mathematics is more complicated.
The only place I know where it's worked out is on of Helmholtz's
appendices. You get a variation in frequency between the two initial
ones, instead of a constant frequency at the average, and the beats are
still there.

> 4b: Regarding 2 pure tones with frequencies A and B (non-equal),
> I dont agree with the text here. The first part makes sense. It
> says that when two tones vary in frequency but have the same
> amplitude, then and only then are interference beats are created,
> which at some points double in amplitude and at others completely
> cancel out. My question here is is the cancellation guaranteed to
> occur regardless of how out-of-phase the two 2 tones are.

If the two frequencies are slightly different, they'll drift in and out of
phase, so the initial phase isn't important. If the frequencies are
related by a simple ratio, you might get "phase locking" but you need
different mathematics to describe that case.

> 4d: when the beat frequency (the absolute value of the difference
> of the two frequencies which determines the amplitude of the tone
> superposition) is below 15-20Hz, what do we hear - individual
> beats or a single fused tone? As the beat frequency increases,
> supposedly we encounter the limit of frequency discrimination and
> the critical band? What is the nature of interference beats from
> 20Hz to the limit of frequency discrimination? What is the nature
> of interference beats from the limit of frequency discrimination
> to the critical band? What is the nature of interference beats
> beyond the critical band?

The low frequencies are what should properly be described as "beats". You
hear a fluctuation in the sound that you can time with a watch. As the
frequency rises, you start to hear "roughness" and this is what the toy
theories say you should avoid to get smooth consonance. If the frequency
rises further still, you hear a difference tone.

The critical band is wider, around 1 or 2 semitones and not the same in
all registers. Pure tones are reported to be dissonant if they sound
together within a critical band.

> <quote>
> When two pure tones of unequal amplitude are sounded together in
> the appropriate frequency relationship, beating takes the form of
> a period amplitude variation, but complete cancellations do not
> occur.
> </quote>
>
> I have two issues with this quote. First, "appropriate frequency
> relationship" is a vague term lacking mathematical precision.
> Second, is it a fact that when two pure tones vary in both
> amplitude and frequency that they can never cancel each other
> out? I find this hard to believe but would be willing to hear
> proof to the contrary.

I don't know what "appropriate" means there. Possibly that the
frequencies don't differ by much. Where two tones differ in amplitude,
the resultant can never have a lower amplitude than the difference between
them. To take an extreme case, you can't cancel out a 50 dB signal with a
1dB signal.

Graham