Notation for mixed-o/u-tonalities

As Paul Erlich showed (17 Jun 1997), just chords are possible where neither the otonal nor utonal representation of the ratios gives the true "limit" of the chord; such as the two varieties of 9-limit just minor seventh chord below, which appear to be 15 and 21-limit respectively. Incidentally the 12-tET minor seventh chord better approximates the right-hand one, which is more consonant anyway.

A C E G A B# E Fx Meantone spelling
10 : 12 : 15 : 18 12 : 14 : 18 : 21 Otonal rep.
1/18:1/15:1/12:1/10 1/21:1/18:1/14:1/12 Utonal rep.
5 6 9 6 7 9 Individual
4 5 6 4 6 7 intervals
2 3 2 3

In an attempt at a notation which is as compact as the otonal and utonal representations, these mixed-o/u-tonalities could be notated as

1/3 :2/5 :1/2 :3/5 2/7 :1/3 :3/7 :1/2
or
1 2 1 3 2 1 3 1
- : - : - : - - : - : - : -
3 5 2 5 7 3 7 2

This certainly reveals their consonance at a glance. However it seems to go too far! It makes them look as if they are only 5-limit and 7-limit respectively. It still does not reveal the existence of the 9:5 and 9:7 to casual inspection.

Of course if you don't believe in odd limits (only prime) there is no problem. However it is clear to me that odd-limits are useful since scales exist where the fifths (or some of them at least) may be tuned independently of the ninths.

I still think the above "mixed-o/u" notation is useful since it does show up the consonance and I find it easier to unpack the individual intervals from it than from the otonal or utonal representations.

Regards,
-- Dave Keenan
http://dkeenan.com

>As Paul Erlich showed (17 Jun 1997), just chords are possible where
neither the otonal nor utonal >representation of the ratios gives the
true "limit" of the chord; such as the two varieties of 9-limit just
>minor seventh chord below, which appear to be 15 and 21-limit
respectively. Incidentally the 12-tET >minor seventh chord better
approximates the right-hand one, which is more consonant anyway.

> A C E G A B# E Fx Meantone spelling
> 10 : 12 : 15 : 18 12 : 14 : 18 : 21 Otonal rep.
>1/18:1/15:1/12:1/10 1/21:1/18:1/14:1/12 Utonal rep.
> 5 6 9 6 7 9 Individual
> 4 5 6 4 6 7 intervals
> 2 3 2 3

>In an attempt at a notation which is as compact as the otonal and
utonal representations, these mixed->o/u-tonalities could be notated as

>1/3 :2/5 :1/2 :3/5 2/7 :1/3 :3/7 :1/2
>or
> 1 2 1 3 2 1 3 1
> - : - : - : - - : - : - : -
> 3 5 2 5 7 3 7 2

Excellent!

Dave, why do you think the 12-tET minor seventh chord better
approximates the right-hand one, and that that one is more consonant? I
would have said the opposite.

I wrote:
>> A C E G A B# E Fx Meantone spelling
>>1/3 :2/5 :1/2 :3/5 2/7 :1/3 :3/7 :1/2 Mixed o/u-tonal rep.

Paul Erlich replied:
>Excellent!
>
>Dave, why do you think the 12-tET minor seventh chord better
>approximates the right-hand one, and that that one is more consonant? I
>would have said the opposite.

Err... Because I still haven't learnt my left from my right. :) And I thought my 5-year-old son (who just started school today (sob!)) had problems. I meant to say the left hand one. Thanks for picking that up Paul (and so tactfully too).

Regards,
-- Dave Keenan
http://dkeenan.com

In my earlier post on this topic I failed to notice that (since they are their own duals) the two Just minor 7th chords could be notated even more simply as

5 5
- : 2 : - : 3 and
3 2

7 7
2 : - : 3 : -
3 2

The point of this, for those who have forgotten, was to make it apparent at a glance that these chords do not involve any interval with more than a 9 in its ratio. This is less apparent when they are notated as 10:12:15:18 and 12:14:18:21 respectively.

I'm not sure if any useful sort of canonical form can be defined, since even a major triad *could* be described as

5
2 : - : 3 which doesn't seem quite as enlightening as 4:5:6.
2

-- Dave Keenan
http://dkeenan.com