properties of the circle of 5ths

Hi

I've been playing around with the circle of 5ths and have found
some interesting things, but have probably only scratched the
surface. Here is what I've got so far:

http://home.austin.rr.com/jmjensen/CircleOf5thsFun.html

Are there any important things that I am missing?

--JEff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> Hi
>
> I've been playing around with the circle of 5ths and have found
> some interesting things, but have probably only scratched the
> surface. Here is what I've got so far:
>
> http://home.austin.rr.com/jmjensen/CircleOf5thsFun.html
>
> Are there any important things that I am missing?

One thing is that if you open it out to become the _chain_ of fifths
then everything you have done still works, but like a slide-rule
instead of a clock, and it gives the correct "spelling" and works in
many other tunings besides 12-ET.

... Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx ...

Other ETs close this into a circle in different ways, e.g. 5-ET, 7-ET,
17-ET and 19-ET.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > Hi
> >
> > I've been playing around with the circle of 5ths and have found
> > some interesting things, but have probably only scratched the
> > surface. Here is what I've got so far:
> >
> > http://home.austin.rr.com/jmjensen/CircleOf5thsFun.html
> >
> > Are there any important things that I am missing?
>
> One thing is that if you open it out to become the _chain_ of fifths
> then everything you have done still works, but like a slide-rule
> instead of a clock, and it gives the correct "spelling" and works in
> many other tunings besides 12-ET.
>
> ... Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
Fx ...
>
> Other ETs close this into a circle in different ways, e.g. 5-ET, 7-
ET,
> 17-ET and 19-ET.

My humble comments on this:

If we write the lists of stepsizes of an n-eq tuning, and discard the
ones that have dividers in common with n (apart from the unity), we
get the intervals that can generate the whole tuning.

For instance, in 12-eq, the only "generators" are stepsizes 1,5,7 and
11. [2,4,6,8,10 share with 12 the divider 2; 3,6,9 the divider 3,
etc. The minor third (3 steps) cannot make a complete circle, eg. C
Eb Gb A C, missing D,E, etc. This, by the way, also forms one of the
simmetric figures Jeff menctiones.]

That is so special of the fifth in 12-eq. Besides the chromatic
stepsizes 1 and 11, it's the only interval that can generate the
whole 12 notes in a circle.

I guess most of the graphical properties analized with the circle of
fifths in 12-eq can be translated to the chromatic circle.

For instance, relative major/minor keys are still 90º apart.

5,7,17 and 19-eq (31-eq) have been previously menctioned. In 19 and
31-eq, for instance, one finds that (19 and 31 being prime numbers)
any interval can generate the whole tuning, and any stepsize forms a
complete circle of 19 or 31 notes. [Same with 5,7, etc.]

Interestingly enough, 15-eq (which has been a hot topic over the past
days) has a "fifth" of 720 cents (9th degree), that cannot generate a
complete circle. But intervals 1,2,4,7,8,11,13 and 14 steps long can.

Jeff menctioned simmetrically shaped chords.

We can draw an equilateral triangle on vertices C,E and G#. That
makes an augmented chord. a square can be drawn with vertices C, Eb,
Gb, A. That would make a diminished 7th chord.

A sequence of augmented chords, in 12-eq, closes fast. For instance,
C+ E+ G+ and C again. And it's difficult to determine the key of such
a key, as Jeff suggests. C+ is just an inversion of E+ and viceversa.

As we can't build those simmetrical figures into an 11-eq 19-eq or 31-
eq circle, it turns out that a sequence of augmented chords in n-eq
(n=17,19,31, etc.) only closes after n steps.

One has for example, in 31-eq, C+ as C E G#. And E+ is E G# B#, which
is not the same. So, here the key can be characterized.

Max.

>If we write the lists of stepsizes of an n-eq tuning, and discard the
>ones that have dividers in common with n (apart from the unity), we
>get the intervals that can generate the whole tuning.
>
>For instance, in 12-eq, the only "generators" are stepsizes 1,5,7 and
>11. [2,4,6,8,10 share with 12 the divider 2; 3,6,9 the divider 3,
>etc. The minor third (3 steps) cannot make a complete circle, eg. C
>Eb Gb A C, missing D,E, etc. This, by the way, also forms one of the
>simmetric figures Jeff menctiones.]
>
>That is so special of the fifth in 12-eq. Besides the chromatic
>stepsizes 1 and 11, it's the only interval that can generate the
>whole 12 notes in a circle.
>
>I guess most of the graphical properties analized with the circle
>of fifths in 12-eq can be translated to the chromatic circle.

All true. However, I would like to point out:

() Notation systems should not be restricted to equal temperaments.

() The nominals of a notation system, taken together, should
*make sense* as a scale, chord, or some other desirable (for any
reason) structure. Merely 'generating' the tuning is not enough.

() Finally, the notation need not be made of a single chain of
generators, nor even be based on chains at all.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >If we write the lists of stepsizes of an n-eq tuning, and discard
the
> >ones that have dividers in common with n (apart from the unity),
we
> >get the intervals that can generate the whole tuning.
> >
> >For instance, in 12-eq, the only "generators" are stepsizes 1,5,7
and
> >11. [2,4,6,8,10 share with 12 the divider 2; 3,6,9 the divider 3,
> >etc. The minor third (3 steps) cannot make a complete circle, eg.
C
> >Eb Gb A C, missing D,E, etc. This, by the way, also forms one of
the
> >simmetric figures Jeff menctiones.]
> >
> >That is so special of the fifth in 12-eq. Besides the chromatic
> >stepsizes 1 and 11, it's the only interval that can generate the
> >whole 12 notes in a circle.
> >
> >I guess most of the graphical properties analized with the circle
> >of fifths in 12-eq can be translated to the chromatic circle.
>
> All true. However, I would like to point out:
>
> () Notation systems should not be restricted to equal temperaments.
>
> () The nominals of a notation system, taken together, should
> *make sense* as a scale, chord, or some other desirable (for any
> reason) structure. Merely 'generating' the tuning is not enough.
>
> () Finally, the notation need not be made of a single chain of
> generators, nor even be based on chains at all.
>
> -Carl

Yep, sure. I was pointing out some stuff basically characteristic of
eq's; but the broader our sight is, the better.

Max.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
> > Hi
> >
> > I've been playing around with the circle of 5ths and have found
> > some interesting things, but have probably only scratched the
> > surface. Here is what I've got so far:
> >
> > http://home.austin.rr.com/jmjensen/CircleOf5thsFun.html
> >
> > Are there any important things that I am missing?
>
> One thing is that if you open it out to become the _chain_ of fifths
> then everything you have done still works, but like a slide-rule
> instead of a clock, and it gives the correct "spelling" and works in
> many other tunings besides 12-ET.
>
> ... Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
Fx ...
>
> Other ETs close this into a circle in different ways, e.g. 5-ET, 7-
ET,
> 17-ET and 19-ET.

Are you saying to write it as a spiral? Like the Pythagorean spiral
where each note is the (rescaled) 3rd harmonic of its predecessor?

I'm not clear on why everything would still work then, since I
would no longer have a symmetric figure, or what
you mean by a "slide rule".

--Jeff

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

>
> My humble comments on this:
>
> If we write the lists of stepsizes of an n-eq tuning, and discard
the
> ones that have dividers in common with n (apart from the unity), we
> get the intervals that can generate the whole tuning.
>
> For instance, in 12-eq, the only "generators" are stepsizes 1,5,7
and
> 11. [2,4,6,8,10 share with 12 the divider 2; 3,6,9 the divider 3,
> etc. The minor third (3 steps) cannot make a complete circle, eg. C
> Eb Gb A C, missing D,E, etc. This, by the way, also forms one of
the
> simmetric figures Jeff menctiones.]

It is not clear to me what it means musically to talk about going
around by step sizes of 2,3,4 etc and whether or not that reaches
all the points on the circle... On the other hand, I have recently
realized (with some help) that going 4 steps clockwise is a major
3rd, going 3 steps counter-clockwise is a minor 3rd etc. I am
guessing that going clockwise always generates major intervals and
the other way always minor intervals... Then we can reflect these
intervals through any line through the center...

>
> That is so special of the fifth in 12-eq. Besides the chromatic
> stepsizes 1 and 11, it's the only interval that can generate the
> whole 12 notes in a circle.
>
> I guess most of the graphical properties analized with the circle
of
> fifths in 12-eq can be translated to the chromatic circle.
>
> For instance, relative major/minor keys are still 90º apart.

Yes, that is interesting...

>
> 5,7,17 and 19-eq (31-eq) have been previously menctioned. In 19 and
> 31-eq, for instance, one finds that (19 and 31 being prime numbers)
> any interval can generate the whole tuning, and any stepsize forms
a
> complete circle of 19 or 31 notes. [Same with 5,7, etc.]
>
> Interestingly enough, 15-eq (which has been a hot topic over the
past
> days) has a "fifth" of 720 cents (9th degree), that cannot generate
a
> complete circle. But intervals 1,2,4,7,8,11,13 and 14 steps long
can.
>
> Jeff menctioned simmetrically shaped chords.
>
> We can draw an equilateral triangle on vertices C,E and G#. That
> makes an augmented chord. a square can be drawn with vertices C,
Eb,
> Gb, A. That would make a diminished 7th chord.
>

This is something that I am presently working on, drawing all
the important chords. My goal is to have an algorithm to be able
to identify strange chords or chord fragments that occur in
the piano music I play (not atonal stuff, just classics like
Beethoven, etc)

> A sequence of augmented chords, in 12-eq, closes fast. For
instance,
> C+ E+ G+ and C again. And it's difficult to determine the key of
such
> a key, as Jeff suggests. C+ is just an inversion of E+ and
viceversa.
>
> As we can't build those simmetrical figures into an 11-eq 19-eq or
31-
> eq circle, it turns out that a sequence of augmented chords in n-eq
> (n=17,19,31, etc.) only closes after n steps.
>
> One has for example, in 31-eq, C+ as C E G#. And E+ is E G# B#,
which
> is not the same. So, here the key can be characterized.
>

That is something that I guess I am not ready for yet, but I am
trying to formulate a theory of chords independent of a particular
tuning, which I hope to post soon.

I still owe you a reply to your input on my questions on Paul
Erlich's 22 tone paper, and maybe one of these days I'll get it
done!

--Jeff

> Max.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> All true. However, I would like to point out:
>
> () Notation systems should not be restricted to equal temperaments.
>

By "notation system" I guess you mean the circle diagram? Trying to
draw a diagram for meantone(s) is causing me a lot of aggravation,
although I think I have something, which I hope to post soon.

> () The nominals of a notation system, taken together, should
> *make sense* as a scale, chord, or some other desirable (for any
> reason) structure. Merely 'generating' the tuning is not enough.
>

This I heartily agree with, if I means what I think.

> () Finally, the notation need not be made of a single chain of
> generators, nor even be based on chains at all.
>
> -Carl

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> > Other ETs close this into a circle in different ways, e.g. 5-ET, 7-
> ET,
> > 17-ET and 19-ET.
>
> Are you saying to write it as a spiral? Like the Pythagorean spiral
> where each note is the (rescaled) 3rd harmonic of its predecessor?

Well you could. But no, I was just saying that sometimes it's very
useful to leave it straight and not have to say whether it is 12-ET or
pythagorean or meantone.

> I'm not clear on why everything would still work then, since I
> would no longer have a symmetric figure,

Sorry. I shouldn't have said _everything_ will still work. Obviously
we do lose those symmetries that rely on the divisibility of 12.
Circles like yours are very good for understanding specific equal
temperaments.

> or what
> you mean by a "slide rule".

Harmonic structures have a fixed pattern on this chain and so by
constructing one example you can then slide it along to read off
others (correctly spelled), just by adding or deleting spaces to the
left of the |---| patterns. e.g.

... Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# ...
!--|--------| major
|--------!--| minor
!-----------|-----------| aug
|--------|--------! dim

I've used "!" instead of "|" to indicate the nominal root of each chord.

>> All true. However, I would like to point out:
>>
>> () Notation systems should not be restricted to equal temperaments.
>
>By "notation system" I guess you mean the circle diagram? Trying to
>draw a diagram for meantone(s) is causing me a lot of aggravation,
>although I think I have something, which I hope to post soon.

By all means, post. By notation system, I guess I mean just anything
let lets you notate music to your satisfaction.

>> () The nominals of a notation system, taken together, should
>> *make sense* as a scale, chord, or some other desirable (for any
>> reason) structure. Merely 'generating' the tuning is not enough.
>
>This I heartily agree with, if I means what I think.

By nominals we mean unaltered (without accidental) notes, C D E...
and so forth. Is that what you thought?

-Carl