Equal divisions of phi (1.618/1)

A long time ago, I proposed the idea of using non-rational logarithmic bases
to measure interval sizes and to create equal temperaments and other scales
without a bias towards any rational interval. I originally proposed e/1
(2.718/1) as a period, and found that, for example, 52 equal divisions of
e/1 is a good approximate of 36-edo.

But a probably better base would be phi, defined as (1+5^0.5)/2, the
infinite limit of the ratio between numbers in a Fibonacci sequence. Phi
squared is equal to phi plus one, while the reciprocal of phi is phi minus
one. In architecture and geometry, it's an important and even sacred number.
So maybe it could have a significance for music. The interval phi/1 is a
sharp minor sixth.

The size of phi/1 is 833.09 cents. Fifty equal divisions of phi produces
72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I call
50/6-GET (50/6-tone Golden Equal Temperament).

Now is there a name for the scale unit of phi/1, the logarithm base phi,
like the decibel scale is based on base-10 logarithms?

> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi produces
> 72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I call
> 50/6-GET (50/6-tone Golden Equal Temperament).

Or 25/3 better yet. GET can also be called 'edgr' (equal divisions of the
golden ratio).

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> A long time ago, I proposed the idea of using non-rational
logarithmic bases
> to measure interval sizes and to create equal temperaments and other
scales
> without a bias towards any rational interval.

What's the point?

> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi produces
> 72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I
call
> 50/6-GET (50/6-tone Golden Equal Temperament).

OK, so you have a 72-et with a slightly flat octave. More logical, to
me, would be one with a slightly sharp octave, but clearly this system
is viable. But what's the point? Why would we want to do this?

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> But a probably better base would be phi, defined as (1+5^0.5)/2, the
> infinite limit of the ratio between numbers in a Fibonacci
sequence. Phi
> squared is equal to phi plus one, while the reciprocal of phi is
phi minus
> one. In architecture and geometry, it's an important and even
sacred number.
> So maybe it could have a significance for music. The interval phi/1
is a
> sharp minor sixth.
>
> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi
produces
> 72-edo with a slightly flat octave (1199.65 cents);

John Chowning's _Stria_ uses eight equal divisions of phi, IIRC.

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

> > 72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I
> call
> > 50/6-GET (50/6-tone Golden Equal Temperament).
>
> OK, so you have a 72-et with a slightly flat octave. More logical, to
> me, would be one with a slightly sharp octave, but clearly this system
> is viable. But what's the point? Why would we want to do this?

Why would anybody want to write music in 11-tet or 13-tet? Weirdness and
experimentalism. But I'm still trying to find a useful purpose for this. Or
at least an alternate measurement system to cents.

There are a couple scales in the Scala archive with an 'octave' of phi, or 1
+ phi.

But what I'm going to end up using in my own music will most likely be
stretched-octave meantones, like the one with the fifth lowered 1/6-comma
and the octave raised 1/6-comma, which approximates 128 equal divisions of
3/1 (close to 81-tet).

I guess you have to go through the homepage at

http://members.aol.com/bpsite/

Rick Tagawa wrote:

> Heinz Bohlen's article "A 833 Cents Scale" is at:
> > http://www.members.aol.com/hpbolen/833cent.html
> > Danny Wier wrote:
> >>> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi produces
>>> 72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I call
>>> 50/6-GET (50/6-tone Golden Equal Temperament).
>>>
>>
>> Or 25/3 better yet. GET can also be called 'edgr' (equal divisions of the
>> golden ratio).
>>
>>
>> snip
> > > >

In a message dated 6/15/2004 7:20:48 PM Eastern Daylight Time,
dawiertx@sbcglobal.net writes:

> Why would anybody want to write music in 11-tet or 13-tet? Weirdness and
> experimentalism.

Gene, more often than not, it is the way the materials are put together,
rather than the materials themselves, that makes the success of a composition. We
already know that using the best materials is not guarantee of compositional
success.

best, Johnny (away at summer camp)

p.s. check out Skip La Plante's Theme and Variations in 13-Equal coming out
soon as part of the new PITCH CDs. It is written more for reasons of fun.

--- In tuning@yahoogroups.com, Afmmjr@a... wrote:
> In a message dated 6/15/2004 7:20:48 PM Eastern Daylight Time,
> dawiertx@s... writes:

> > Why would anybody want to write music in 11-tet or 13-tet?
Weirdness and
> > experimentalism.
>
> Gene, more often than not, it is the way the materials are put
together,
> rather than the materials themselves, that makes the success of a
composition. We
> already know that using the best materials is not guarantee of
compositional
> success.

11-equal has some of the same harmonic resources as 22-equal, of
course. In particular, four steps is an excellent 9/7, and one step a
fine 16/15. Three steps gives the same minor third as 22-equal, two
steps the same 8/7, and five steps the same 11/8. Put this all
together and it looks like

1--16/15--8/7--6/5--9/7--11/8--16/11--14/9--5/3--7/4--15/8

Various chords are possible, such as 1--11/8--7/4 and the
1--9/7--5/3 augmented triad.

13-equal seems more limited, but six steps is an excellent 11/8 and
3 steps a decent 7/6. The 0-3-6 chord is therefore a consideration,
and the 0-3-6-9 chord has a 9 which serves as 13/8, so this is an
approximate 1--7/6--11/8--13/8 bit of magic which should have
something of a diminished seventh sound if someone wants to try it.

On the Bohlen-Pierce site Heinz Bolen in the article from September 18, 1999 "A 833 Cents Scale" actually discusses 36ET. See link "More Unusual Scales."

I was just listening to it and it's really beautiful. Since the 72ET approximation of the 13th harmonic is also 833� then this interval can serve a dual purpose.

--rt

Afmmjr@aol.com wrote:

> In a message dated 6/15/2004 7:20:48 PM Eastern Daylight Time, > dawiertx@sbcglobal.net writes:
> > > Why would anybody want to write music in 11-tet or 13-tet? Weirdness and
> experimentalism.
> > > > Gene, more often than not, it is the way the materials are put together, > rather than the materials themselves, that makes the success of a > composition. We already know that using the best materials is not > guarantee of compositional success.
> > best, Johnny (away at summer camp)
> > p.s. check out Skip La Plante's Theme and Variations in 13-Equal coming > out soon as part of the new PITCH CDs. It is written more for reasons > of fun.
> > You can configure your subscription by sending an empty email to one
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Hi there.

> > Danny Wier wrote:
> >
> >>> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi
produces
> >>> 72-edo with a slightly flat octave (1199.65 cents).

Great! So what about equal multiplications? I was quite surprised when I
first discovered that multiplying Phi by itself (i.e. Phi, Phi^2, Phi^3,
ETC.) makes a scale that is an equal tuning and an addition chord at the
same time (I hate using the term "equal temperament" in these cases). I was
also experimenting with this scale and I found lots of interesting things
about it. Yet I'm just wondering how many of these multiplications I should
make to get an interval not too far from an octave (I mean no more than a
kleismatic or schismatic difference).
Petr

> > Danny Wier wrote:
> >
> >>> The size of phi/1 is 833.09 cents. Fifty equal divisions of phi
produces
> >>> 72-edo with a slightly flat octave (1199.65 cents); 12-edo is what I
call
> >>> 50/6-GET (50/6-tone Golden Equal Temperament).

Yooooh!
I'd be quite glad to know where you got the idea of dividing Phi right into
50 equal steps, not less, not more. Now it seems to me that this idea of
dividing Phi must have actually come from the opposite - that is
multiplication. About 3 hours ago, I was trying to find an integer power of
Phi that is suitably close to any integer power of 2. After multiplying Phi
by itself for the 36th time (at that time I was thinking of giving it all
up), I got just under 2^25. When I converted it, I found a difference of
only about 8.7 cents. Hmmm, great. So if Phi^36 is close to 2^25, then 25/36
of an octave is close to the factor of Phi itself. In fact, this is just the
same as what you're speaking about! I'm pretty surprised, where did you get
it?
Petr