12-tone harmonic scale thoughts

I was just thinking about altering the rather standard 12-tone harmonic
scale that we've talked about before. Maybe somebody else has already done
the same thing, but here are my thoughts. The "standard" scale is:

C C# D D# E F F# G G# A A# B
16 17 18 19 20 21 22 24 26 27 28 30

In C this gives you harmonics 1 through 22 without skipping any.

In G it gives you harmonics 1 through 10 without skipping any.

So my thought was that by changing C# from 17 to 33/2, you lose 17 in C, but
then you get harmonics 1 through 12 in G. Having the 11 available in 2 keys
seems useful.

So then the question is, as long as the sequence in C was broken before 17,
it might be worth giving up 19 also, which is not used for the C 1-16
sequence or the G 1-12 sequence. What would be the most useful pitch to put
there (on D#, currently 19)? Maybe I could leave the otonal scenario to use
it to complete the C minor triad. But maybe there is some other interesting
chord that would be available. Or maybe someone has some different thoughts
entirely.

Actually I'm thinking about this putting this tuning on my harpsichord. I'm
also thinking of rooting this scale on F instead of C, so I get C and F as
the good keys. Then I was also thinking of tuning the bottom octave and
half (from F down) in a subharmonic sequence, the mirror of this harmonic
sequence. Then I have the possibility of a utonal base with an otonal
treble, so to speak--something that Twining used a lot in the Crysalid
Requiem. And around the split point a bunch of other interesting chords
become possible, like the augmented triad.

Any thoughts?

Thanks,
Kurt

Kurt wrote...

>I'm also thinking of rooting this scale on F instead of C, so I
>get C and F as the good keys.

It's more typical to favour the sharp keys on keyboard instruments,
though for accompanying singers the flat keys are often nice to
have.

>Then I was also thinking of tuning the bottom octave and half
>(from F down) in a subharmonic sequence, the mirror of this
>harmonic sequence. Then I have the possibility of a utonal
>base with an otonal treble,

Not sure I follow. You'll have a utonal scale in the bass, but
you won't have a diamond -- you won't be able to walk otonal
chords down a utonal scale, for instance.

Denny would do the coolest piano tunings, interweaving harmonic
and subharmonic scales like from 20-40 and such, up and down the
keyboard. They always sounded fantastic.

One of his 12-tone repeating scales is:

!
Denny Genovese's superposition of harmonics 8-16 and subharmonics 6-12.
12
!
12/11
9/8
6/5
5/4
4/3
11/8
3/2
13/8
12/7
7/4
15/8
2/1
!

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> I was just thinking about altering the rather standard 12-tone harmonic
> scale that we've talked about before. Maybe somebody else has already done
> the same thing, but here are my thoughts. The "standard" scale is:
>
> C C# D D# E F F# G G# A A# B
> 16 17 18 19 20 21 22 24 26 27 28 30
>
> In C this gives you harmonics 1 through 22 without skipping any.
>
> In G it gives you harmonics 1 through 10 without skipping any.
>
0> So my thought was that by changing C# from 17 to 33/2, you lose 17 in C, but
> then you get harmonics 1 through 12 in G. Having the 11 available in 2 keys
> seems useful.
>
> So then the question is, as long as the sequence in C was broken before 17,
> it might be worth giving up 19 also, which is not used for the C 1-16
> sequence or the G 1-12 sequence. What would be the most useful pitch to put
> there (on D#, currently 19)? Maybe I could leave the otonal scenario to use
> it to complete the C minor triad. But maybe there is some other interesting
> chord that would be available. Or maybe someone has some different thoughts
> entirely.

Well, if you wanted G to the 13th you could go ahead and replace 17/16 and 19/16
with 33/32 and 39/32 - the latter would end you up with a C neutered triad...and a
nice small step between that and 5/4.

I have tried quite a bit of harmonics 12-24 to the octave - last year I retuned a little
3-octave reed organ to it. I ended up breaking the lowest C and having to go Db to
Db, which were tuned to D's (harmonic series in G). Quite fun; less functional
harmony than in yours, though.

> Then I was also thinking of tuning the bottom octave and
> half (from F down) in a subharmonic sequence, the mirror of this harmonic
> sequence. Then I have the possibility of a utonal base with an otonal
> treble, so to speak--something that Twining used a lot in the Crysalid
> Requiem. And around the split point a bunch of other interesting chords
> become possible, like the augmented triad.

That sounds very cool. I'll take this opportunity to present a scale I recently made.
Originally I wanted 4 6:7:9:11 tetrads, separated by 6:7:9:11. I tried to make it with a
Scala function but it somehow added two notes, 14/11 and 18/11. I don't know how
they got there. It's worth tuning up:

! tetratetra.scl
!
tetratetradic scale on 6:7:9:11! 22:27:33 on degree 11! spooky pentatonic!
12
!
77/72
12/11
9/8
7/6
14/11
11/8
3/2
18/11
121/72
7/4
11/6
2/1

peace,
Jacob

on 8/23/04 2:01 AM, Carl Lumma <ekin@lumma.org> wrote:

> Kurt wrote...
>> I'm also thinking of rooting this scale on F instead of C, so I
>> get C and F as the good keys.
> It's more typical to favour the sharp keys on keyboard instruments,
> though for accompanying singers the flat keys are often nice to
> have.

I guess I'm not worried about what is typical. It is a practical matter for
me that C be one of the good keys. I can choose to root that tuning I
showed in F or C and achieve this. F root gives me F and C as the good
keys; C root gives me C and G. My original thought was that I want the
otonal/utonal transition to happen at the root of the tuning, and I wanted
the root to be F so I have almost 1.5 octaves of "otonal space" at the
bottom. I don't want it to be 1 octave or 2 octaves. That was my thinking.
But I might not have to make the otonal/utonal transition on the bottom note
of the tuning. It might work to have the tuning in C (C and G good keys)
and have the u/o transition on G. It is worth looking into that possibility
to see how it works, because there are interactions between the utonal and
the otonal sides and I should try all the major possibilities. Basically
the choices allow these possibilities:

scale root C allows any combination of:

utonal C guide-tone otonal C root

utonal F guide-tone otonal G root

scale root F allows any combination of:

utonal F guide-tone otonal F root

utonal Bb guide-tone otonal C root

This follows from the fact that I plan to simply invert the scale vertically
in the utonal section of the keyboard. I hope this is clear enough since I
didn't explain it well.

The two choices amount to about the same thing except for the relative
positions of the otonal and utonal portions of chords that are built from
both components. So that's what I want to check out in advance.

Maybe there are even other interesting possible choices for where to put the
transition.

>> Then I was also thinking of tuning the bottom octave and half
>> (from F down) in a subharmonic sequence, the mirror of this
>> harmonic sequence. Then I have the possibility of a utonal
>> base with an otonal treble,
> Not sure I follow. You'll have a utonal scale in the bass, but
> you won't have a diamond -- you won't be able to walk otonal
> chords down a utonal scale, for instance.

I can't have *much* of a diamond in 12 notes. For now I want to have major
triads in 2 keys since it is possible. Later I can try other things but
this is my current experiment.

> Denny would do the coolest piano tunings, interweaving harmonic
> and subharmonic scales like from 20-40 and such, up and down the
> keyboard. They always sounded fantastic.
>
> One of his 12-tone repeating scales is:
> !
> Denny Genovese's superposition of harmonics 8-16 and subharmonics 6-12.

I'm pretty sure this is one of the scales you gave me a year ago. If so, I
never ended up using it. I'll try it some day, but for now I want two
"good" keys and I can't have good otonal in 2 keys if I have good utonal
also woven in, I don't think, not in 12 notes.

Personally I found even the 20-some-tone diamond subset (complete through 9,
I think) very limiting, and even the 30-some subset (which included 15).
(I'm talking about the tunings on Norman Henry's instruments. Sorry my
memory is so bad about the details but I don't have time to correct this
now.) I would want to expand it more, as some have described I think having
2 instances of the diamond a fifth apart. That would do the trick, I think.
That's kind of what I'm approaching (weakly) with this tuning idea. I think
I can have some utonal and otonal choices in 2 keys by limiting what part of
the keyboard the utonal and otonal appear in. I really like what Twining
did with utonal bass and otonal soprano. I can't achieve that freedom of
modulation but having 2 keys is a great start.

-Kurt

>I really like what Twining id with utonal bass and otonal
>soprano.

Any time indexes you'd cite as good examples?

>I can't achieve that freedom of modulation but
>having 2 keys is a great start.

Sure.

-Carl

on 8/23/04 11:58 AM, Jacob <jbarton@rice.edu> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> I was just thinking about altering the rather standard 12-tone harmonic
>> scale that we've talked about before. Maybe somebody else has already done
>> the same thing, but here are my thoughts. The "standard" scale is:
>>
>> C C# D D# E F F# G G# A A# B
>> 16 17 18 19 20 21 22 24 26 27 28 30
>>
>> In C this gives you harmonics 1 through 22 without skipping any.
>>
>> In G it gives you harmonics 1 through 10 without skipping any.
>>
>> So my thought was that by changing C# from 17 to 33/2, you lose 17 in C,
>> but
>> then you get harmonics 1 through 12 in G. Having the 11 available in 2 keys
>> seems useful.
>>
>> So then the question is, as long as the sequence in C was broken before 17,
>> it might be worth giving up 19 also, which is not used for the C 1-16
>> sequence or the G 1-12 sequence. What would be the most useful pitch to put
>> there (on D#, currently 19)? Maybe I could leave the otonal scenario to use
>> it to complete the C minor triad. But maybe there is some other interesting
>> chord that would be available. Or maybe someone has some different thoughts
>> entirely.
>
> Well, if you wanted G to the 13th you could go ahead and replace 17/16 and
> 19/16
> with 33/32 and 39/32 - the latter would end you up with a C neutered
> triad...and a
> nice small step between that and 5/4.

Of course, I didn't think of that. However, I was also thinking of giving
up the 13 in C. So freeing up G# to be used for another purpose. I think
making it through the 11 is giving you a lot already, and the rest of the
12-tone resources can be directed elsewhere, *if* there's anything to be
gained. But otherwise having 13 in both C and G is probably good.

> I have tried quite a bit of harmonics 12-24 to the octave - last year I
> retuned a little
> 3-octave reed organ to it. I ended up breaking the lowest C and having to go
> Db to
> Db, which were tuned to D's (harmonic series in G). Quite fun; less
> functional
> harmony than in yours, though.

Yes, I've been wondering about the question of retuning reed organs. Also
about building them (or finding someone to build one) with different
keyboard layouts.

>> Then I was also thinking of tuning the bottom octave and
>> half (from F down) in a subharmonic sequence, the mirror of this harmonic
>> sequence. Then I have the possibility of a utonal base with an otonal
>> treble, so to speak--something that Twining used a lot in the Crysalid
>> Requiem. And around the split point a bunch of other interesting chords
>> become possible, like the augmented triad.
>
> That sounds very cool. I'll take this opportunity to present a scale I
> recently made.
> Originally I wanted 4 6:7:9:11 tetrads, separated by 6:7:9:11.

Ah, you mean a diamond of sorts? How about separated by 11:9:7:6?

> I tried to
> make it with a
> Scala function but it somehow added two notes, 14/11 and 18/11. I don't know
> how
> they got there. It's worth tuning up:

Thanks, I'll try it.

-Kurt

> ! tetratetra.scl
> !
> tetratetradic scale on 6:7:9:11! 22:27:33 on degree 11! spooky pentatonic!
> 12
> !
> 77/72
> 12/11
> 9/8
> 7/6
> 14/11
> 11/8
> 3/2
> 18/11
> 121/72
> 7/4
> 11/6
> 2/1
>
> peace,
> Jacob

on 8/23/04 6:29 PM, Carl Lumma <ekin@lumma.org> wrote:

>> I really like what Twining id with utonal bass and otonal
>> soprano.
>
> Any time indexes you'd cite as good examples?

He does it in a few places. Listen for a descending bass "melody" which
also accumulates into a chord (an arpegio that accumulates), with each
successive note getting closer to the previous. I think one occurrence is a
6-note chord like this IIRC:

1/(8:9:10:11:12:13)

which starts with just the 1/8 then becomes 1/(8:9) etc. Maybe it even goes
to the 1/14 (7 notes).

I'd say you can't miss it, but I'll try to point it out sometime, or note
the track/time next time I play it, and post it. The 1/1 article also talks
about it, but I think there was also an editing error in relation to the
topic.

-Kurt

> Yes, I've been wondering about the question of retuning reed organs. Also
> about building them (or finding someone to build one) with different
> keyboard layouts.

Get me some of that too. I just acquired yet anot

> > Originally I wanted 4 6:7:9:11 tetrads, separated by 6:7:9:11.
>
> Ah, you mean a diamond of sorts? How about separated by 11:9:7:6?

After a little investigation, what I wanted was a "corner" CPS: any 2, counting
duplicates, of 1/1 7/6 3/2 11/6. A diamond, yes!

on 8/23/04 8:13 PM, Jacob <jbarton@rice.edu> wrote:

>> Yes, I've been wondering about the question of retuning reed organs. Also
>> about building them (or finding someone to build one) with different
>> keyboard layouts.
>
> Get me some of that too. I just acquired yet anot

I've been looking for a field organ, one of the portable pump ones. Got any
extras with an 8' and 4' rank?

>>> Originally I wanted 4 6:7:9:11 tetrads, separated by 6:7:9:11.
>>
>> Ah, you mean a diamond of sorts? How about separated by 11:9:7:6?
>
> After a little investigation, what I wanted was a "corner" CPS: any 2,
> counting
> duplicates, of 1/1 7/6 3/2 11/6. A diamond, yes!

After thinking some more, it is like a tonality diamond except by being
based on products rather than ratios. That is you are taking all possible
products of 6,7,9,11 with 6,7,9,11 (and then reducing and octave adjusting
the results), whereas a diamond takes for example all ratios of 1,3,5,7,9
with 1,3,5,7,9. The good thing is you immediately get only half as many
distinct pitches because of the commutativity of multplication.

The funny thing is I went back to check your scale against your original
description and I have more comments...

on 8/23/04 11:58 AM, Jacob <jbarton@rice.edu> wrote:

> That sounds very cool. I'll take this opportunity to present a scale I
> recently made. Originally I wanted 4 6:7:9:11 tetrads, separated by 6:7:9:11.
> I tried to make it with a Scala function but it somehow added two notes, 14/11
> and 18/11. I don't know how they got there.

Besides that I don't know how the 12/11 got there and why the 49/36 is
missing (the way I did it). I started by multiplying to get all the
combinations, just as large numbers. You might have started by multiplying
some ratios, but this should yield the same end result. So here's a little
multiplication table in which I left out the duplicate numbers:

* 6 7 9 11
6 36 42 54 66
7 49 63 77
9 81 99
11 121

So dividing each of those 10 resulting numbers by 72 or 36 or 18 to get the
correct octave results yields (unsorted):

36/18 = 2/1
42/36 = 7/6
54/36 = 3/2
66/36 = 11/6
49/36 = 49/36 (missing from tetratetra.scl)
63/36 = 7/4
77/36 = 77/36
81/72 = 9/8
99/72 = 11/8
121/72 = 121/72

Comparing to your list:

> 77/72
> 12/11
> 9/8
> 7/6
> 14/11
> 11/8
> 3/2
> 18/11
> 121/72
> 7/4
> 11/6
> 2/1

reveals the extra 12/11 (in addition to 14/11 and 18/11) and the missing
49/36.

Gee I wonder if you (via scala) separated them by 11:9:7:6 like I was
suggesting as an alternative (did you notice?) instead of 6:7:9:11? If so
you really *did* make a diamond. This would explain the 11 denominators.
But that's an exercise for another day.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
That is you are taking all possible
> products of 6,7,9,11 with 6,7,9,11 (and then reducing and octave
adjusting
> the results), whereas a diamond takes for example all ratios of
1,3,5,7,9
> with 1,3,5,7,9.

Which makes it a combination product set. It's in the Monzopedia.