Re: Barbershop and 17

There was some talk of barbershop tuning diminished 7'th
chords as 10:12:14:17. Not having an audio source right
now, there seems, on MY paper at least, some other tunings
that may be more probable.

15:18:21:25 looks good if you think the diminished seventh
should pun the major sixth
25:30:35:42 stretches the diminished seventh and also
looks smoother than either of the other two.

Both of these are 7-prime-limit interpretations. I also
have a bias towards making the outer intervals fall into
a major triad in second inversion. The 17/10 seems 'too
wide', the 25/15 (5/3) is perhaps 'too smooth' for this
kind of chord. The 42/25 looks 'just right' implying
a mildly strident 75:(100):126 major triad, (hey! thats
our old friend 12tet).

For what its worth, my 'looks smoother' comment came from
a psuedo harmonic entropy measure, which is geomean of all
internal interval terms. So for

10 : 12 : 14 : 17
6/5 7/6 17/14
7/5 17/12
17/10

PHE = geomean( 6, 5, 7, 6, 17, 14, 7, 5, 17, 12, 17, 10 )
= 9.192307692

The table for all the candidates I played with follows

10:12:14:17 9.192307692
15:18:21:25 8.566928432 // due to smoother outer interval
25:30:36:42 10.18785224 // 6/5 6/5 7/6 (9/5 seventh)
125:150:180:216 17.01901335 // 6/5 stack
25:30:35:42 7.77263187 // 6/5 7/6 6/5
7/5 7/5
42/25 = 1.68

Perhaps this is all in the realm of tempering...

Bob Valentine

Robert Valentine wrote,

>I also
>have a bias towards making the outer intervals fall into
>a major triad in second inversion. The 17/10 seems 'too
>wide', the 25/15 (5/3) is perhaps 'too smooth' for this
>kind of chord. The 42/25 looks 'just right' implying
>a mildly strident 75:(100):126 major triad, (hey! thats
>our old friend 12tet).

I don't understand what you mean by "fall into" and "implying" here . . .

> From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
>
> Robert Valentine wrote,
>
> >I also
> >have a bias towards making the outer intervals fall into
> >a major triad in second inversion. The 17/10 seems 'too
> >wide', the 25/15 (5/3) is perhaps 'too smooth' for this
> >kind of chord. The 42/25 looks 'just right' implying
> >a mildly strident 75:(100):126 major triad, (hey! thats
> >our old friend 12tet).
>
> I don't understand what you mean by "fall into" and "implying" here . . .
>

E G Bb Db

My bias is that I would like to stick an A in the middle of this
chord and have the E and Db "fall into" an approximately major triad
in second inversion. It may be a 12tet bias, but it is something I
expect from this chord.

10:12:14:17 implies a 30:[40]:51 major triad, the third of 1.275
very wide.

15:18:21:25 implies a 3:[4]:5 major triad, obviously lovely, but
perhaps too stable??? For one thing, it is a diminished seventh
rather than major sixth, but I'm also toying here with ideas left
over from the Eskelin threads regarding a 'high third'. I believe
this to be a real phenomena, at least in leading tone situations,
and I would think that barbershop would both want to settle-in
harmonically AND keep some sort of leading-tone-like tension going.

25:30:35:42 implies a 75:[100]:126 major triad. The fact that this
simultaneously gives a stretched major triad AND provides greater
simplicity to all the other internal intervals was why I thought
it may be an interesting choice.

Bob Valentine

Bob Valentine wrote,

> E G Bb Db

>My bias is that I would like to stick an A in the middle of this
>chord and have the E and Db "fall into" an approximately major triad
>in second inversion.

How about 12:14:17:20?