nonoctave ET: 14th root of 12/7

Hi Jacky and all!

OK, this is my 2nd installment about my nonoctave ETs.
These two scales are my favorite nonoctave ETs and taken
together, I think you can easily figure out one of my
major approaches to nonoctave ET scale design. (It's the
one that I've found makes the best scales -- 'best'
meaning that I personally like it.)

So, here is the 2nd scale.

Take the 14th root of 12:7 (the supermajor sixth).
This gives a chromatic step size of 66.652078 cents.
Form the scale from the following pattern of steps:
[3,1,2,3,2,1,1,1].

So here is the result:

ratio cents
------------- ----------
(12:7)^(0/14) 0.000000
(12:7)^(3/14) 199.956235
(12:7)^(4/14) 266.608313
(12:7)^(6/14) 399.912469
(12:7)^(9/14) 599.868704
(12:7)^(11/14) 733.172860
(12:7)^(12/14) 799.824938
(12:7)^(13/14) 866.477017
(12:7)^(14/14) 933.129095

(and here is the next repetition of the scale pattern:)

(12:7)^(17/14) 1133.085329
(12:7)^(18/14) 1199.737407
(12:7)^(20/14) 1333.041563
(12:7)^(23/14) 1532.997798
(12:7)^(25/14) 1666.301954
(12:7)^(26/14) 1732.954032
(12:7)^(27/14) 1799.606111
(12:7)^(28/14) 1866.258189

And here is the scale as a list of nearest
13 odd limit ratios:

00 1/1
01 9/8 - 3.953767 cents
02 7/6 - 0.262593 cents
03 5/4 + 13.598755 cents
04 7/5 + 17.356511 cents
05 3/2 + 31.217859 cents
06 8/5 - 13.861348 cents
07 18/11 + 13.884957 cents
08 12/7
09 13/7 + 61.383574 cents
10 2/1 - 0.262593 cents
11 13/6 - 5.531097 cents
12 12/5 + 17.356511 cents
13 13/5 + 12.088006 cents
14 11/4 - 18.363910 cents
15 14/5 + 17.093918 cents
16 3/1 - 35.696812 cents
17 10/3 - 18.144290 cents

OK -- now before you all hit [Reply] to
tell me off, let me defend myself a bit in advance.

I know what some people are thinking:

"Jeff, that is an ABSURD and OUTLANDISH claim to
be making that that is a nonoctave scale. Why,
you yourself _admit_ that it contains an interval
of 1199.737407 cents. I mean, come on now!!"

"Jeff, your 'new' scale is nothing more than 18tET.
I hate to break it to you, but it's been done before!"

"Jeff, that scale of yours may not be exactly the
same as a variation on sixth-tone tuning, but it is
so close that few synths have the tuning resolution
to tell the difference!"

... and I definately know what you mean.
It's one of those wacky number things that
makes:

(2:1)^(2/12) ~= (12:7)^(3/14)

Crazy, huh? There's a ton of those types of
equations, by the way.

But that does not matter! OK, I admit it, for most intents
and purposes this scale DOES have the interval of an
octave present. But because of the subsetting, the octave
is not present everywhere. Instead, _12:7_ is present
everywhere (and what a nice interval it is too I think we
can agree). And there is a darn close 7:6 (the octaval
complement of 12:7) in there too. There's a 12tET major
third at 399.9 cents. And the other intervals are all kind
of out there. See, that's where the fun begins. This
tuning has a lot of interesting beat patterns in it. Try
it. It is fun. It is a good tuning. I like this tuning.
No, I am not playing a joke. I really do see this as a
nonoctave tuning and a good one at that. The interesting
coincidence that there are some 12tET class whole tones in
there, even extending in 6 note whole tone scale chains
everywhere that stop before they can hit an octave, makes
it _really_ interesting. You can do cool things with this
scale.

OK, here are my notes for the scale:

"Name: 'Celestial Interference' or 'SM6/14'

Designed by considering Supermajor 6th & Subminor 3rd.

Has very little 'dissonance' as I define the word.

Similar to 18tET or 36tET (the so-called 'sixth-tone'
scale), but does not repeat at octave.

Adjectives:
Shimmering, Crystalline, Peaceful,
Holy, Restive, Spherical crystal enclosure,
Mystical, Levitating, In the Groove of Intonation

Good Timbres:
Full Tines, Voices, Tuba Violin"

Ok guys, now you can let me have it!
(Here is where I duck under the desk.)

- Jeff

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:
> Hi Jacky and all!
>
> OK, this is my 2nd installment about my nonoctave ETs.
> These two scales are my favorite nonoctave ETs and taken
> together, I think you can easily figure out one of my
> major approaches to nonoctave ET scale design. (It's the
> one that I've found makes the best scales -- 'best'
> meaning that I personally like it.)

Jeff,

It's fantastic! So many rich melodic and harmonic possibilites.

This just immediately inspired thematic composition. Listening to my
results now.

>
> So, here is the 2nd scale.
>
> Take the 14th root of 12:7 (the supermajor sixth).
> This gives a chromatic step size of 66.652078 cents.
> Form the scale from the following pattern of steps:
> [3,1,2,3,2,1,1,1].
>
> So here is the result:
>
> ratio cents
> ------------- ----------
> (12:7)^(0/14) 0.000000
> (12:7)^(3/14) 199.956235
> (12:7)^(4/14) 266.608313
> (12:7)^(6/14) 399.912469
> (12:7)^(9/14) 599.868704
> (12:7)^(11/14) 733.172860
> (12:7)^(12/14) 799.824938
> (12:7)^(13/14) 866.477017
> (12:7)^(14/14) 933.129095
>
> (and here is the next repetition of the scale pattern:)
>
> (12:7)^(17/14) 1133.085329
> (12:7)^(18/14) 1199.737407
> (12:7)^(20/14) 1333.041563
> (12:7)^(23/14) 1532.997798
> (12:7)^(25/14) 1666.301954
> (12:7)^(26/14) 1732.954032
> (12:7)^(27/14) 1799.606111
> (12:7)^(28/14) 1866.258189
>
>
> OK -- now before you all hit [Reply] to
> tell me off, let me defend myself a bit in advance.
>
> I know what some people are thinking:
>
> "Jeff, that is an ABSURD and OUTLANDISH claim to
> be making that that is a nonoctave scale. Why,
> you yourself _admit_ that it contains an interval
> of 1199.737407 cents. I mean, come on now!!"

Well - we'll have to let you slide on this little "artifact" of
mathematics.

>
> "Jeff, your 'new' scale is nothing more than 18tET.
> I hate to break it to you, but it's been done before!"
>
> "Jeff, that scale of yours may not be exactly the
> same as a variation on sixth-tone tuning, but it is
> so close that few synths have the tuning resolution
> to tell the difference!"
>
> ... and I definately know what you mean.
> It's one of those wacky number things that
> makes:
>
> (2:1)^(2/12) ~= (12:7)^(3/14)
>
> Crazy, huh? There's a ton of those types of
> equations, by the way.
>
> But that does not matter! OK, I admit it, for most intents
> and purposes this scale DOES have the interval of an
> octave present. But because of the subsetting, the octave
> is not present everywhere. Instead, _12:7_ is present
> everywhere (and what a nice interval it is too I think we
> can agree). And there is a darn close 7:6 (the octaval
> complement of 12:7) in there too. There's a 12tET major
> third at 399.9 cents. And the other intervals are all kind
> of out there. See, that's where the fun begins. This
> tuning has a lot of interesting beat patterns in it. Try
> it. It is fun. It is a good tuning. I like this tuning.
> No, I am not playing a joke. I really do see this as a
> nonoctave tuning and a good one at that. The interesting
> coincidence that there are some 12tET class whole tones in
> there, even extending in 6 note whole tone scale chains
> everywhere that stop before they can hit an octave, makes
> it _really_ interesting. You can do cool things with this
> scale.

I'm convinced that there's some kind of "secret octave" reference
here. He, he! A sort of 2/1 Freudian Slip! : )

>
> OK, here are my notes for the scale:
>
> "Name: 'Celestial Interference' or 'SM6/14'
>
> Designed by considering Supermajor 6th & Subminor 3rd.
>
> Has very little 'dissonance' as I define the word.

I would agree, and was able to instantly come up with cool melodies
and harmonies. Very nice - even with that...er...well never mind.

> Adjectives:

Noble, Majestic, Just

>
> Good Timbres:
> Full Tines, Voices, Tuba Violin"

Strings, Synth Brass.

>
> Ok guys, now you can let me have it!
> (Here is where I duck under the desk.)

No - don't go away, we've got more scales to share! I really like
this one.

Thanks,

Jacky Ligon

Hi Jacky!

> Jeff,

> It's fantastic! So many rich melodic and harmonic possibilites.

> This just immediately inspired thematic composition. Listening
> to my results now.

Thank you so much for taking the time to try it out.

You know, it's one thing to be off in the corner doing
something so cool but so incredibly different and, well,
'off in the corner' from the mainstream -- and know that
it really is cool and interesting even if no one has any
idea what you're doing...

It's something else entirely to have the experience of
someone else checking you out in the corner and coming
over and saying, "Hey that's cool what you're doing there
mixing the green and yellow paint together! I like the
symbolism there where you're tying together sunflowers and
wagon wheels suggesting the new hope of the pioneers!"

> I'm convinced that there's some kind of "secret octave"
> reference here. He, he! A sort of 2/1 Freudian Slip! : )

Yes octaves are hard to avoid since we are so well trained
on them. They are not a _bad_ interval at all... it's just
that they don't do much for you so why set them in cement
and work around them? Why not set _other_ things in cement
instead?

Now, if some 'wild' octaves happen to start growing here
and there -- between the cracks so to speak -- well, let
them be! I just worry about letting them take over the
whole garden since they can tend to choke out some of the
other plants when there's octaves sprawling everywhere.

Everyone could do to carry around an octave hoe, you know
-- to prevent them from getting out of control. A little
weeding goes a long way!

>> Has very little 'dissonance' as I define the word.

> I would agree, and was able to instantly come up with cool
> melodies and harmonies. Very nice - even with that...er...well
> never mind.

Yes! I am so happy you are having fun with it!

>> Adjectives:
> Noble, Majestic, Just
> ... don't go away, we've got more scales to share! I really
> like this one.

> Thanks,

> Jacky Ligon

You are so welcome and thank you, Jacky!

- Jeff

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:

/tuning/topicId_20309.html#20315

> Yes octaves are hard to avoid since we are so well trained
> on them. They are not a _bad_ interval at all... it's just
> that they don't do much for you so why set them in cement
> and work around them? Why not set _other_ things in cement
> instead?
>
> Now, if some 'wild' octaves happen to start growing here
> and there -- between the cracks so to speak -- well, let
> them be! I just worry about letting them take over the
> whole garden since they can tend to choke out some of the
> other plants when there's octaves sprawling everywhere.
>
> Everyone could do to carry around an octave hoe, you know
> -- to prevent them from getting out of control. A little
> weeding goes a long way!
>

Well, I couldn't resist making the comment that, for traditional
performers, if one is writing for such, the use of octaves makes a
BIG difference in terms of comprehensibility and ease of use of a
scale...at least that's how it seems to me at the moment. I'm a
great fan of non-octave work for solo electronics, however...

__________ ______ _______
Joseph Pehrson