Sub-minor question

I have a question for those of you who like to tune minor keys in
sub-minor, i.e., 6:7:9 rather than the more tradional 10:12:15.

I think the sub-minor has a nice dark sound. But there seems to be
a serious problem created by its use. Consider the minor chord A,C,E.
We tune this in 6:7:9 (ignore, for the moment, any concerns about
absolute placement in tuning space). Suppose we then add the note
G above E. This is a classic minor-minor seventh chord, but how shall
it be tuned? We're almost forced to tune the G a fifth above C, but
when we do this, we have formed a "C-super-major" chord, C,E,G, tuned
14:18:21 (intervals 9/7 and 7/6). It becomes all the more stark if,
after some time with all four notes sounding, we drop the A, leaving
only C,E,G.

The super-major is an ugly chord, does anybody disagree? It makes a
12-tET major chord sound lovely by comparison.

Is use of the sub-minor effectively confined to pieces which are
carefully constructed, without many otherwise desirable, rich, and
tunable chords? If no, does anybody have a midi sequence (or other
music example) illustrating how to solve this problem?

JdL

John A. deLaubenfels wrote:

> Is use of the sub-minor effectively confined to pieces which are
> carefully constructed, without many otherwise desirable, rich, and
> tunable chords? If no, does anybody have a midi sequence (or other
> music example) illustrating how to solve this problem?

I show some 7-limit scales on my website, including melodic examples (still
no chords). Using subminor thirds in a tonal context looks like trouble.
Using them with adaptive tuning really means more than 12 notes per octave
to me. This is why I'm not so interested in adaptive tuning. With a
meantone, I know exactly where the subminor thirds are should I want to use
them. Diminished triads aren't my favourite 7-limit chords, as well.

Graham
http://www.cix.co.uk/~gbreed/

Graham!
Just a note about your reference to Wilson Hexany. Wilson considers
this and his other CPS as harmonic/lattice constructs and not a scale
which he sees as melodic construct. He sees (as I understand it) the
need to solve both when choosing pitch material .
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Hi John!

John A. deLaubenfels wrote:
> I have a question for those of you who like to tune minor
> keys in sub-minor, i.e., 6:7:9 rather than the more
> tradional 10:12:15.

Well you caught me just in time. I started doing that
only last week. It all started after reading the email
about 9:8 vs 8:7 which got me to experimenting. Before
I knew it I had decided that my new love for this
month is the following fifth-repeating scale I discovered:

"seven and five" tuning:

1/1, 9/8, 9/7, 4/3, 3/2
=
0.0, 203.910, 435.084, 498.045, 701.955 (cents)

This has a real medieval feel and is also quite
Pythagoreanish on account of the base being a pure fifth.
Lots of subminor and supermajor thirds; this is a
root-position septimal triad lover's wet dream. Or
at least a quick fling. We'll see if it lasts.

For those of you who (like me) cringe at the static
dullness of pure just intervals (they sound like fingernails
on a blackboard to me - anyone with me on this other
than the Balinese?), stretch the scale until the fifth
is around 1.507 or so:

"Nuevo Renaissance" tuning (in cents):

0.0, 206.251, 440.080, 503.764, 710.015

Oooh! Now the supermajor third is at 440.1 cents - saucy!
[I recommend mapping these to the vanilla keys if you are
using a vanilla & chocolate style keyboard]

> I think the sub-minor has a nice dark sound.

I would call it voluptuous.

> The super-major is an ugly chord, does anybody disagree?

I disagree with you 100%. You're just not treating her
right. She's rather saucy and strident. You can have a
lot of fun with her if you let *her* show *you* where to
put your hands.

> Is use of the sub-minor effectively confined to pieces
> which are carefully constructed, without many otherwise
> desirable, rich, and tunable chords?

No, but you must overcome your addiction to the octave.

> If no, does anybody have a midi sequence (or other
> music example) illustrating how to solve this problem?

You must follow your own path, Luke!
(I do have about 40 minutes of songs in the above
tuning if you're curious, but I doubt it would answer
any of your questions.)

- Jeff

>John A. deLaubenfels wrote:

>> The super-major is an ugly chord, does anybody disagree?

J. Scott wrote,
>
>I disagree with you 100%. You're just not treating her
>right. She's rather saucy and strident. You can have a
>lot of fun with her if you let *her* show *you* where to
>put your hands.

The super-major chord 1/9:1/7:1/6 is a subset of the 9-limit utonality,
e.g., 1/9:1/7:1/6:1/5:1/4. If played _very_ delicately, with the upper notes
doubled in higher octaves (adding 1/3, 1/2, and 1/1), some consonance can be
discerned in this chord. But use the slighest bit of force, and she will
break (due to combination tones, which tend to fill up the entire frequency
spectrum). Otonal chords such as the sub-minor (6:7:9) are, by contrast,
aided by combination tones, which instead of filling up the entire frequency
spectrum, only occur at dicrete frequencies, namely the other overtones of
the missing fundamental that the chord tones are the 6th, 7th, and 9th (or
whatever) overtones of.

John A. deLaubenfels wrote,

>I have a question for those of you who like to tune minor keys in
>sub-minor, i.e., 6:7:9 rather than the more tradional 10:12:15.

>I think the sub-minor has a nice dark sound. But there seems to be
>a serious problem created by its use. Consider the minor chord A,C,E.
>We tune this in 6:7:9 (ignore, for the moment, any concerns about
>absolute placement in tuning space). Suppose we then add the note
>G above E. This is a classic minor-minor seventh chord, but how shall
>it be tuned? We're almost forced to tune the G a fifth above C, but
>when we do this, we have formed a "C-super-major" chord, C,E,G, tuned
>14:18:21 (intervals 9/7 and 7/6). It becomes all the more stark if,
>after some time with all four notes sounding, we drop the A, leaving
>only C,E,G.

>The super-major is an ugly chord, does anybody disagree? It makes a
>12-tET major chord sound lovely by comparison.

>Is use of the sub-minor effectively confined to pieces which are
>carefully constructed, without many otherwise desirable, rich, and
>tunable chords? If no, does anybody have a midi sequence (or other
>music example) illustrating how to solve this problem?

John, this is the world of 9-limit harmony. Many JI composers (Micheal
Harrison comes to mind)have used the scale of three sub-minor chords in a
chain of fifths: 1/1 9/8 7/6 4/3 3/2 14/9 7/4. I call it the sub-minor
scale. In typical meantone a very close approximation of the scale can be
found: Bb C C# Eb F G G#. Though the three sub-minor chords are nice, there
is a very ugly 21:16 between 4/3 and 7/4 (in meantone, an augmented third -
yuck!). 21:16 sounds like 4:3 but is out-of-tune by a septimal comma, 64:63.
The best solution (apart from adaptive tuning) is to use a temperament where
the septimal comma vanishes, such as 22-tET. The sub-minor scale, in 22-tET
steps, is 4 1 4 4 1 4 4.

In any case, one has the "super-major" chords to contend with. As they
convey none of the flavor of traditional major chords, I would say that few
traditional pieces can be played in the sub-minor scale. It's a great scale
for non-traditional music, though, and one of the many reasons I like
22-tET.

The tuning of the A-C-E-G chord above (12:14:18:21) was given by Kami
Rousseau as his preferred tuning of a C6 chord. That would be a super-major
chord with an added large sixth. I responded that I would prefer
10:12:15:18. These two chords are examples of saturated 9-limit chords;
i.e., chords where all intervals are within the 9-limit but no note can be
added without disturbing this property. The other two examples are the
9-limit otonality (4:5:6:7:9) and the 9-limit utonality
(1/9:1/7:1/6:1/5:1/4).

Back in td147, Graham Breed responded to my sub-minor question:

> I show some 7-limit scales on my website, including melodic examples
> (still no chords). Using subminor thirds in a tonal context looks
> like trouble. Using them with adaptive tuning really means more than
> 12 notes per octave to me.

Adaptive tuning works best, I agree, when one has the whole pitch
spectrum available.

> This is why I'm not so interested in adaptive tuning.

Really??? I had gotten a different idea from the exchanges we've had
apart from this list!! Why does it matter that you have more than 12
notes per octave, especially if your INPUT is still 12/octave?

> With a meantone, I know exactly where the subminor thirds are should I
> want to use them. Diminished triads aren't my favourite 7-limit
> chords, as well.

Meantone is fine for pieces that don't modulate a whole lot. My problem
is, when I'm playing I don't want to be confined to a narrow set of
keys; I want to modulate freely. So, for me, adaptive JI is a
necessity.

JdL

I want to acknowledge and thank everyone who responded to my sub-minor
question not long back.

To Kami Rousseau, td145.23, thanks for the reference! I listened to
the midi sequence at your site, and can say that my ear definitely needs
time to accept this tuning in places. Fascinating sounds, however!

To Graham Breed, td147.02, I've already responded.

To Jeff Scott, td147.18, she (the super-major) is definitely "saucy and
strident", as you say. Maybe I just need time to like it...
> You must follow your own path, Luke!
I do intend to!

Special thanks go to Brett Barbaro and Paul Erlich, who BOTH responded
in the same digest (td148.08 and td148.16)! You both offered inciteful
comments, for which I am appreciative.

JdL