7-limit vs. 5-limit

Gerald (et al),

It's a pleasure to meet you over-the-net! I love your book, "Lies My Music
Teacher Told Me"; I like your writing style. (I'm not much of a "reader",
so if a book is able to keep my attention, it must have something going for
it, in my humble opinion...)

At 05:17 PM 5/7/99 -0700, you wrote:
>... I can't seem to find what you mean by "seven limit." In what sense
>did I give the impression in my book "Lies" that I "use it"?

Okay, I dug up your book, and found that you do not refer to your tuning by
the name "seven limit". I'm using the term "limit" in the following sense
(quoting my own web page at <http://www.pacificnet.net/~nowitzky/justint>):

"The 5-Limit Just Intonation tuning system is confined to
ratios of integers which are products of 2, 3, and 5 (prime
numbers up to 5). Increasing the limit to 7 would include
the 7th harmonic, which sounds flat to Western ears."

Another definition of limit may be found at
<http://www.ixpres.com/interval/dict/limit.htm>. Our hero Helmholtz' book
"On the Sensations of Tone" refers to 7-limit as "septimal" harmony, and
5-limit as "tertian" harmony (due to pure major third intervals). (See
"septimal" in his index.)

In your chapter entitled "Lie #7: Key Name is First Step", on p.76 (first
printing), you tune a V7 chord as 4:5:6:7. When you refer to singers' notes
"locking in", I imagine you're referring to this tuning of a V7. And
elsewhere in your book you consider even higher limits, such as the 11th and
13th partials.

>I am convinced that a simple major triad is not tuned (by even amateur
>singers) as 4:5:6. I have observed that when C is sounded and sung by basses
>and women are asked to sing E they tune it somewhat below tempered E,
>probably 4:5. When tenors join the basses C with a well-tuned G (presumably
>a 2:3 ratio), the women (without coaching) tune their E above tempered
>pitch. This "experiment" never fails. Your comments, please.

That's pretty puzzling. I don't have an explanation for that phenomenon.
But it reminds me of something I recently heard about string players.
Supposedly they sometimes play a little sharp so they can hear themselves
apart from the rest of the section. I guess trombonists like myself don't
have that problem, since our whole head is pretty much rattling to whatever
note we're playing.

>Regarding the tendency of the dom7 chord to resolve, I believe it has much
>to do with the acoustic (by ear) tuning of the tritone (5:7 and 7:10). While
>the tempered tritone (thus the term) can by heard enharmonically in two
>keys, the well-tuned tritone is clearly in one or the other. In other words,
>B-F clearly pulls to the key of C (major and/or minor), while B-E# clearly
>pulls to the key of F# (major and/or minor). The augmented fourth is
>significantly larger than the diminished fifth. The two "versions" can be
>"tuned" by ear as clearly as any other small-ratio combination. (The more
>experience, the better, naturally.) Comments?

I agree that the diminished fifth (B-F) should be tuned differently than the
tritone (B-E#). In my obsession to keep things in 5-limit just intonation,
though, I'd tune B-F as 45:64, and B-E# as 32:45.

equal 7-limit 5-limit
temp. (your tuning) (my tuning)
----- ------------- -------------
B-F 1.414 5:7 = 1.4 45:64 = 1.422
B-E# 1.414 7:10 = 1.429 32:45 = 1.406

So that's pretty wild - you and I both advocate two flavors of the interval,
but my B-F is the wider one, whereas your B-E# is your wide one. I'll have
to look into that! (The rationale for my tuning goes "a little something
like this": In B-F, I tune both notes a perfect fourth away from notes in
the resolution chord: B is 3/4 of E, and F is 4/3 of C.)

I'm sure I'll have replies to the rest of your email, but for now I better
get some sleep! I love the topic of musical tunings, and I really
appreciate your input on it. Thanks again,

--Mark

P.S.: I live in Glendale, so I'm right in your area. As a matter of fact,
I'm gonna attend the "Microfest" concert at Pierce College Sunday (5/9/99)
at 7pm. You're probably already aware of the concert - they use all sorts
of so-called "microtonal" tunings. Among others, I believe your fellow
faculty member John Schneider will be playing guitar. You oughta go to it!

P.P.S.: You wrote:
> If you have the time and interest, my web site is
> <http://home.earthlink.net/~stg3music/>

Been there - that's where I found your email address...

+------------------------------------------------------+
| Mark Nowitzky |
| email: nowitzky@alum.mit.edu AIM: Nowitzky |
| www: http://www.pacificnet.net/~nowitzky |
| "If you haven't visited Mark Nowitzky's home |
| page recently, you haven't missed much..." |
+------------------------------------------------------+