Coherant difference tones.

Hello. For chords such as 3:7:10, and 4:6:10, where the difference
tones land on the chord tones (ie, 10-7=3, 10-6-4; Helmholtz commented
on these, and I like them too!)...I had a question. This is probably
axiomatic to the math adept (but an exciting mystery to me!):what math
principle is it that accounts for the fact that if you take 3 adjacent
numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the middle one
(to get 4,6,10...and 3,7,10), that you get 10-7=3, 10-6=4? (Don't
laugh...)

A similiar question, is: take the chord 12:15:20. You have 5/4 x 3 =
15:12, and 4/3 x 5 = 20/15. Then, 15-12 equals the difference tone 3,
and 20/15 equals the differnece tone 5. It just seems neat that 5/4 x
3 = 15/12 (the position of the interval in that particular chord), and
that the difference between the two numbers (15 and 12) is the same
number with which the lower fraction (5/4) was multiplied. So why are
the differnce tone and the 'interval height' the same?

thanks, Kelly

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
> Hello. For chords such as 3:7:10, and 4:6:10, where the
> difference tones land on the chord tones (ie, 10-7=3, 10-6-4;
> Helmholtz commented on these, and I like them too!)...I had
> a question. This is probably axiomatic to the math adept
> (but an exciting mystery to me!):what math principle is it
> that accounts for the fact that if you take 3 adjacent
> numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the
> middle one (to get 4,6,10...and 3,7,10), that you get 10-7=3,
> 10-6=4? (Don't laugh...)

Not laughing, Kelly; it's an interesting pattern.

Let's take your first example:
Sequence: 4, 5, 6
Double the middle term:
Sequence : 4, 10, 6
Rearrange: 4, 6, 10
so that : 4 + 6 = 10
Is that what you mean?

More generally, if we call the middle element m,
and the common difference d, we have:
Sequence: (m-d), m, (m+d)
Double the middle term:
Sequence : (m-d), 2m, (m+d)
Rearrange: (m-d), (m+d), 2m
so that : (m-d) + (m+d) = 2m

Yes, the pattern holds for any values of m and d
that are integers.

As I said, an interesting pattern. But I can't
help but ask myself: what does this have to do
with Harmonic Entropy?

I almost think you need to join - or start! - a
list on numerology. ;-)

Regards,
Yahya

hi Yahya,

--- In harmonic_entropy@yahoogroups.com, "yahya_melb" <yahya@...> wrote:
>
>
> --- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
> >
> > Hello. For chords such as 3:7:10, and 4:6:10, where the
> > difference tones land on the chord tones (ie, 10-7=3, 10-6-4;
> > Helmholtz commented on these, and I like them too!)...I had
> > a question. This is probably axiomatic to the math adept
> > (but an exciting mystery to me!):what math principle is it
> > that accounts for the fact that if you take 3 adjacent
> > numbers in a sequence (eg, 4,5,6...or 3,5,7) and double the
> > middle one (to get 4,6,10...and 3,7,10), that you get 10-7=3,
> > 10-6=4? (Don't laugh...)
>
> Not laughing, Kelly; it's an interesting pattern.
>
> Let's take your first example:
> Sequence: 4, 5, 6
> Double the middle term:
> Sequence : 4, 10, 6
> Rearrange: 4, 6, 10
> so that : 4 + 6 = 10
> Is that what you mean?

Yes.

> More generally, if we call the middle element m,
> and the common difference d, we have:
> Sequence: (m-d), m, (m+d)
> Double the middle term:
> Sequence : (m-d), 2m, (m+d)
> Rearrange: (m-d), (m+d), 2m
> so that : (m-d) + (m+d) = 2m
>
> Yes, the pattern holds for any values of m and d
> that are integers.
>
> As I said, an interesting pattern. But I can't
> help but ask myself: what does this have to do
> with Harmonic Entropy?

Chords of the type 3:7:10 have the coherant difference tones --which
is pertinant to the list, yes?--- and are useful ways to arrange
chords, and wanted to know how the math worked. Or maybe I just need
a math tutor? Anyway, can you, personally, separate the love of math
(or numbers) from love of music?!

> I almost think you need to join - or start! - a
> list on numerology. ;-)
>
> Regards,
> Yahya
>

Kelly,

--- In harmonic_entropy@yahoogroups.com, "traktus5" wrote:
>
[snip]
> >
> > Let's take your first example:
> > Sequence: 4, 5, 6
> > Double the middle term:
> > Sequence : 4, 10, 6
> > Rearrange: 4, 6, 10
> > so that : 4 + 6 = 10
> > Is that what you mean?
>
> Yes.

Good!

> > More generally, if we call the middle element m,
> > and the common difference d, we have:
> > Sequence: (m-d), m, (m+d)
> > Double the middle term:
> > Sequence : (m-d), 2m, (m+d)
> > Rearrange: (m-d), (m+d), 2m
> > so that : (m-d) + (m+d) = 2m
> >
> > Yes, the pattern holds for any values of m and d
> > that are integers.

Did you follow this reasoning? Do you see
that it means that "double the middle term
in an arithmetic progression is always equal
to the sum of the two outside terms"?

In other words, we've proved that the pattern
you suspected really does hold - always.

> > As I said, an interesting pattern. But I can't
> > help but ask myself: what does this have to do
> > with Harmonic Entropy?
>
> Chords of the type 3:7:10 have the coherant
> difference tones --which is pertinant to the
> list, yes?

Yes, I see.

> --- and are useful ways to arrange chords, and wanted
> to know how the math worked.

So, I hope it is clear now?

> Or maybe I just need a math tutor?

That'll be 5 bucks an answer, then ...! ;-)

> Anyway, can you, personally, separate the love of math
> (or numbers) from love of music?!

Oh yes, easily. They both draw on my love of pattern,
true, but each has also other dimensions and aspects
that the other doesn't share. And maths doesn't make
me dance!

Regards,
Yahya