Arithmetic Scale Tuning

Has anyone on the list ever played around with this scale:

1:1 13:12 7:6 5:4 4:3 17:12 3:2 19:12 5:3 7:4 11:6 23:12 2:1

This is actually just the 12th to the 24th harmonic in the harmonic
series, and the frequency difference between each adjacent tone
(fundamental difference tone) is identical; hence the characterization
as an arithmetic tuning (perhaps, it could be dubbed the "12-tat"
scale). Of course, the difference frequency between _any_ two tones
will be an exact harmonic of the fundamental difference tone. Actually,
the above scale has the feel of the dominant mode, with the tonic mode
starting on the 4:3 tone. Recasting the scale on this note, it becomes:

1:1 17:16 9:8 19:16 5:4 21:16 11:8 23:16 3:2 13:8 7:4 15:8
2:1

Both modes have the 7:4 dyad, which gives it a very "bluesy" quality.
All chords have a very blended (consonant) sound, even tone clusters.
However, it doesn't lend itself to modulation of tonal center; it could
be said that it has "super"tonality.

Has anyone explored the possibilities of this scale, or written in it?
Or is it so restrictive that it is a dead-end path?

Fred Reinagel

How would I go about getting these tones from using these fractions?

Micah

> -----Original Message-----
> From: Fred Reinagel [SMTP:violab@wny.com]
> Sent: Thursday, May 27, 1999 1:15 PM
> To: tuning@onelist.com
> Subject: [tuning] Arithmetic Scale Tuning
>
> From: Fred Reinagel <violab@wny.com>
>
> Has anyone on the list ever played around with this scale:
>
> 1:1 13:12 7:6 5:4 4:3 17:12 3:2 19:12 5:3 7:4 11:6 23:12 2:1
>
> This is actually just the 12th to the 24th harmonic in the harmonic
> series, and the frequency difference between each adjacent tone
> (fundamental difference tone) is identical; hence the characterization
> as an arithmetic tuning (perhaps, it could be dubbed the "12-tat"
> scale). Of course, the difference frequency between _any_ two tones
> will be an exact harmonic of the fundamental difference tone. Actually,
> the above scale has the feel of the dominant mode, with the tonic mode
> starting on the 4:3 tone. Recasting the scale on this note, it becomes:
>
> 1:1 17:16 9:8 19:16 5:4 21:16 11:8 23:16 3:2 13:8 7:4 15:8
> 2:1
>
> Both modes have the 7:4 dyad, which gives it a very "bluesy" quality.
> All chords have a very blended (consonant) sound, even tone clusters.
> However, it doesn't lend itself to modulation of tonal center; it could
> be said that it has "super"tonality.
>
> Has anyone explored the possibilities of this scale, or written in it?
> Or is it so restrictive that it is a dead-end path?
>
> Fred Reinagel
>
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Fred Reinagel writes:

> Has anyone on the list ever played around with this scale:
>
> 1:1 13:12 7:6 5:4 4:3 17:12 3:2 19:12 5:3 7:4 11:6 23:12 2:1
>
> This is actually just the 12th to the 24th harmonic in the harmonic
[snip]
>
> Has anyone explored the possibilities of this scale, or written in it?
> Or is it so restrictive that it is a dead-end path?

I've played around with it a bit. I wrote two short pieces
in it a couple of years ago. It has a very interesting set
of strengths and weaknesses. I really like having the 11:6
and 13:12 on either side of the 12:12 tonic, sort of fat
semitones. One of the pieces I wrote uses a bass ostinato
that emphasizes that characteristic (picked bass guitar in
straight 8th note rhythm):

12-13-12-11-12-13-12-11-12-11-12-11-12-13-12-11 repeat ad
infinitum

Makes me feel like I'm half weightless after a couple of
minutes of it, which is good since I wrote it for a custom
level of a computer game set in space. Strange that a
tuning that looks so "natural" on the surface would easily
sound so un-earthly. I guess it's because the primes are
out of proportion with their occurrence in nature? The way
I used it, I mean, not necessarily the tuning itself.

Actually, now that I look at the sequence more carefully, I
see that I added 27:12 (9:8) to the mix, as well, and used
it to help achieve a more triadic/modulating sound,
especially in the bridge.

You're right, of course, about the high degree of
consonance, as would be expected for a set of relatively low
harmonic relationships.

Dead end path? I guess it depends on where you're trying to
get to. I have yet to find a tuning that I'd describe that
way. As with any other, try to compose to its strengths and
make creative use of its weaknesses and you're bound to get
somewhere interesting, though it might not be where you
expected to end up.

I don't have a full-blown website at the moment but I'll put
up a page with just those two MIDI files on it, and maybe
add Real Audio versions after the weekend/holiday. The URL
will be:

http://www.tcinternet.net/users/jfinnamore/df/12tat.html

I like that suggestion, 12-tat! :-) The page should be up
within a few hours if I can find the old
SoundBlaster-compatible versions.

David J. Finnamore
Just tune it!

>Has anyone on the list ever played around with this scale:
>
>1:1 13:12 7:6 5:4 4:3 17:12 3:2 19:12 5:3 7:4 11:6 23:12

Yes, I've improvised extensively on this scale. I prefer the above mode to the one you list later on. BTW, Fred, there's a little convention on the list: the colon for an interval (distance), the slash for a note (frequency). So...

1/1 24/23 13/12 7/6 5/4 4/3 17/12 3/2 19/12 5/3 7/4 11/6

...would be the preferred way to write the scale.

>Has anyone explored the possibilities of this scale, or written in it?
>Or is it so restrictive that it is a dead-end path?

It isn't the most versatile scale you'll ever use. But it is a fairly interesting way to use tune 12 notes in extended JI. Usually, modes of the harmonic series starting on a power of 2 are better, however, while it is subharmonic modes that tend to work best when started on a 3 (or octave extension thereof). If 12 is the number, this leads to the following scale...

1/1 12/11 9/8 6/5 5/4 4/3 11/8 3/2 13/8 12/7 7/4 15/8

...which we owe to Denny Genovese. This contains harmonics 8-16 and subharmonics 6-12.

-C.

Fred Reinagel wrote on Thu, 27 May 1999 16:15:25:

"Has anyone on the list ever played around with this scale:

1:1 13:12 7:6 5:4 4:3 17:12 3:2 19:12 5:3 7:4 11:6 23:12
2:1"

I made a struck brass pipe metalophone instrument once with 24 tones,
the 13th through the 26th partials of A-110. It was the only instrument
I had with that tuning, and I made it just to experiment with higher
partials. Its best characteristic was demonstrated when I dropped it.
First was the shock value of a musician throwing an instrument on the
floor, with the loud SMACK as it hit the cement, then a perceptions of a
tone cluster of very loud high tones, then suddenly a very pronounced
low tone, made of the difference tones of the cluster. It was clearly a
low A-110, like a low hum. Made a nice ending for our 80's space music
concerts in San Francisco.

Prent Rodgers
Mercer Island, WA