That city block again

Paul Erlich wrote:

>>>The city block distance has the very important property that if an
>>>interval arises most simply as the sum of two simpler intervals, the
>>>metric of the first interval is the sum of the metrics of the other
two.

>>On a triangular lattice? What kind of sum do you mean?

>Well, 15:8 is the result of stacking a 5:4 and a 3:2, which is an
>intervallic sum as would be normally understood. On a triangular _or_
>rectangular lattice, the city block metric makes this the sum of the
>metric of 5:4 and the metric of 3:2.

So what about 5:1 and 1:3? Or do you have a circular definition of
simplicity?

>>>>The city block distance has the very important property that if an
>>>>interval arises most simply as the sum of two simpler intervals, the
>>>>metric of the first interval is the sum of the metrics of the other
two.

>>>On a triangular lattice? What kind of sum do you mean?

>>Well, 15:8 is the result of stacking a 5:4 and a 3:2, which is an
>>intervallic sum as would be normally understood. On a triangular _or_
>>rectangular lattice, the city block metric makes this the sum of the
>>metric of 5:4 and the metric of 3:2.

>So what about 5:1 and 1:3? Or do you have a circular definition of
>simplicity?

I don't know quite what you mean, but I would say that while 5:3 is a
strong interval quite independently of construction from 5:4 and 4:3,
and so the metric of 5:3 is less than the sum of the metrics of 5:4 and
of 4:3, 16:15 is not and its metric is no less than the sum of the
metrics of 5:4 and of 4:3. Stacking two of the same interval always
leads to an interval with twice the metric of the original interval.

[Erlich:]
> I would say that while 5:3 is a strong interval quite > independently
of construction from 5:4 and 4:3,
> and so the metric of 5:3 is less than the sum
> of the metrics of 5:4 and of 4:3, 16:15 is not
> and its metric is no less than the sum of the
> metrics of 5:4 and of 4:3.

I've been thinking for quite a while that harmonics
of the "subdominant" [= 3^-1] could function as
identities of chords in a similar way as harmonics
of the "tonic" or "root" of the chord itself,
1/1 [= n^0].

5/3 makes a very nice "6th" in the "major 6th" chord,
and 4/3 works well as a "4th" in "suspended" chords,
or as an "11th" in "11th" chords.

Last year Paul Erlich pointed out that 4/3 also works
well in a type of chord alternative-rockers use a lot,
a "tonic" chord with "dominant 7th" structure, with
a 4/3 added in: 1/1 - 5/4 - 4/3 - 3/2 - 7/4.

I wrote a piece based on this idea, using in addition to
4/3 the rational interpretations Paul discarded: 21/16,
11/8, and even 43/32. They all sounded "right" to me,
each with its own unique contribution to the overall
effect of the chord, thus my inspiration for writing the
piece. But the 4/3 certainly did fit in with the harmonics
4:5:6:7. I frequently used the 4/3 *as* the bass when
the piece cycled around to this particular version,
giving a chord of 8:12:15:16:18:21, but it didn't sound
as good in the bass as the 1/1 [i.e., 12 in these proportions].

And Paul also suggested, in discussion of the actual
subject under debate, that the most likely rational
interpretation for the "sharp 9th" in the "Jimi Hendrix
chord" was 7/6, whereas I had said 19/16.

I still prefer the sound of the 19/16 in my "sharp 9th"
chords, but I had to admit that Paul was right (as usual),
and that based on guitar and finger mechanics, Jimi's
"sharp 9th" could be called most likely a 7/6.

And recently, I've posted about my favorite "major 7th",
11/6. Since I said that here, and discussion has been
agreeing that 11/8 sounds less consonant (or whatever
is the proper term) than 13/8, and Erlich's Harmonic
Entropy shows "stronger" minima at 11/6 than at 11/8,
I've been comparing my feelings when hearing 11/8
and 11/6, and have decided definitely that 11/6 is
going to be more useful in my music than 11/8.

Haven't tried 13/12 yet, but all this leads me to
think that the proper lattice metric for these harmonics
of 3^-1 would connect them also with n^0 [1/1], and,
measuring from n^0, be shorter than the higher-numbered
(more-complex) harmonics of n^0 that are nearby in pitch.

-Monzo

|\=/|.-"""-. Joseph L. Monzo...................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\ c/o Sonic Arts, PO Box 620027, San Diego, CA, USA
_U//_/-/__/// | "...I broke thru the lattice barrier..." |
/monz\ ((jgs; | - Erv Wilson |

___________________________________________________________________
You don't need to buy Internet access to use free Internet e-mail.
Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
or call Juno at (800) 654-JUNO [654-5866]

> Message: 19
> Date: Thu, 25 Mar 1999 17:43:44 -0500
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
> Subject: Re: Digest Number 121

[Erlich, TD 122-19:]
> [me:]
> >Last year Paul Erlich pointed out that 4/3 also works
> >well in a type of chord alternative-rockers use a lot,
> >a "tonic" chord with "dominant 7th" structure, with
> >a 4/3 added in: 1/1 - 5/4 - 4/3 - 3/2 - 7/4.
>
> >I wrote a piece based on this idea, using in addition to
> >4/3 the rational interpretations Paul discarded: 21/16,
> >11/8, and even 43/32.
>
> I don't know if I can be said to have "accepted" 1/1-5/4-4/3-3/2-7/4
and
> "discarded" 1/1-5/4-21/16-3/2-7/4. Both chords contain a 21/16 interval
> that is best tempered away. Even better than 12-tET for this is
> (surprise) 22-tET, due to its better 7/6 and 7/4. Alternative rockers
of
> the future, take note!

OK, possibly I described your "evaluation" or description
of these chords the wrong way, but, lacking the original
reference right now, I *can* quote you from memory as saying:

> "what is the 4th? Is it 11/8? 21/16? 43/32. No - it's
> just plain old 4/3 . . ."

The "no" sounded like a discard to me.

And, since rational interpretation of ET notes was the
subject of the original discussion (and not the other way
around, which is what you describe above), my point was
that *all* of those ratios "work" as a "4th" in a JI
version of that chord. I proved it (at least to myself)
by writing the piece.

But I *would* love to hear some real alternative rock in 22-ET . . .
how 'bout it, Paul?

And what do you mean by "a 21/16 interval that is best tempered
away"? Is that implying that there's something intrinsically
wrong with the JI versions? Or that the ET versions are intrinsically
better?

>
> >Haven't tried 13/12 yet, but all this leads me to
> >think that the proper lattice metric for these harmonics
> >of 3^-1 would connect them also with n^0 [1/1], and,
> >measuring from n^0, be shorter than the higher-numbered
> >(more-complex) harmonics of n^0 that are nearby in pitch.
>
> So are you starting to lean towards a triangular lattice? (That would
> accomplish what you're saying).
>

I think I *started* to lean towards a triangular lattice before
I even created my first lattice, back when I started drawing
3^x * 5^y matrix diagrams (similar, as it turned out later,
to the ones in the Tipple & Frye book) based on Partch's
diagonal Diamonds.

I only modified it to rectangular because of my theories
on the importance of the prime-factors in musical perception.

I'd say that I've been leaning ever heavier towards triangular
ever since my realizations of the consonance of these 3^-1 harmonies
*in a chord built on n^0* a few years ago, and particularly
strongly since our discussion of last year.

At the same time, tho, I feel that my lattice diagrams
have reached a level of complexity where the triangular
connections are best left off, because I think it would
look too confusing if I included them.

I generally don't think higher than a 13-prime-limit
because of the complexity of the diagram, with the
exception that I like 19/16 a lot, so I use it.
(Also, as I pointed out here, 19 pops up in a lot
of historical tuning systems, which I've diagrammed)

But if I tried to show all the triangular connections
on even an 11-prime-limit lattice, the density of connecting
vectors would go contrary to my purpose for using the
lattice in the first place - its excellence at portraying
the many varied numerical relationships at a glance.
It would look like scribbling.

Some of the lattices I've done of Sims's and Young's
tuning systems already resemble that, even without
triangulation.

And with my recent idea on how to incorporate
ETs, JI, and octave-reduction into the same lattice,
it's quite packed with geometry and text already.

I don't disagree with you about the relevance of
composites *or* triangular connections in regard
to sonance of dyads, and perhaps I can even agree
in regard to larger harmonic structures over a 1/1.

But in regard to understanding a particular tuning
system, *as a whole*, especially one with many notes,
or indeed to understanding *all* tuning systems
taken together (in terms of their JI implications),
IMO, prime-factoring can't be beat.

And from that, I extrapolate to the idea that
prime-factors have some kind of harmonic significance.
Perhaps it's along the lines of what I said above,
regarding multiple dimensions of sonance.

-monz
|\=/|.-"""-. Joseph L. Monzo...................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\ c/o Sonic Arts, PO Box 620027, San Diego, CA, USA
_U//_/-/__/// | "...I broke thru the lattice barrier..." |
/monz\ ((jgs; | - Erv Wilson |

___________________________________________________________________
You don't need to buy Internet access to use free Internet e-mail.
Get completely free e-mail from Juno at http://www.juno.com/getjuno.html
or call Juno at (800) 654-JUNO [654-5866]