Fokker and ED?

Hi there.
After some hours of learning how to understand Fokker's way of describing
intervals, I finally managed to read through his article about unison
vectors ("www.xs4all.nl/~huygensf/doc/fokkerpb.html"). In his calculations
he often substitutes the actual resulting ratios with their nearest
epimores. For example: One would expect that the 7-limit lattice coordinates
of "(1, 0, 3)" comes out as 1029/1024. Instead, he puts the result as
206/205. An unaware reader might think: "Is that a mistake?" Not a mistake,
just the nearest epimoric factor. Fokker goes on like this almost allover
the article. Even instead of 2048/2025 for the diaschisma, he writes 89/88
(that is pretty close indeed). After I reached the end of the paper, I tried
to think a bit about Fokker's ideas once again. Suddenly I realized a
strange coincidence between the way he approximated intervals there and the
way I used to simplify my calculations some time ago. I think it could be
quite useful to say something more about this in the following text.
As far as I could find out, it was in March 2002 when I first described
an interval in a way I've never seen before. And I often used this method
again since. The value used for this is what I call "ED" (or "Epimoric
Denominator"). You can convert a linear factor to ED by doing "ED = 1/(F-1)"
(where F is the factor) and the opposite by "F = (ED+1)/ED". It shows how
close a factor is to an epimoric ratio. If the factor itself is actually
epimoric, then ED is the denominator and therefore an integer. If ED is not
an integer, then the interval is either not epimoric or not JI at all.
Nonetheless, it always serves as a possible denominator for a ratio with the
numerator exactly 1 higher, no matter if integer or not. And this is the
main thing that's so important about ED. For example: ED for an octave is 1
(i.e. the factor = 2/1), ED for a fifth is 2 (i.e. the factor is 3/2), ED
for a major sixth is 1.5 (i.e. "5/3" is the same as "2.5/1.5"). As you can
see, the smaller the interval, the larger the ED (i.e. the closer to
Infinity). In fact, a unison can't be expressed as ED at all (this would
require a value of 1/0). But it does not need to be a rational number.
Though ED is mainly useful for comparing JI intervals, it can, of course,
describe simply any possible interval you just think of. Moreover, it can
also distinguish between raising and lowering intervals. While a major third
upwards has an ED of 4 (i.e. the factor is 5/4), the ED for a major third
downwards is -5 (i.e. "4/5" is the same as "(-4)/(-5)"). Working with ED can
be useful in various situations. First of all, if you consider the ED to be
an absolute frequency in Hz, adding a tone 1Hz higher makes the desired
interval. That's why one can make a good image about the interval quite
nicely. For example: The regular 5-limit schisma has an ED of ~885.62162
(quite a large value and therefore just a small detuning), while the ED for
the Pythagorean comma is ~73.29624 (a significantly smaller value so the
detuning is remarcably stronger). Next, ED can help in finding the part by
which to alter the frequency in order the tone would rise or fall by the
given interval. The part in question is nothing else than the inversion
(i.e. the reciprocal) of the ED. For example: A major third upwards has an
ED of 4, while a major third downwards has an ED of -5. That means that a
tone gets a major third higher if you increase the frequency by its 1/4,
while decreasing the frequency by its 1/5 shifts the tone a major third
lower. Okay, let's stop talking about the usefulness of ED. Now the main
question - why do I say all this? At the time I first used ED, I found no
resources showing any evidence of such a system used earlier by someone
else. So I thought I just devised something new. And now, two years later, a
surprise is here. Though Fokker limits his epimoric approximations just to
integer epimores, his concept of substituting intervals with epimores is
very similar to ED. If it is like this, I can nothing but wonder if
something like ED was ever used even earlier.
A good way of averaging JI intervals is to take their EDs and find the
arithmetic mean of these. The result will then be used as the ED for the
average interval. You can also get the same result by converting the linear
factors to fractions with always the same difference between "num" and "den"
and then finding the arithmetic mean of these. For example: To average 3/1
and 2/1 this way, you can either use the arithmetic mean of 0.5 and 1 as the
resulting ED of 0.75, or convert the factors to 6/2 and 8/4 and then do the
simple average (i.e. between 2 and 4 is 3, between 6 and 8 is 7, so between
6/2 and 8/4 is 7/3). After consulting this with Manuel, I finally agreed to
call it "epimoric mean" for using in Scala. But, as the time passes, I got a
strange impression (which I just can't get out of my mind) that I must have
read somewhere about a thing called "partial mean" which was exactly this
(i.e. between 2/1 and 5/3 is 9/5 because it's the same as between 8/4 and
10/6). Maybe some of you could help me. Do you think the term "partial mean"
is my invention, or was it perhaps already known earlier? Thanks.
Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@w...> wrote:

Suddenly I realized a
> strange coincidence between the way he approximated intervals there
and the
> way I used to simplify my calculations some time ago.

The way he is doing it, so far as I can tell, is to take the last
convergent of the continued fraction which is an epimore. This is
reasonably easy to accomplish, though it does leave the question of
why you want to do it. One answer might be comparison of the sizes of
intervals, a kind of rational alternative to cents.

hi Gene and Petr,

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

> --- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@w...> wrote:
>
> > Suddenly I realized a strange coincidence between the
> > way he approximated intervals there and the way I
> > used to simplify my calculations some time ago.
>
> The way he is doing it, so far as I can tell, is to take
> the last convergent of the continued fraction which is
> an epimore. This is reasonably easy to accomplish, though
> it does leave the question of why you want to do it. One
> answer might be comparison of the sizes of intervals, a
> kind of rational alternative to cents.

this kind of thing was common in the tuning literature
in the days before calculators and computers.

in Ellis's translation of Helmholtz, both Helmholtz in
the main text and Ellis in his massive appendix used
epimoric approximations of large intervals. in most
cases, Helmholtz substitutes the epimoric approximations
without any comment, then Ellis explains the correct values
in the appendix.

i've seen this in a lot of German theoretical works
and journals from c.1850-1950.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> this kind of thing was common in the tuning literature
> in the days before calculators and computers.
>
> in Ellis's translation of Helmholtz, both Helmholtz in
> the main text and Ellis in his massive appendix used
> epimoric approximations of large intervals. in most
> cases, Helmholtz substitutes the epimoric approximations
> without any comment, then Ellis explains the correct values
> in the appendix.
>
> i've seen this in a lot of German theoretical works
> and journals from c.1850-1950.

in fact, it seems to me that it was pretty standard
to just say that the schisma = 887/886 ratio and the
Pythagorean comma = 74/73 ratio, in this literature.

of course, using these numbers is fine if one knows
that they are an approximation to the correct results
... but generally there is no commentary explaining this.

things have changed today, thanks to computers.

i do most of my calculations in Excel using floating-point
to 10-digit accuracy. these are the values which go
in the Tuning Encyclopaedia webpages, and in many cases
i note small-integer fractional approximations, whether
epimoric or not.

in fact, upon writing this post, i saw that i had not
done this for the Pythagorean-comma or skhisma, and so
i've added that now.

-monz

From: "Gene Ward Smith" <gwsmith@s>

> The way he is doing it, so far as I can tell, is to take the last
> convergent of the continued fraction which is an epimore. This is
> reasonably easy to accomplish, though it does leave the question of
> why you want to do it. One answer might be comparison of the sizes of
> intervals, a kind of rational alternative to cents.

Hmmm, great. I have nothing to say. You got it perfectly. Yes, I more or
less mean it to be used like this. As you can see from my previous message,
I see the main "plus" of ED in the fact that it does not have to be an
integer and therefore not only epimores can be measured as EDs but also any
other intervals. This could help, for example, to examine the properties of
some equal tunings for intervals not too far from epimores. I don't think I
need to mention again the way ED can help in shifting a tone by a desired
interval, I've written about it in my previous text. Moreover, I often find
the "partial mean" useful when generating intervals with a "ring modulator"
or effects like that. Let's give an example here. If the carrier input is a
sine wave of 400Hz and the modulator is a sine of 100Hz, the output contains
two sines mixed of 300Hz and 500Hz (i.e. the difference and the sum). So I
get a major sixth as a result. Now I'll shift the carrier frequency 100Hz
higher (i.e. to 500Hz) and leave the modulator unchanged. What I get then
is a perfect fifth of 400Hz and 600Hz. When I set the carrier frequency
right in-between (i.e. to 450Hz), I get 350Hz and 550Hz as the result. So
indeed, the partial mean between 5/3 and 3/2 is 11/7 as it is the same as
between 10/6 and 12/8. Yet one other
thing is here to be said. I never tried to find out how many people were
given the ability to distinguish absolute pitches just by ear. But since I
was lucky enough to be one of these, I often find it useful to imagine an
examined interval as a
pair of two particular tones. In the case of intervals like dieses or
commas, the ED value can easily be used as an absolute frequency of a tone
in the range of human hearing. Then the numerator is exactly 1Hz higher so
it will always make beats one per second. And it's quite clear that low
tones have to be detuned more than the higher ones in order to get a beat
rate of one per second. So in this case it helps me a lot because I just
think of a tone that comes out as the ED and imagine some beats added to it
with a rate of 1 per second. For example, if the ED is close to 100, I just
know: "Aha, it's something like 101/100. And 100Hz is a bit higher than G2."
So I imagine a tone a bit higher than G2 with the beats of one per second.
And it's quite clear that adding beats to a tone of around 100Hz makes a
different sound than adding beats to a tone of around, I don't know, 960Hz,
for example.
Some more ideas?
Petr

it turns out that i was a little harsh
on Professor Helmholtz. he has a lot of
fans around here, so before one of you
lays into me, i'll correct my errors myself.

Helmholtz does use epimoric rational approximations
for the error amounts he describes, but he does
indeed mention in passing that they are approximations.

the pages of his book in question are p 312-313
in the 1954 Dover reprint.

on p 313, Helmholtz gives a table comparing
some basic 5-limit JI intervals with their
12edo and Pythagorean counterparts. to this
table, Ellis has added integer cents values
for all three sets of intervals.

the error between Pythagorean and JI is given
in all cases as 81/80, the syntonic comma, and
this is exactly the correct value.

in the middle column, which compares 12edo and JI,
Helmholtz again uses epimoric approximations, to
the irrational error values. i give the correct
values below, and summarize Helmholtz's approximations
in the bottom-most ratio for each interval.

(on the Yahoo web interface, this will look all
messed up. click the "reply" button to see it
formatted properly.)

Helmholtz 1954, p 313:

>> The 3rds and 6ths of the equal temperament are
>> nearer the perfect intervals than are the Pythagorean.

---- monzo ----
2 3 5

| 1/3 0 0 > 2^(4/12) 400
- | 2 0 1 > 5/4 ~386.3137139
------------------
| 7/3 0 -1 > ~13.68628614

= ~6096382/6048377
= ~127/126

| 3 0 -1 > 8/5 ~813.6862861
- | 2/3 0 0 > 2^(8/12) 800
------------------
| 7/3 0 -1 > ~13.68628614

= ~6096382/6048377
= ~127/126

| 1 1 -1 > 6/5 400
- | 1/4 0 0 > 2^(3/12) ~386.3137139
------------------
| 3/4 1 -1 > ~15.641287

= ~9913531/9824368
= ~122/121

| 3/4 0 0 > 2^(9/12) 400
- | 0 -1 1 > 5/3 ~386.3137139
------------------
| 3/4 1 -1 > ~15.641287

= ~9913531/9824368
= ~122/121

| 4 -1 -1 > 16/15 400
- | 1/12 0 0 > 2^(1/12) ~386.3137139
------------------
|47/12 -1 -1 > ~11.73128527

= ~2976153/2956054
= ~148/147

so the actual errors are two instances each
of (2^(7/3))/5 and (2^(3/4))*(3/5), and one of
(2^(47/12))/15.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > this kind of thing was common in the tuning literature
> > in the days before calculators and computers.
> >
> > in Ellis's translation of Helmholtz, both Helmholtz in
> > the main text and Ellis in his massive appendix used
> > epimoric approximations of large intervals. in most
> > cases, Helmholtz substitutes the epimoric approximations
> > without any comment, then Ellis explains the correct values
> > in the appendix.
> >
> > i've seen this in a lot of German theoretical works
> > and journals from c.1850-1950.
>
>
>
> in fact, it seems to me that it was pretty standard
> to just say that the schisma = 887/886 ratio and the
> Pythagorean comma = 74/73 ratio, in this literature.
>
> of course, using these numbers is fine if one knows
> that they are an approximation to the correct results
> ... but generally there is no commentary explaining this.
>
> things have changed today, thanks to computers.
>
> i do most of my calculations in Excel using floating-point
> to 10-digit accuracy. these are the values which go
> in the Tuning Encyclopaedia webpages, and in many cases
> i note small-integer fractional approximations, whether
> epimoric or not.
>
> in fact, upon writing this post, i saw that i had not
> done this for the Pythagorean-comma or skhisma, and so
> i've added that now.
>
>
>
> -monz