Work For the Night is Coming

[David C Keenan:]
> In other words "between a 9/8, a 10/9 or a 49/44 version".

Yes.

>What intervals or chords does the 10/9 take part in?

OK, I dug the parts out off the mothballs (dustballs actually), and
things only get more complicated... The first verse is scored for
oboe, bass clarinet, cello, and guitar quartet (two electric's, and
two steel string acoustics). The piece is also what I'll call a
polytonal modal composite, where the electric guitars take C as the
1/1, while the oboe, bass clarinet, and acoustic guitars take F (4/3)
as the 1/1, and the cello takes D (9/8) as the 1/1. So as the basic
step structure of the electric guitars is 10/9, 21/20, 9/8, 8/7,
28/27, 9/8, 8/7, i.e., LsLLsLL:

1/1, 10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1

While the step structure of the oboe, bass clarinet, and acoustic
guitars is sLLLsLL:

1/1, 28/27, 7/6, 4/3, 40/27, 14/9, 7/4, 2/1

And the step structure of the cello is LLsLLsL:

63/32, 9/8, 5/4, 21/16, 189/128, 27/16, 7/4, 63/32

Which results in a VI, III, V modal composite. (I'd be more than happy
to mail this score, etc. off to someone who's more comfortable with
"traditional" technical analysis, of course they'd have to be an
overly generous, good samaritan gluten for unrewarded punishment
though... Joe Monzo?)

>is it for melodic reasons that it needs to be 10/9?

Yes, but also for the intentionally roughhewn "harmonic" integrity of
the composite... perhaps the 5120/5103, and the 225/224:

63/32 9/8 5/4 21/16 189/128 27/16 7/4 63/32
9:10 6:7 16:21 2:3 9:14 4:7
20:21 160:189 20:27 5:7 40:63 320:567
8:9 7:9 3:4 16:27 8:15
7:8 27:32 3:5
27:28 24:35 32:49 256:441
64:81 32:45 128:189 1024:1701 128:243

would be good candidates for a distribution a la microtuning?

Dan

Earlier I wrote:

"...perhaps the 5120/5103, and the 225/224 would be good candidates
for a distribution a la microtuning?"

Indeed(!), "microtuning" should have read microtempering.

Dan

[Dan Stearns, TD 525.6]
>[David C Keenan:]
>>What intervals or chords does the 10/9 take part in?
>
>OK, I dug the parts out off the mothballs (dustballs actually), and
>things only get more complicated...
...

I couldn't find the answer to my question from what you wrote and it now looks way too hard and I've lost interest. Sorry.

>>is it for melodic reasons that it needs to be 10/9?
>
>Yes, but also for the intentionally roughhewn "harmonic" integrity of
>the composite... perhaps the 5120/5103, and the 225/224:
>
>63/32 9/8 5/4 21/16 189/128 27/16 7/4 63/32
> 9:10 6:7 16:21 2:3 9:14 4:7
> 20:21 160:189 20:27 5:7 40:63 320:567
> 8:9 7:9 3:4 16:27 8:15
> 7:8 27:32 3:5
> 27:28 24:35 32:49 256:441
> 64:81 32:45 128:189 1024:1701 128:243
>
>would be good candidates for a distribution a la microtempering?

I don't understand this table, but either of those commas individually are excellent candidates for microtempering. However they pull too much in opposite directions on the 2's and 3's, so any temperament that distributes them both, isn't quite "micro".

5120 2^10 * 5
---- = -------- = 5.76 c
5103 3^6 * 7

225 3 * 3 * 5 * 5
--- = ------------- = 7.71 c
224 2^5 * 7

Assuming we don't want to (or can't) temper the octave, the first comma alone would have you widen the 2:3's by 0.72 c and widen the 4:7's by 1.44 c. The second alone would have you narrow the 2:3's by 1.29 c and narrow the 4:5's by 2.57 c.

Distributing the two simultaneously, it looks like it might be best to widen the 2:3's by 0.27 c and narrow the 4:5's by 4.13 c and have Just 4:7's. So 5:7's are 4.13 c narrow, 3:5's are 4.40 c narrow, 6:7's are 0.27 c narrow, 7:9's are 0.54 c wide. This is not quite the minimax optimum, but close enough.

So the scale
1/1, 10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1 becomes, in rounded cents
0 178 267 471 702 764 969 1200

Of course this makes it worse as far as using that 10/9 as a 9/8.

By the way Dan, I must commend you on the enormous strides you've made in learning and using standard ASCII math notation since we first e-met. I hope that doesn't sound patronising. The only oversight I noticed recently (and I'm a bloody-minded pedant) was something like "6/17" where you apparently meant six seventeenths of an octave (I don't remember the actual example). It would be safer to write "6/17 octave" or preferably "2^(6/17)".

Regards,

-- Dave Keenan
http://dkeenan.com

In-Reply-To: <3.0.6.32.20000211051211.009826c0@uq.net.au>
Dave Keenan wrote:

> I don't understand this table, but either of those commas individually
> are excellent candidates for microtempering. However they pull too much
> in opposite directions on the 2's and 3's, so any temperament that
> distributes them both, isn't quite "micro".
>
> 5120 2^10 * 5
> ---- = -------- = 5.76 c
> 5103 3^6 * 7
>
> 225 3 * 3 * 5 * 5
> --- = ------------- = 7.71 c
> 224 2^5 * 7

Aha! I recognize these!! They're septimal schismas!!!!!!!

So these are what Dan wants tempering out? That defines a schismic
tuning. See:

http://x31eq.com/schismic.htm#matrix

The difference between them should be a regular schisma.

> Assuming we don't want to (or can't) temper the octave, the first comma
> alone would have you widen the 2:3's by 0.72 c and widen the 4:7's by
> 1.44 c. The second alone would have you narrow the 2:3's by 1.29 c and
> narrow the 4:5's by 2.57 c.
>
> Distributing the two simultaneously, it looks like it might be best to
> widen the 2:3's by 0.27 c and narrow the 4:5's by 4.13 c and have Just
> 4:7's. So 5:7's are 4.13 c narrow, 3:5's are 4.40 c narrow, 6:7's are
> 0.27 c narrow, 7:9's are 0.54 c wide. This is not quite the minimax
> optimum, but close enough.

Yep, 0.27 cent wide fifths are bang on for the septimal schismic
temperament with just 4:7.

[David C Keenan:]
> I couldn't find the answer to my question from what you wrote

Yes, sorry. The problem was that the question doesn't really have an
answer in the music (the music was largely operating under a different
set of parameters).

>and it now looks way too hard and I've lost interest. Sorry.

Can't blame you a bit.

>Assuming we don't want to (or can't) temper the octave,

The way the music is operating, it's really a bit difficult to say...
but on the surface of it I don't see why not really. As I better
understand what your doing now Dave, I'll just try some of these
myself when I get a chance. Usually I would say that most of these
microtempering differences would probably just be lost in the ebb and
flow of naturally occurring errors - by that I'm meaning both
unintended tuning errors in performance and whatnot, and sensitive,
and variable intonation adjustments made by non-fixed pitch
instruments (I also think that these sensitive and intended micro
adjustments of pitch are sometimes adumbrated, or hinted at, by the
dynamics and attacks and placements of certain notes by fixed pitch
instruments as well). However my recollection of this piece was that
certain intervals were very sensitive to slight tuning alterations, so
microtempering seems just the thing to try.

>By the way Dan, I must commend you on the enormous strides you've
made in learning and using standard ASCII math notation since we first
e-met.

Well as I've said before, though I don't have much of a desire to iron
out every single wrinkle and quirk of the way I was used to going
about things, I very much DO appreciate what I've been able to learn,
better understand, and perhaps better present, from the time I've
spent this previous year at the TD. I got the computer and all that
goes with it (electronic tuning forums, standard ASCII math notation,
etc., etc,) all at once last year, and it was all completely uncharted
territory for me. So I had to get my feet wet in a bunch of different
(and usually embarrassing!) ways all at once.

>The only oversight I noticed recently (and I'm a bloody-minded
pedant) was something like "6/17" where you apparently meant six
seventeenths of an octave (I don't remember the actual example). It
would be safer to write "6/17 octave" or preferably "2^(6/17)".

Joe Monzo has also made this point to me a couple of times
previously... and I guess the truth is, that I personally just find it
unnecessarily bulky - especially if I'm going to be repeatedly writing
it... I guess I've just never had much of a problem understanding that
when the numerator is less than the denominator here, chances are
pretty fair that it's going to mean a fraction of an octave. But I
also do understand the desire for a more sound or precise
nomenclature... so I'll think it over again.

Dan

[Graham Breed:]
> Aha! I recognize these!! They're septimal schismas!!!!!!!

Here the 225/224 is the difference between the 45/32 and the 7/5, and
the 5120/5103 is the difference between the 7-limit intervals offset
by an 81/80 and the 3-limit intervals.

>That defines a schismic tuning. See:
> http://x31eq.com/schismic.htm#matrix

Thanks Graham, I'll check it out.

> Yep, 0.27 cent wide fifths are bang on for the septimal schismic
temperament with just 4:7.

Just don't try to get those "0.27 cent wide fifths" too "bang on" on
your fretless guitar (even your bassoon for that matter!)...

Dan

Dan Stearns wrote (Fri, 11 Feb 2000 13:21:42 -0500)

> > Yep, 0.27 cent wide fifths are bang on for the septimal schismic
> temperament with just 4:7.
>
> Just don't try to get those "0.27 cent wide fifths" too "bang on" on
> your fretless guitar (even your bassoon for that matter!)...

All intervals in a schismic temperament are defined by the size of the
fifth. The fact that Dave Keenan got the right numbers to two decimal
places means he must have derived the same scale. The fact that
everything is determined by the fifth means that it isn't a
microtemperament. Or does it? I don't find a definiiton in Monzo's
dictionary. Here's an attempt.

A temperament where each just interval approximates to a single tempered
interval, and more than two intervals (including the octave) are required
to define the tuning.

If a just interval approximates to different tempered intervals, you have
a well temperament.

If only two intervals, usually an octave and a fifth, are required to
define the tuning, you have a linear temperament. Examples are meantone
and schismic scales.

Kudos to Dave Keenan for blazing this trail.

For an octave-invariant scale, the number of bridge intervals for a
microtemperament is two less than the number of harmonic axes. For
7-prime limit, there are 3 harmonic axes: 3, 5 and 7. So only 1 bridge is
required for a microtemperament. Two bridges therefore must give you a
linear temperament, as indeed they do here.

I also disagree with Monzo's definition of temperament: "a tuning which is
/not/ a just-intonation; that is, the intervals are /not/ small-integer
ratios". I would prefer "a tuning which /approximates/ just-intonation;
that is, the intervals are /close to/ small-integer ratios, and the scale
is somehow simplified relative to its just equivalent". Some tunings make
no attempt to approximate JI, and so should not be considered
temperaments.

>For an octave-invariant scale, the number of bridge intervals for a
>microtemperament is two less than the number of harmonic axes.

What if the number is three less, four less, etc.? Still could be a
microtemperament, I would say. And if the number is one less? Many linear
temperaments _are_ examples of microtemperament, like Helmholtz's and
Groven's schismatic temperaments.

In-Reply-To: <CE80F17667E4D211AE530090274662729C4A1B@acadian-asset.com>
Paul Erlich wrote:

> >For an octave-invariant scale, the number of bridge intervals for a
> >microtemperament is two less than the number of harmonic axes.
>
> What if the number is three less, four less, etc.? Still could be a
> microtemperament, I would say.

Yes, my oversight.

> And if the number is one less? Many
> linear
> temperaments _are_ examples of microtemperament, like Helmholtz's and
> Groven's schismatic temperaments.

Well, that's why we need a definition.