Mathematical proof sought

Hi.
I get a great deal of insight about tuning systems from the
observation that:

(3/2)^12 != (2/1)^7

that is, twelve fifths does not exactly equal seven octaves. In fact,
it seems that there are no M and N such that

(3/2)^M != (2/1)^N

but I can't prove it. Other "near misses" occur at, for example,
M=53, N=31.

Is anyone aware of a proof of the general case? All leads
appreciated.

--- In tuning@y..., "tunerguy2002" <tunerguy2002@y...> wrote:
> Hi.
> I get a great deal of insight about tuning systems from the
> observation that:
>
> (3/2)^12 != (2/1)^7
>
> that is, twelve fifths does not exactly equal seven octaves. In
fact,
> it seems that there are no M and N such that
>
> (3/2)^M != (2/1)^N
>
> but I can't prove it. Other "near misses" occur at, for example,
> M=53, N=31.
>
> Is anyone aware of a proof of the general case? All leads
> appreciated.

This is a very simple consequence of the Fundamental Theorem of
Arithmetic. I'm sure Gene can give you the most concise proof of
this. You should ask the question at

tuning-math@yahoogroups.com

--- tunerguy2002 <tunerguy2002@yahoo.com> wrote:
> Hi.
> I get a great deal of insight about tuning systems
> from the
> observation that:
>
> (3/2)^12 != (2/1)^7
>
> that is, twelve fifths does not exactly equal seven
> octaves. In fact,
> it seems that there are no M and N such that
>
> (3/2)^M != (2/1)^N

> but I can't prove it. Other "near misses" occur at,
> for example,
> M=53, N=31.
>
> Is anyone aware of a proof of the general case? All
> leads
> appreciated.
>
>
Hi Anonymous TunerGuy,

The proof you seek is pretty easy to grasp. After the
zeroth power, 1 in both cases, no power of 3 (or of
3/2) will ever equal any power of 2 (or of 2/1).

2^0 = 1 3^0 = 1
2^1 = 2 3^1 = 3
2^2 = 4 3^2 = 6
2^3 = 8 3^3 = 9
2^4 = 16 3^4 = 27
2^5 = 32 3^5 = 81
2^6 = 64 3^6 = 243
2^7 = 128 3^7 = 729
2^8 = 256 3^8 = 2187
2^9 = 512 3^9 = 6561
2^10 =1024 3^10 =19683

After 1, the powers of 2 are always even numbers.
After 6, the powers of 3 are always odd numbers.

Note that after 6, the final digits of the powers of 3
begin to repeat 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3,
etc. If all the powers of 3 above 6 will end in 9, or
7, or 1, or 3, then all the powers of 3 above 6 will
be odd numbers.

Since no power of 2 happens to equal 6, and since no
odd number can ever be equal to any even number, these
two facts taken together constitute proof that no
power of 2 can ever be equal to any power of 3.

Also, by the way, the list of better and better near
misses goes on infinitely!

This subject of near misses is related to the equal
temperaments, which are also known as the equal
divisions of the octave. One can make a list of those
equal divisions of the octave whose nearest scale
steps are better and better approximations of one or
more musical intervals (scale steps). If, for
example, 3/2, 5/4, and 6/5 are chosen as the target
scale step ratios, then the list of equal divisions of
the octave whose nearest scale steps form better and
better approximations to 3/2 and 5/4 and 6/5 looks as
follows:

Equal
Divisions
of the Octave
whose nearest scale
steps are closer and closer
approximations to 3/2, 5/4, and 6/5

1
2
3
5
7
12
19
31
34
53
118
171
289
323
441
612
730
1171
1783
2513
4296, etc., etc., into the millions.

These are equal temperaments which are better and
better approximations of 5-limit just intonation.

Many people have worked in his area, including Sir
Isaac Newton, Joseph Wurschmidt, Joseph Yasser,
Bosanquet, Ivor Darreg, Erv Wilson, John Chalmers,
Larry Hanson, Siemen Terpstra, Kees van Prooijen, and
me.

All the best to you,
- Mark Rankin

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--- In tuning@y..., Mark Rankin <markrankin95511@y...> wrote:
>
> --- tunerguy2002 <tunerguy2002@y...> wrote:
> > Hi.
> > I get a great deal of insight about tuning systems
> > from the
> > observation that:
> >
> > (3/2)^12 != (2/1)^7
> >
> > that is, twelve fifths does not exactly equal seven
> > octaves. In fact,
> > it seems that there are no M and N such that
> >
> > (3/2)^M != (2/1)^N
>
>
> > but I can't prove it. Other "near misses" occur at,
> > for example,
> > M=53, N=31.
> >
> > Is anyone aware of a proof of the general case? All
> > leads
> > appreciated.
> >
> >
> Hi Anonymous TunerGuy,
>
> The proof you seek is pretty easy to grasp. After the
> zeroth power, 1 in both cases, no power of 3 (or of
> 3/2) will ever equal any power of 2 (or of 2/1).
>
>
> 2^0 = 1 3^0 = 1
> 2^1 = 2 3^1 = 3
> 2^2 = 4 3^2 = 6
> 2^3 = 8 3^3 = 9
> 2^4 = 16 3^4 = 27
> 2^5 = 32 3^5 = 81
> 2^6 = 64 3^6 = 243
> 2^7 = 128 3^7 = 729
> 2^8 = 256 3^8 = 2187
> 2^9 = 512 3^9 = 6561
> 2^10 =1024 3^10 =19683
>
>
> After 1, the powers of 2 are always even numbers.
> After 6, the powers of 3 are always odd numbers.
>
> Note that after 6, the final digits of the powers of 3
> begin to repeat 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1, 3,
> etc. If all the powers of 3 above 6 will end in 9, or
> 7, or 1, or 3, then all the powers of 3 above 6 will
> be odd numbers.
>
> Since no power of 2 happens to equal 6, and since no
> odd number can ever be equal to any even number, these
> two facts taken together constitute proof that no
> power of 2 can ever be equal to any power of 3.
>
> Also, by the way, the list of better and better near
> misses goes on infinitely!
>
> This subject of near misses is related to the equal
> temperaments, which are also known as the equal
> divisions of the octave. One can make a list of those
> equal divisions of the octave whose nearest scale
> steps are better and better approximations of one or
> more musical intervals (scale steps). If, for
> example, 3/2, 5/4, and 6/5 are chosen as the target
> scale step ratios, then the list of equal divisions of
> the octave whose nearest scale steps form better and
> better approximations to 3/2 and 5/4 and 6/5 looks as
> follows:
>
>
> Equal
> Divisions
> of the Octave
> whose nearest scale
> steps are closer and closer
> approximations to 3/2, 5/4, and 6/5
>
> 1
> 2
> 3
> 5
> 7
> 12
> 19
> 31
> 34
> 53
> 118
> 171
> 289
> 323
> 441
> 612
> 730
> 1171
> 1783
> 2513
> 4296, etc., etc., into the millions.
>
> These are equal temperaments which are better and
> better approximations of 5-limit just intonation.
>
> Many people have worked in his area, including Sir
> Isaac Newton, Joseph Wurschmidt, Joseph Yasser,
> Bosanquet, Ivor Darreg, Erv Wilson, John Chalmers,
> Larry Hanson, Siemen Terpstra, Kees van Prooijen, and
> me.
>
> All the best to you,
> - Mark Rankin

Anyone really interested in this stuff is encouraged to join the list

tuning-math@yahoogroups.com

where we have studied the behaviors of these approximations in great
depth and have discovered many new and unexpected features.