What is Frequency?

Hi all,

there's a point which I think is important for tuning issues but which in the whole "rational verses irrational" debate is being lost in translation somewhere. So I thought I'd start afresh just to mark the occasion. Because people often assume that they already know what a frequency is, it is then assumed that we can experiment with different tunings without regard for this question. However, I suspect that "what is frequency?" and "what is in-tune?" are in fact the same question. In any case, I'd like to put it out there.

Why is it that we can play the same note on a variety of different instruments which might not have anything else in common? Standard scientific answer; because two instruments may share the same fundamental (sine wave) frequency, but they are distinguished by the number and intensity of their overtones. However, this is not the whole picture. Frequency can be more generally defined as the class of all periodic functions which share a common period. And if we look at the behaviour of waves, there is one subtle but vital difference between these two definitions. For the second definition does not require a sine wave corresponding to the fundamental. Simple E.g. the frequencies 9Hz and 6Hz when added produce a frequency corresponding to their highest common factor 3Hz. Since 9/6 = 3/2 then these are the 2nd and 3rd harmonics to the fundamental 1. While two violins (say) will each produce a sine wave corresponding to their respective fundamentals, it is
quite possible that their combination will produce a third frequency with no sine wave corresponding to it.

Why is this so important? 1. Because the very definition of frequency must admit rationality (or something near to it) to begin with. Intervals are composed of frequencies and frequencies of intervals. The definitions are inter-dependent and dialectical. 2. We no longer need "boundary conditions" to produce harmonics as a mere "special case". Irrespective as to questions of how a wave is produced, all that is required for harmony to exist is that two waves occupy the same place and time. It therefore applies to all waves including light and free matter. Scientists thought that by explaining the physics of an individual isolated instrument they were explaining the whole of musical harmony. But a symphony orchestra is more than the sum of its parts. They did not consider the very meaning of musical harmony, notes in combination. 3. This HCF frequency seems to explain musical tonality since the tonic is octave equivalent to this fundamental. If we trust the
model, then to play an untempered 4: 5 as E: G together will produce a C note. It was THIS which led me to the realization that the minor 3rd was better approximated by 16: 19 because 5: 6 as Eb: G gives something like a B tonic (D# being the 5th harmonic or maj 3 to B).

Of course this model might be over simplistic and it might end up that no pure frequency exists, that real waves are only ever "almost" periodic or something. But the set of whole numbers is by no means a territory that's been fully explored and any new discovery, however small, might have many far reaching consequences. Thoughts anybody?

NB: It is currently believed in physics that not all waves obey this principle, that so-called "dispersive" waves give different ratios for different observers. But once an addition has taken place then this is the (single) wave and this is the frequency. The upshot of this is that universal harmony has been misrepresented and music becomes less important than mechanics and mathematics. But dispersive waves rely on sums and differences between frequency levels and therefore are beat frequencies. It is a gross misinterpretation. (How could "they" get it so wrong??? Such is the world of advertising!).

Rick

Get the name you always wanted with the new y7mail email address.
www.yahoo7.com.au/mail

Say Rick, I think I should point out that both Carl and Tom, whether
intentionally or no, each recently posted a "test" which actually
answers, in the realm of "real life", the fundamental questions in
the recent "off topic" debates, insofar as they pertain to actual
music making and appreciation. Carl posted the link to an online
interval discernment test and Tom posted a question about a
rectangular table (notice that "rectangular" is given, and so is the
fact that "you are sitting at it").

-Cameron Bobro

--- In tuning@yahoogroups.com, rick ballan <rick_ballan@...> wrote:
>
> Hi all,
>
> there's a point which I think is important for tuning issues but
which in the whole "rational verses irrational" debate is being lost
in translation somewhere. So I thought I'd start afresh just to mark
the occasion. Because people often assume that they already know
what a frequency is, it is then assumed that we can experiment with
different tunings without regard for this question. However, I
suspect that "what is frequency?" and "what is in-tune?" are in fact
the same question. In any case, I'd like to put it out there.
>
> Why is it that we can play the same note on a variety of different
instruments which might not have anything else in common? Standard
scientific answer; because two instruments may share the same
fundamental (sine wave) frequency, but they are distinguished by the
number and intensity of their overtones. However, this is not the
whole picture. Frequency can be more generally defined as the class
of all periodic functions which share a common period. And if we
look at the behaviour of waves, there is one subtle but vital
difference between these two definitions. For the second definition
does not require a sine wave corresponding to the fundamental.
Simple E.g. the frequencies 9Hz and 6Hz when added produce a
frequency corresponding to their highest common factor 3Hz. Since 9/
6 = 3/2 then these are the 2nd and 3rd harmonics to the fundamental
1. While two violins (say) will each produce a sine wave
corresponding to their respective fundamentals, it is
> quite possible that their combination will produce a third
frequency with no sine wave corresponding to it.
>
> Why is this so important? 1. Because the very definition of
frequency must admit rationality (or something near to it) to begin
with. Intervals are composed of frequencies and frequencies of
intervals. The definitions are inter-dependent and dialectical. 2.
We no longer need "boundary conditions" to produce harmonics as a
mere "special case". Irrespective as to questions of how a wave is
produced, all that is required for harmony to exist is that two
waves occupy the same place and time. It therefore applies to all
waves including light and free matter. Scientists thought that by
explaining the physics of an individual isolated instrument they
were explaining the whole of musical harmony. But a symphony
orchestra is more than the sum of its parts. They did not consider
the very meaning of musical harmony, notes in combination. 3. This
HCF frequency seems to explain musical tonality since the tonic is
octave equivalent to this fundamental. If we trust the
> model, then to play an untempered 4: 5 as E: G together will
produce a C note. It was THIS which led me to the realization that
the minor 3rd was better approximated by 16: 19 because 5: 6 as Eb:
G gives something like a B tonic (D# being the 5th harmonic or maj 3
to B).
>
> Of course this model might be over simplistic and it might end up
that no pure frequency exists, that real waves are only ever
"almost" periodic or something. But the set of whole numbers is by
no means a territory that's been fully explored and any new
discovery, however small, might have many far reaching consequences.
Thoughts anybody?
>
> NB: It is currently believed in physics that not all waves obey
this principle, that so-called "dispersive" waves give different
ratios for different observers. But once an addition has taken place
then this is the (single) wave and this is the frequency. The upshot
of this is that universal harmony has been misrepresented and music
becomes less important than mechanics and mathematics. But
dispersive waves rely on sums and differences between frequency
levels and therefore are beat frequencies. It is a gross
misinterpretation. (How could "they" get it so wrong??? Such is the
world of advertising!).
>
> Rick
>
>
>
> Get the name you always wanted with the new y7mail email
address.
> www.yahoo7.com.au/mail
>