just diminished (digest 1362, topic 1)

Hi Bob,

(I was trying to get this into the bottom of Digest 1363 - "Missed it by
'that much'...".)

At 09:07 AM 3/23/98 -0800, you wrote (Digest 1362, Topic 1):
>From: "Bob Lee"
>Subject: Well temperaments, just diminished
>...
>On another subject, some of the music I play contains "diminished 7th"
>chords (stacked minor thirds). Is this chord a modern invention? It seems
>to me that such a thing is impossible in anything but equal temperament.
>I'm having a hard time getting acceptable diminished chords out of my E9
>pedal steel using the standard JI tuned pedals. I know I could tune a pedal
>specifically for these chords, but that's really an awkward solution.
>Comments?

My thoughts are based on these two assumptions:

1) If you want to avoid "preparing" your instrument by altering it normal
tuning, you should stay with "5 limit" just intonation. Your Nashville E9th
pedal steel tuning chart, at
http://www.wco.com/~quasar/articles/just_e9.html, uses 5-limit ratios, even
with the compensators.

2) The tuning of diminished chords depends on the subsequent chord.

For purpose of discussion, suppose you're basically in the key of E, so
you're heading for an E major chord. Here's your E triad (4:5:6):
E: E (1/1) G# (5/4) B (3/4)

Here's a possibility for tuning a D# diminished triad (45:54:64=5:6:4*16/9):
D#dim: D# (15/8) F# (9/8) A (4/3)

You can get it by taking the top three notes of a B7 chord
(36:45:54:64)=(4:5:6:4*16/9):
B7: B (3/4) D# (15/8) F# (9/8) A (4/3)

To replace a B7 with a C full diminished seventh, you could raise the B by a
"leading tone" ratio (16/15), arriving at this
(192:225:270:320)=(4*16/15:5:6:4*16/9):
Cdim7: C (4/5) D# (15/8) F# (9/8) A (4/3)

But there's no C on your tuning chart. Suppose you transpose the whole
thing up a fifth (to get F#7 with the F# raised to a G):
Gdim7: G (3/5=6/5) A# (45/32) C# (27/32) E (1/1)

Now the C# (27/32) is not on the instrument. The only C# on the chart is 5/6.

No matter where you transpose to, it turns out you'll always be missing one
of the four notes. Here missing notes are shown in square brackets ([]),
and "FX" is "F double sharp":
Cdim7: [C (4/5)] D# (15/8) F# (9/8) A (4/3)
Edim7: E (1/1) [FX (75/64)] A# (45/32) C# (5/6)
Gdim7: G (6/5) A# (45/32) [C# (27/32)] E (1/1)

The rarest of the necessary intervals on the instrument is 75/64, as it
appears only at G (6/5) - A# (45/32).

Perhaps the best solution is to leave out one of the four notes, depending
on which ones are available at the time. Or get an instrument with even
more strings and pedals - But it'll start looking like a harp!

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

Yes, CDs
--------

You might be asking, why CDs? Why not just use plain, simple cassette
tapes? Clearly audio-quality isn't the reason; after all, it's mostly
just narration. Why did I spent $50 to get each of three DAT masters
transferred to "one-off" CDs (more properly called "CDRs" - those green,
recordable jobbies)? The answer is one very important word:

* CONTROL *

Just like working out in the gym, you need control over how you flex those
ear muscles! As you know, CDs do track-seeking, and that lets you do very
important things like:

* Jump immediately straight to, say, leap-destination identification
exercises, if that's what I need practice in at the moment.

* With the press of a single button, immediately jump back to the
beginning of a series of exercises, if after doing them all I'm
concerned that I didn't really get them down well enough.

* Jump past a section if it's just proving too easy (then again,
next time it might turn out exactly the opposite!).

* Put it on "shuffle" (random) play to cause me to switch off between
skills less predictably.

Let me also point out that typical cassette track-seek mechanisms won't
work for this. They search for a few seconds of silence between a blocks
of a few minutes of mostly continuous sound. These CDs consist almost
entirely gaps of silence. A cassette track-search mechanism stops at every
one of those gaps of silence, rendering it useless.

Believe me; CDRs are almost ideal for personalized ear training practice!

Also, CDR transfers are probably cheaper now. Lord knows the technology
to do them certainly is! In the three years since I did my 88CET
ear-training CDs, the cost of the equipment to do those CDR transfers
(using a PC) has dropped to about what just those three CDR transfers alone
cost me!