Reciprocals of triangular numbers cite

Here are urls connected to this business of expressing musical
intervals as sums of reciprocals of triangular numbers:

http://encyclopedia.thefreedictionary.com/Tertius%20minor

http://encyclopedia.thefreedictionary.com/Ditono

I'm still wondering if we can track this down to some ancient source.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Here are urls connected to this business of expressing musical
> intervals as sums of reciprocals of triangular numbers:
>
> http://encyclopedia.thefreedictionary.com/Tertius%20minor
>
> http://encyclopedia.thefreedictionary.com/Ditono
>
> I'm still wondering if we can track this down to some
> ancient source.

i haven't been following this thread, but i think
the Sumerians and Babylonians knew about triangular numbers.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> > I'm still wondering if we can track this down to some
> > ancient source.

> i haven't been following this thread, but i think
> the Sumerians and Babylonians knew about triangular numbers.

The Babylonians dealt with fractions by expressing them as numbers
base 60, and they knew that this sometimes lead to infinte expansions
which required approximating. The Egyptians dealt with them by
expressing them as a sum of reciprocals of integers ("Egyptian
fractions"), a habit picked up by the Greeks and thence the Romans. I
would guess this business traces back to expressing musical ratios as
Egyptian fractions.

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > > I'm still wondering if we can track this down to some
> > > ancient source.
>
> > i haven't been following this thread, but i think
> > the Sumerians and Babylonians knew about triangular numbers.
>
> The Babylonians dealt with fractions by expressing them
> as numbers base 60, and they knew that this sometimes lead
> to infinte expansions which required approximating. The
> Egyptians dealt with them by expressing them as a sum of
> reciprocals of integers ("Egyptian fractions"), a habit
> picked up by the Greeks and thence the Romans.

the Babylonians (who most likely got it from the Sumerians)
were well-versed in using reciprocals, but you're correct
that they expressed them in sexagesimal form as base-60
digits.

they also had excellent approximations for such irrational
numbers as SQRT(2) and PI ... much better, in fact, than
the decimal approximations that most of us use today.

> I would guess this business traces back to expressing
> musical ratios as Egyptian fractions.

that would be my guess too.

my years of research into ancient tuning-theory tells me
that *lots* of mathematical and numerical properties were
discovered by examining musical pitches, and also
astronomical data.

in fact, in antiquity, the mathematics of music and
astronomy were often considered to be the same.

-monz

and from the citation Gene cites (which I originally stumbled upon!),
can anyone explain the passage "the ratio 6/5 between a pair of
frequencies or, equivalently, the ratio 5:6 betwee a pair of
wavelenths..."

regarding the distintion between frequency and wavelenth,and 6/5 vs
5/6? thanks, Kelly

> Here are urls connected to this business of expressing musical
> intervals as sums of reciprocals of triangular numbers:
>
> http://encyclopedia.thefreedictionary.com/Tertius%20minor
>
> http://encyclopedia.thefreedictionary.com/Ditono
>
> I'm still wondering if we can track this down to some ancient
source.

hi Kelly,

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> and from the citation Gene cites (which I originally
> stumbled upon!), can anyone explain the passage "the
> ratio 6/5 between a pair of frequencies or, equivalently,
> the ratio 5:6 betwee a pair of wavelenths..."
>
> regarding the distintion between frequency and wavelenth,
> and 6/5 vs 5/6? thanks, Kelly

frequency and wavelength are inversely proportional
to each other. in other words, the longer the wavelength,
the lower the frequency ... and therefore the shorter
the wavelength, the higher the frequency.

let's examine our standard reference tone A-440 Hz.
this means that the sound waves which produce that pitch
vibrate in such a way ("periodic") that they repeat
exactly, 440 times per second.

now let's take the "A" an 8ve above that. the ratio
of frequencies for the 8ve is 2:1. this means that
the note "A" an 8ve higher than A-440, will have a
waveform which repeats 880 times per second.

now, the time unit has remained constant: one second
in both cases. so if the lower note's frequency is
1/2 that of the higher note, can you see that the
*length* of its wave (the part which repeats exactly)
will be twice as long as the higher note's?

so that's why ratios which describe frequency relationships
must be inverted to describe the wavelength relationships.

now ... are you clear on the use of ratios in discussing
tunings? if not, you'll need a refresher course on that
... which i could give you, but another time.

PS -- Kelly, if you want to continue discussing stuff
that's mostly math, it will irritate folks on this list.
we have a list just for that:

/tuning-math/

-monz

thanks Monz. I've never come across the acoustical explanation
(involving frequency and wavelength) of inverted fractions. I
assume this is unrelated to 'subharmonic chords' etc, where you also
see 'inverted fractions' for intervals? To be complete, I've read in
several tracts that is somewhat arbitrary that the usual JI intervals
for the harmonic series have the higher numbered harmonic in the
numerator? thanks, Kelly (I get your point about math threads. You
held your irritation in check pretty well!)

> hi Kelly,
>
> --- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> > and from the citation Gene cites (which I originally
> > stumbled upon!), can anyone explain the passage "the
> > ratio 6/5 between a pair of frequencies or, equivalently,
> > the ratio 5:6 betwee a pair of wavelenths..."
> >
> > regarding the distintion between frequency and wavelenth,
> > and 6/5 vs 5/6? thanks, Kelly
>
>
>
> frequency and wavelength are inversely proportional
> to each other. in other words, the longer the wavelength,
> the lower the frequency ... and therefore the shorter
> the wavelength, the higher the frequency.
>
> let's examine our standard reference tone A-440 Hz.
> this means that the sound waves which produce that pitch
> vibrate in such a way ("periodic") that they repeat
> exactly, 440 times per second.
>
> now let's take the "A" an 8ve above that. the ratio
> of frequencies for the 8ve is 2:1. this means that
> the note "A" an 8ve higher than A-440, will have a
> waveform which repeats 880 times per second.
>
> now, the time unit has remained constant: one second
> in both cases. so if the lower note's frequency is
> 1/2 that of the higher note, can you see that the
> *length* of its wave (the part which repeats exactly)
> will be twice as long as the higher note's?
>
> so that's why ratios which describe frequency relationships
> must be inverted to describe the wavelength relationships.
>
>
> > -monz

hi Kelly,

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> thanks Monz. I've never come across the acoustical
> explanation (involving frequency and wavelength) of
> inverted fractions. I assume this is unrelated to
> 'subharmonic chords' etc, where you also see
> 'inverted fractions' for intervals? To be complete,
> I've read in several tracts that is somewhat arbitrary
> that the usual JI intervals for the harmonic series have
> the higher numbered harmonic in the numerator? thanks,
> Kelly (I get your point about math threads. You
> held your irritation in check pretty well!)

well, it's not exactly arbitrary.

the same kind of inversion occurs when comparing the
ratios of frequencies with the ratios of string-lengths.

from ancient times until the 1500s, music-theorists
always measured the ratios of musical intervals by
use of a monochord, which was simply a box with a
string stretched across it. several bridges would
be placed at particular carefully-measured distances
along the string, to produce the different intervals.

then when scientists discovered how to measure the
actual frequencies of pitches, most theorists switched
to using the ratios to describe frequencies instead of
string-lengths ... and the two procedures are inversely
related.

for example, i said in my last post to you that the
standard ratio of the 8ve is 2:1, meaning that a pitch
an 8ve higher than the reference pitch will have a
waveform which repeats exactly twice as often as that
of the reference pitch.

but in terms of string-length, that same 8ve-higher pitch
would be exactly 1/2 of the string.

therefore, for a reference pitch with string-length s
and frequency f, a pitch an 8ve higher will have
1/2 s = 2 * f .

as for "subharmonic chords", they are simply a way of
creating an arithmetical series which is an exact
mirror-image of the harmonic series. so all the
relationships are exactly the inverse of those i gave
above.

thus, a pitch an 8ve *lower* than the reference would
be 2 * s = 1/2 f .

the 5-limit just-intonation "minor-3rd" above the
reference is 5/6 s = 6/5 f.

the corresponding 5-limit JI "minor-3rd" below the
reference is 6/5 s = 5/6 f.

got it?

so what *is* arbitrary is whether one decides to
describe ratios in terms of frequency or in terms
of string-length. but once that decision is made,
everything else is consistent, because as long as
the inversion is taken into consideration, the two
methods are equivalent.

-monz