more on e-pi ratios

About my last post...

The stretched-octave 53-tone scale can also be derived from 76 equal divisions of e, or 87 equal divisions of pi. The three most important intervals are:

pi/e ("semifourth") ~ 250.5613 cents
pi^7/e^8 ("comma") ~ 22.6951 cents
e^87/pi^76 ("schisma") ~ 0.9153 cents

So who wants to have fun with these?

~Danny~

I do, I do. Can we build maqams with these?

Cordially,
Ozan
----- Original Message -----
From: Danny Wier
To: tuning-math@yahoogroups.com
Sent: 23 Nisan 2005 Cumartesi 21:18
Subject: [tuning-math] more on e-pi ratios

About my last post...

The stretched-octave 53-tone scale can also be derived from 76 equal
divisions of e, or 87 equal divisions of pi. The three most important
intervals are:

pi/e ("semifourth") ~ 250.5613 cents
pi^7/e^8 ("comma") ~ 22.6951 cents
e^87/pi^76 ("schisma") ~ 0.9153 cents

So who wants to have fun with these?

~Danny~

Ozan Yarman wrote (about my e/pi tuning):

> I do, I do. Can we build maqams with these?

Since it's close to 53-EDO, you can in the Turkish system. The octave is stretched by 7.42 cents, which might be a lot, unless you're tuning a piano. The fifth is 4.34 cents sharp and the fourth is 3.08 cents sharp. This may create a problem when multiple octaves are piled on, unless you temper down a 0.92-cent "schisma" every tone or something.

Maqam as-Saba (the full minor tenth range) in Arabic tuning, treating the E half-flat as two commas flat, could be tuned: 0.00 (Dugah/D/Re), 159.78, 295.95, 410.34, 683.60, 797.07, 1002.25, 1116.64, 1389.89, 1503.37.

Monz, Danny, Ozan,

Why use pi and e as bases for intervals? I don't see the point.
They don't actually _resonate_! :-)

Now, I know that when I touch the midpoint of my guitar string,
I get a clean harmonic. 1:2 - now _that_ resonates ...

Yahya

Danny wrote:

> pi/e ("semifourth") ~ 250.5613 cents
> pi^7/e^8 ("comma") ~ 22.6951 cents
> e^87/pi^76 ("schisma") ~ 0.9153 cents

> So who wants to have fun with these?

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Brother Yahya,

Why should we seek resonating intervals all the time? Meantone temperaments certainly do not. I'm sure there are many other varieties of tuning such as the Arabic quarter-tone system that do not as well.

Perhaps the question should rather be `who is resonating to what intervals`?

Cordially,
Ozan

----- Original Message -----
From: Yahya Abdal-Aziz
To: tuning-math@yahoogroups.com
Sent: 24 Nisan 2005 Pazar 19:32
Subject: [tuning-math] RE: more on e-pi ratios

Monz, Danny, Ozan,

Why use pi and e as bases for intervals? I don't see the point.
They don't actually _resonate_! :-)

Now, I know that when I touch the midpoint of my guitar string,
I get a clean harmonic. 1:2 - now _that_ resonates ...

Yahya

Danny wrote:

> pi/e ("semifourth") ~ 250.5613 cents
> pi^7/e^8 ("comma") ~ 22.6951 cents
> e^87/pi^76 ("schisma") ~ 0.9153 cents

> So who wants to have fun with these?

Yahya Abdal-Aziz wrote:

> Monz, Danny, Ozan,
>
> Why use pi and e as bases for intervals? I don't see the point.
> They don't actually _resonate_! :-)

It's intended to be pointless (except for the 53-tone thing), or a more extreme case of xenharmonia. Like writing music in 11- or 13-equal. Many of the pitches are no less resonant than some intervals in 12-tone anyway.

(All of this is part of my own work on tunings that temper the octave and not just the fifth, as well as more experimental thing, like this.)

~Danny~

Yahya Abdal-Aziz wrote:
> Monz, Danny, Ozan,
> Why use pi and e as bases for intervals? I don't see the point.
> They don't actually _resonate_! :-)

Ozan replied:
> Why should we seek resonating intervals all the time?

[YA] No reason to use them at all, except that we want to
find some harmonies. When we need discords as an expressive
tool, non-resonant intervals will do fine ...

Danny replied:
> It's intended to be pointless (except for the 53-tone thing),
> or a more extreme case of xenharmonia. Like writing music
> in 11- or 13-equal. Many of the pitches are no less resonant
> than some intervals in 12-tone anyway.

[YA] So it's just a more sophisticated version of having fun
by drumming on a large empty coffee can? Just another tonal
resource, perhaps? I can buy that.

When was the last time any of you went to, or took part in, a
concert using only improvised instruments? It can be a lot of
fun!

In the early days of Australian settlement by Europeans, many
a bush band did exactly this. The most famous instruments
resulting were the lagerphone - a broomstick to which large
numbers of beer bottle tops were nailed, played as a rhythm
instrument by striking the floor - and the tea-chest bass -
using a large teachest (plywood box in which loose tea was
shipped) as a resonating chamber, a broomstick as the neck,
and a stretched cord, usually from hide, as the single string;
the player would alter the note by pulling the top of the neck
away from the teachest to which it was fastened, and pluck
the resulting open stretched string. Simple stuff, but the
backbone of many a memorable night's dancing.

There are times when I really enjoy playing around with
essentially untunable instruments; focusses the attention on
what other musical resources we can find besides pitch. But
perhpas this is not the list on which to be advocating this. :-)

Regards,
Yahya

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Further to having fun with transcendental tuning,
here's some pie in your face!

1. pie = pi . e = 2. pi. (e/2). This is the circumference of
a circle (in Euclidean space) with radius e/2, or diameter e.
Suppose we draw this circle in any 2-D lattice with one of its
unit vectors of length proportional to e, say e/k, and the
other - based on interval x - of length (log x)/k. Then the
circumference of this circle is pie; the circle includes exactly
k steps of e/k - with taxicab distance e - and
n = [k. (e/k) / ((log x)/k)] = [k . e / log x] = [k . log_x e]
steps of the interval x. If e = x ^ y, y = log_x e (using base x),
and n = [k . y]. (Oh dear! Looks like we've got some k . y pie!
Quick - can anyone suggest an "l" that might help us turn it into
a k . y. l . e pie instead?)

Anyway, our "pie" circle includes k e-steps and [k.y] log x-steps.
Choosing base x for all the logs, the two intervals are x, with
size 1/k, and x^e, with size e/k. Because e is not an integer,
this does not equate to e integer steps of interval x; because e
is not a rational p/q, it does not equate to p integer steps of
interval x^(1/q); and so the lattice does not degenerate to a
one-dimensional case. The circle with circumference pi.e
contains [k . log_x e] steps of interval x and k steps of interval
x^e. This works for any interval x which is not a power of e.

2. Can we vary the preceding by using the area of the circle
instead? The area of a circle of radius r is A = pi.r^2. This could
be pi.e, if we take radius r = 1/e^2.

3. Can we generalize the preceding to more than 2 dimensions?
The volume of a sphere of radius r is 4/3.pi.r^3. This could be pi.e,
if we take radius r = (3/(4.e))^3 = 3^3.4^-3.e^-3.

4. When we had a glut of fresh seasonal fruit, Mum
used to bake us some fruit pies to use up the excess. More
exactly, she'd bake fruit tarts with a lattice pastry crust.
I thought the connection poetic ... if remote!

What is the pie lattice? A 2-D lattice with orthogonal unit
vectors of lengths proportional to logs of pi and e.

5. Can we get a TOP pie transcendental tuning? If we take
as comma (e/pi), its HD = Harmonic Distance = log_2 (pi . e).
Tempering this out in any prime-limit lattice gives the TOP-
tempered prime interval:
TOPcents(p) = cents(p) +/- cents(e/pi) . log(p)/log(pi . e)
as our TO tempering of the interval p:1, for each prime
p < limit.

Enough nonsense for today - I really must do some _work_
now.

Regards,
Yahya

-----Original Message-----

Yahya Abdal-Aziz wrote:
> Monz, Danny, Ozan,
> Why use pi and e as bases for intervals? I don't see the point.
> They don't actually _resonate_! :-)

Ozan replied:
> Why should we seek resonating intervals all the time?

[YA] No reason to use them at all, except that we want to
find some harmonies. When we need discords as an expressive
tool, non-resonant intervals will do fine ...

Danny replied:
> It's intended to be pointless (except for the 53-tone thing),
> or a more extreme case of xenharmonia. Like writing music
> in 11- or 13-equal. Many of the pitches are no less resonant
> than some intervals in 12-tone anyway.

[YA] So it's just a more sophisticated version of having fun
by drumming on a large empty coffee can? Just another tonal
resource, perhaps? I can buy that.

When was the last time any of you went to, or took part in, a
concert using only improvised instruments? It can be a lot of
fun!

In the early days of Australian settlement by Europeans, many
a bush band did exactly this. The most famous instruments
resulting were the lagerphone - a broomstick to which large
numbers of beer bottle tops were nailed, played as a rhythm
instrument by striking the floor - and the tea-chest bass -
using a large teachest (plywood box in which loose tea was
shipped) as a resonating chamber, a broomstick as the neck,
and a stretched cord, usually from hide, as the single string;
the player would alter the note by pulling the top of the neck
away from the teachest to which it was fastened, and pluck
the resulting open stretched string. Simple stuff, but the
backbone of many a memorable night's dancing.

There are times when I really enjoy playing around with
essentially untunable instruments; focusses the attention on
what other musical resources we can find besides pitch. But
perhaps this is not the list on which to be advocating this. :-)

Regards,
Yahya

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--- In tuning-math@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
> Further to having fun with transcendental tuning,
> here's some pie in your face!

Sadly, there is not even a proof available to show pie *is*
transcendental, though of course everyone believes it is true, and
that pi and e are algebraically independent. It is trivial to prove
that between pi+e and pi*e, at least one must be transcendental.

>> Further to having fun with transcendental tuning,
>> here's some pie in your face!
>
>Sadly, there is not even a proof available to show pie *is*
>transcendental,

I thought it had, so I checked mathworld, and it cited
Lindemann 1882.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> Further to having fun with transcendental tuning,
> >> here's some pie in your face!
> >
> >Sadly, there is not even a proof available to show pie *is*
> >transcendental,
>
> I thought it had, so I checked mathworld, and it cited
> Lindemann 1882.

Hermite proved e is transcendental, and Lindemann proved that pi is
transcendental. If you check the page history for the article
"Lindemann-Weierstrass theorem" on Wikipedia you may find a familiar
name, by the way. That both e and pi are transcendental does not prove
pie = pi*e is transcendental, however. You need to show pi and e are
algbebraically independent, which is one of those things which are
obviously true but hard to prove. However, since pi and e are both
transcendental, the coefficients of (x-pi)(x-e) = x^2 - (pi+e)x + p*e
cannot both be algebraic, or pi and e would both be algebraic. Hence
at least one of p+e and pi*e must be transcendental. From the
Gelfond-Schneider theorem, (-1)^(-i) = exp(pi*i*(-i)) = e^pi is
transcentdental, but pi^e is not so easily dealt with.

What this has to do with music is beyond me.

>> >> Further to having fun with transcendental tuning,
>> >> here's some pie in your face!
>> >
>> >Sadly, there is not even a proof available to show pie *is*
>> >transcendental,
>>
>> I thought it had, so I checked mathworld, and it cited
>> Lindemann 1882.
>
>Hermite proved e is transcendental, and Lindemann proved that pi is
>transcendental. If you check the page history for the article
>"Lindemann-Weierstrass theorem" on Wikipedia you may find a familiar
>name, by the way. That both e and pi are transcendental does not prove
>pie = pi*e is transcendental, however. You need to show pi and e are
>algbebraically independent, which is one of those things which are
>obviously true but hard to prove. However, since pi and e are both
>transcendental, the coefficients of (x-pi)(x-e) = x^2 - (pi+e)x + p*e
>cannot both be algebraic, or pi and e would both be algebraic. Hence
>at least one of p+e and pi*e must be transcendental. From the
>Gelfond-Schneider theorem, (-1)^(-i) = exp(pi*i*(-i)) = e^pi is
>transcentdental, but pi^e is not so easily dealt with.
>
>What this has to do with music is beyond me.

Sorry; thought "pie" was a typo for "pi", but thanks for the
clarification.

-Carl