Octatonic, diminished, symmetry

"Octatonic" refers to any scale with 8 tones, which can be the diminished scale of 12-tet, 8-tet or any other MOS or non-MOS with 8 tones to the interval of repetition.

"Diminished" is a term open to controversy since it can refer both to intervals, chords or scale systems. As for scales, i interpret a "diminished scale" as one based upon the diminished chord created by stacking three minor thirds. This can be a _quadrasymmetrical_ scale repeating at 1/4 of the octave, thus constituting the diminshed chord regardless of what other pitch(es) the generator lead to.

Or it can be an octave based scale using an interval _close to_ 1/4 of an octave as a generator. If it higher, as in 19TET, MOS are of 7 and 10 notes, while a <300 cent generator leads to a 9-note MOS, as in 17TET and 31TET. In these cases, the "diminished chord" is no longer 4-equal, yet unmistakably a diminished chord, which is why i use "diminished scale" to refer all three of the above scale types.

"Symmetrical" in scale terms does usually not refer to inversional symmetry, which is a feature of all generator chains, but to a scale that repeats at a fraction of the octave. A scale that repeats at half the octave (such as Paul Erlich's 22TET decatonics) are simply called symmetrical, while "trisymmetrical" and "quadrasymmetrical"* are suitable terms for those who repeat at thirds and fourths. The last of which includes the 12-tet diminished scale, with the particular properties that have been discussed here recently.

*)Highly usable trisymmetrics are found in 12 and 24,and there are also a "pentasymmetric" in 15TET, etc. The "Double equal temperament" tunings of Vicentino (two full 19TET or 31TET sets a just 3/2 apart) can be seen as an extreme form of sub-octavic symmetry.

-Mats �ljare
(always healthy, always here)

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--- In tuning@y..., "Mats Öljare" <oljare@h...> wrote:
>
> "Octatonic" refers to any scale with 8 tones, which can be the
diminished
> scale of 12-tet, 8-tet or any other MOS or non-MOS with 8 tones to
the
> interval of repetition.
>
> "Diminished" is a term open to controversy since it can refer both
to
> intervals, chords or scale systems. As for scales, i interpret
a "diminished
> scale" as one based upon the diminished chord created by stacking
three
> minor thirds. This can be a _quadrasymmetrical_ scale repeating at
1/4 of
> the octave, thus constituting the diminshed chord regardless of
what other
> pitch(es) the generator lead to.
>
> Or it can be an octave based scale using an interval _close to_ 1/4
of an
> octave as a generator. If it higher, as in 19TET, MOS are of 7 and
10 notes,
> while a <300 cent generator leads to a 9-note MOS, as in 17TET and
31TET. In
> these cases, the "diminished chord" is no longer 4-equal, yet
unmistakably a
> diminished chord, which is why i use "diminished scale" to refer
all three
> of the above scale types.
>
> "Symmetrical" in scale terms does usually not refer to inversional
symmetry,
> which is a feature of all generator chains, but to a scale that
repeats at a
> fraction of the octave. A scale that repeats at half the octave
(such as
> Paul Erlich's 22TET decatonics) are simply called symmetrical,
while
> "trisymmetrical" and "quadrasymmetrical"* are suitable terms for
those who
> repeat at thirds and fourths.

well, the diminished and augmented scales are those most commonly
referred to as simply "symmetrical scales".

in general i agree with you completely, mats, and it is in this
spirit that i suggested the more colorful terms "trefoil"
and "quatrefoil". these terms didn't go over very well -- perhaps
your more technical sounding "trisymmetrical" and "quadrasymmetrical"
will fare better -- one could assume that the fifth is the generator
as a default.

Paul,

Hmm, why would one "assume that the fifth is the generator as a
default"?

The way I look at, in 12-tet the generator for the diminished scale
would be the whole step and the fractional period would be the minor
third, and for the augmented scale, the generator would be the minor
third and the fractional period would be the major third.

If you take these as Fibonacci series generators, the diminished would
be a narrow major second at ~185�, and the augmented would be a wide
super major second at ~247�.

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Friday, March 08, 2002 1:29 PM
Subject: [tuning] Re: Octatonic, diminished, symmetry

--- In tuning@y..., "Mats �ljare" <oljare@h...> wrote:
>
> "Octatonic" refers to any scale with 8 tones, which can be the
diminished
> scale of 12-tet, 8-tet or any other MOS or non-MOS with 8 tones to
the
> interval of repetition.
>
> "Diminished" is a term open to controversy since it can refer both
to
> intervals, chords or scale systems. As for scales, i interpret
a "diminished
> scale" as one based upon the diminished chord created by stacking
three
> minor thirds. This can be a _quadrasymmetrical_ scale repeating at
1/4 of
> the octave, thus constituting the diminshed chord regardless of
what other
> pitch(es) the generator lead to.
>
> Or it can be an octave based scale using an interval _close to_ 1/4
of an
> octave as a generator. If it higher, as in 19TET, MOS are of 7 and
10 notes,
> while a <300 cent generator leads to a 9-note MOS, as in 17TET and
31TET. In
> these cases, the "diminished chord" is no longer 4-equal, yet
unmistakably a
> diminished chord, which is why i use "diminished scale" to refer
all three
> of the above scale types.
>
> "Symmetrical" in scale terms does usually not refer to inversional
symmetry,
> which is a feature of all generator chains, but to a scale that
repeats at a
> fraction of the octave. A scale that repeats at half the octave
(such as
> Paul Erlich's 22TET decatonics) are simply called symmetrical,
while
> "trisymmetrical" and "quadrasymmetrical"* are suitable terms for
those who
> repeat at thirds and fourths.

well, the diminished and augmented scales are those most commonly
referred to as simply "symmetrical scales".

in general i agree with you completely, mats, and it is in this
spirit that i suggested the more colorful terms "trefoil"
and "quatrefoil". these terms didn't go over very well -- perhaps
your more technical sounding "trisymmetrical" and "quadrasymmetrical"
will fare better -- one could assume that the fifth is the generator
as a default.

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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> Hmm, why would one "assume that the fifth is the generator as
a
> default"?
>
> The way I look at, in 12-tet the generator for the diminished
scale
> would be the whole step

fifth generator is exactly equivalent to that -- simply subtract three
periods.

and the fractional period would be the minor
> third, and for the augmented scale, the generator would be the
minor
> third

ditto -- this time add one period.

Paul,

Hmm, I still don't know what you mean! There's 4 fractional periods in
the diminished and 3 in the augmented, so what's 3 got to do with the
diminished and what's 1 got to do with the augmented that's relevant
besides giving a fifth?

The only way I can see these Messiaen type class of scales being de
facto fifth generated scales is if you allow that they're all
fractions of the Phi fifth, which they would be if the period is
always a 2/1.

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 10, 2002 12:40 PM
Subject: [tuning] Re: Octatonic, diminished, symmetry

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > Hmm, why would one "assume that the fifth is the generator as
> a
> > default"?
> >
> > The way I look at, in 12-tet the generator for the diminished
> scale
> > would be the whole step
>
> fifth generator is exactly equivalent to that -- simply subtract
three
> periods.
>
> and the fractional period would be the minor
> > third, and for the augmented scale, the generator would be the
> minor
> > third
>
> ditto -- this time add one period.
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> Hmm, I still don't know what you mean! There's 4 fractional
periods in
> the diminished and 3 in the augmented, so what's 3 got to do
with the
> diminished and what's 1 got to do with the augmented that's
relevant
> besides giving a fifth?

that's it.

>
> The only way I can see these Messiaen type class of scales
being de
> facto fifth generated scales is if you allow that they're all
> fractions of the Phi fifth,

i don't see how the phi fifth comes in. this should be no more
perplexing than the statement that 3/1 generates the diatonic
scale, given the octave period.

Paul,

But saying the 3/1 generates the diatonic scale, given the octave
period immediately makes it clear in 12-tet that it's a mod 12
structure where the same would be true only of it's compliment. With
the augmented scale the analogy would be a mod 4 structure, and the
fifth could be any odd numbered 12-tet generator... and with the
diminished scale the analogy would be a mod 3 structure, and you could
just as easily say the generator is anything but the period or one of
its multiples!

What I was getting at with the Phi fifth is that symmetrical scales of
this sort always have the same amount of each of the two stepsizes, so
the stepsizes divided by the GCD is always 1, and a Fibonacci
generator would therefore always be some fraction of the Phi fifth.

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 10, 2002 5:42 PM
Subject: [tuning] Re: Octatonic, diminished, symmetry

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > Hmm, I still don't know what you mean! There's 4 fractional
> periods in
> > the diminished and 3 in the augmented, so what's 3 got to do
> with the
> > diminished and what's 1 got to do with the augmented that's
> relevant
> > besides giving a fifth?
>
> that's it.
>
> >
> > The only way I can see these Messiaen type class of scales
> being de
> > facto fifth generated scales is if you allow that they're all
> > fractions of the Phi fifth,
>
> i don't see how the phi fifth comes in. this should be no more
> perplexing than the statement that 3/1 generates the diatonic
> scale, given the octave period.
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> But saying the 3/1 generates the diatonic scale, given the
octave
> period immediately makes it clear in 12-tet that it's a mod 12
> structure where the same would be true only of it's
compliment. With
> the augmented scale the analogy would be a mod 4 structure,
and the
> fifth could be any odd numbered 12-tet generator... and with the
> diminished scale the analogy would be a mod 3 structure, and
you could
> just as easily say the generator is anything but the period or
one of
> its multiples!

well, this is all very true, but of course you need not be in
12-equal or any equal temperament, and in such cases the
'default' generator of fifth could actually be a useful convention.

>
> What I was getting at with the Phi fifth is that symmetrical
scales of
> this sort always have the same amount of each of the two
stepsizes,

right . . .

> so
> the stepsizes divided by the GCD is always 1,

the gcd of what and what?

> and a Fibonacci
> generator would therefore always be some fraction of the Phi
>fifth.

you lost me.

Paul,

By taking the GCD of any two stepsize scale you have the amount that
you'd fractionalize the period by (it's usually the octave, but it
theoretically could be anything). This is also what you'd scale the
stepsize index by--and by stepsize index I simply mean the amount of
each stepsize. So if it's the augmented scale you have an index of 3
small steps and 3 large steps, and a 2/1.

Using the standard augmented scale as an example, the GCD of 3 and 3
is 3, so P/3 gives the fractional periodicity of 1/3 of an octave, and
scaling the index by the GCD gives an index of 1 small step and 1
large step, and that's why you'd have an omnipresent Phi fifth if you
choose to look at these scales in this way.

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Sunday, March 10, 2002 8:21 PM
Subject: [tuning] Re: Octatonic, diminished, symmetry

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > But saying the 3/1 generates the diatonic scale, given the
> octave
> > period immediately makes it clear in 12-tet that it's a mod 12
> > structure where the same would be true only of it's
> compliment. With
> > the augmented scale the analogy would be a mod 4 structure,
> and the
> > fifth could be any odd numbered 12-tet generator... and with the
> > diminished scale the analogy would be a mod 3 structure, and
> you could
> > just as easily say the generator is anything but the period or
> one of
> > its multiples!
>
> well, this is all very true, but of course you need not be in
> 12-equal or any equal temperament, and in such cases the
> 'default' generator of fifth could actually be a useful convention.
>
> >
> > What I was getting at with the Phi fifth is that symmetrical
> scales of
> > this sort always have the same amount of each of the two
> stepsizes,
>
> right . . .
>
> > so
> > the stepsizes divided by the GCD is always 1,
>
> the gcd of what and what?
>
> > and a Fibonacci
> > generator would therefore always be some fraction of the Phi
> >fifth.
>
> you lost me.
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> By taking the GCD of any two stepsize scale you have the
amount that
> you'd fractionalize the period by (it's usually the octave, but it
> theoretically could be anything).

so you're taking the gcd of what and what?

This is also what you'd scale the
> stepsize index by--and by stepsize index I simply mean the
amount of
> each stepsize. So if it's the augmented scale you have an index
of 3
> small steps and 3 large steps, and a 2/1.
>
> Using the standard augmented scale as an example, the GCD
of 3 and 3
> is 3,

oh, so you're talking the gcd of the two numbers in the 'index'?

so P/3 gives the fractional periodicity of 1/3 of an octave, and
> scaling the index by the GCD gives an index of 1 small step
and 1
> large step, and that's why you'd have an omnipresent Phi fifth if
you
> choose to look at these scales in this way.

where did phi come in? i'm lost.

Paul,

What I mean is the GCD of the two stepsizes. If the numbers are
relatively prime, the there's no fractional periodicity. If they're
not, the P (the octave or what have you) is fractionalized into GCD
(a,b) parts. If (a,b) were the Messiaen decatonic, then the GCD would
be 2. So, (8,2) becomes (4,1), and 2/1 becomes 600�.

The Phi fifth only applies to scales like the diminished and the
augmented that have the same number of each of the two stepsizes.
That's because they'd all reduce to (1,1), and this Fibonacci series
when converted to adjacent fractions always gives the Phi fifth or
some fraction of it.

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, March 11, 2002 9:49 AM
Subject: [tuning] Re: Octatonic, diminished, symmetry

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > By taking the GCD of any two stepsize scale you have the
> amount that
> > you'd fractionalize the period by (it's usually the octave, but it
> > theoretically could be anything).
>
> so you're taking the gcd of what and what?
>
> This is also what you'd scale the
> > stepsize index by--and by stepsize index I simply mean the
> amount of
> > each stepsize. So if it's the augmented scale you have an index
> of 3
> > small steps and 3 large steps, and a 2/1.
> >
> > Using the standard augmented scale as an example, the GCD
> of 3 and 3
> > is 3,
>
> oh, so you're talking the gcd of the two numbers in the 'index'?
>
> so P/3 gives the fractional periodicity of 1/3 of an octave, and
> > scaling the index by the GCD gives an index of 1 small step
> and 1
> > large step, and that's why you'd have an omnipresent Phi fifth if
> you
> > choose to look at these scales in this way.
>
> where did phi come in? i'm lost.
>
>
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--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> What I mean is the GCD of the two stepsizes.

in cents? in semitones?

> If the numbers are
> relatively prime, the there's no fractional periodicity. If they're
> not, the P (the octave or what have you) is fractionalized into GCD
> (a,b) parts. If (a,b) were the Messiaen decatonic, then the GCD
would
> be 2. So, (8,2)

these are the stepsizes? in what units? aren't the stepsizes just a
semitone and a whole tone? the above sure looks more like what you
call the "index", than the "stepsizes".

> becomes (4,1), and 2/1 becomes 600¢.

yup.

> The Phi fifth only applies to scales like the diminished and the
> augmented that have the same number of each of the two stepsizes.
> That's because they'd all reduce to (1,1), and this Fibonacci series

which fibonacci series? applied how?

> when converted to adjacent fractions always gives the Phi fifth or
> some fraction of it.

please give an example. it'll help me (and perhaps others) to
understand what you're talking about.

happy day,
paul

Paul,

Okay, I'll use this symmetrical decatonic that I've used in 20-tet as
an example:

bababababa

It has 5 small steps (a) and 5 large steps (b) where the period (P) is
1200 cents. I call that a stepsize index of (5,5). The GCD of (5,5) is
5. Scaling P and the index by the GCD gives (1,1) and 240 cents.

All two-stepsize scales where a = b, like the diminished and the
augmented, will reduce to this same (1,1) index. To find the generator
for a two-stepsize index, you could either seed a Stern-Brocot tree
with a two-stepsize index converted into adjacent fractions, or, as
with the Phi fifth, you could start a Fibonacci series with the
two-stepsize index converted into adjacent fractions:

0 1 1 2 3 5 8
-, -, -, -, -, -, --, ...
1 1 2 3 5 8 13

So because these two-stepsize scales where a = b always reduce to this
same (1,1) index, the Golden generator is always the Phi fifth or some
fraction of it. Using the 20-tet example I gave we'd have 1/5 of the
Phi fifth:

240/(1+Phi*1)*(0+Phi*1)

So the Golden generator for the (5,5) scale is 1/5 ~742 cents, or ~148
cents.

Here are the resulting rotations:

0 148 240 388 480 628 720 868 960 1108 1200
0 92 240 332 480 572 720 812 960 1052 1200

Hope that helps?

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, March 11, 2002 2:39 PM
Subject: [tuning] Re: Octatonic, diminished, symmetry

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> What I mean is the GCD of the two stepsizes.

in cents? in semitones?

> If the numbers are
> relatively prime, the there's no fractional periodicity. If they're
> not, the P (the octave or what have you) is fractionalized into GCD
> (a,b) parts. If (a,b) were the Messiaen decatonic, then the GCD
would
> be 2. So, (8,2)

these are the stepsizes? in what units? aren't the stepsizes just a
semitone and a whole tone? the above sure looks more like what you
call the "index", than the "stepsizes".

> becomes (4,1), and 2/1 becomes 600�.

yup.

> The Phi fifth only applies to scales like the diminished and the
> augmented that have the same number of each of the two stepsizes.
> That's because they'd all reduce to (1,1), and this Fibonacci series

which fibonacci series? applied how?

> when converted to adjacent fractions always gives the Phi fifth or
> some fraction of it.

please give an example. it'll help me (and perhaps others) to
understand what you're talking about.

happy day,
paul

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