Michael's Phi section scale: Scala file in cents

Dear Michael,

Please let me thank you for the information about your Phi
section tuning, and share with you a Scala file I calculated for
it, inviting any corrections or suggestions:

! sheiman_michael-phi_section.scl
!
Michael Sheiman's Phi Section scale, from Tuning List
9
!
149.464
235.774
273.024
366.910
466.181
560.566
597.316
683.627
833.090

One thing I wonder is whether you have Scala (freely available on
the Web, thanks to the generosity of Manuel Op de Coul) or some
other software to convert decimal ratios to cents. While I can
understand lots of JI ratios, and can recognize a few of them
like 1.5 (i.e. 3/2) in decimal form, cents for irrational ratios
are much clearer; and when JI ratios are given in cents, I can
usually recognize them fairly easily. Thus I'd suggest that, when
in doubt, you use cents. For JI ratios, the fractional form is
fine, but the decimal form not so transparent, even with
something as simple as 9/8 vs. 1.125.

As to the scale, I might offer a few comments with the big
caution that while I recognize lots of the sizes as in familiar
regions for me, the idea of a scale with a periodicity at Phi
rather than 2/1 is something I've heard about but not used. What
I'm looking at here is more tetrachords and pentachords, which
put me on a more familiar ground.

You have a very interesting tetrachord 0-236-274-466, which in JI
terms might be rather like 1/1-8/7-7/6-21/16 (0-231-267-471). The
very small "semitone" or diesis between your large tone and small
minor third at 236 and 274 cents -- the latter of which could be
interpreted as 75/64 also in terms of size -- at around 38 cents
might in JI terms be a 49/48.

This is where having your step at 236 cents rather than the Noble
Mediant near 243 cents does make a difference, melodically: you
have enough space, 38 cents rather than a bare 30 cents, to have
the feeling of a distinct albeit very small "semitone" rather
than maybe only a large comma.

Robert the Inventor used this 49/48 step some years ago in a
"sad" European Dorian mode: 1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
(0-231-267-498-702-933-969-1200 cents). See if you don't like the
special melodic feeling of 0-236-273-466 cents in your scale, or
maybe a pentachord adding your narrow fifth, 0-236-273-466-684
cents.

A comment about your 367-cent interval: unless you're a spccial
16/13 fan, I'd say that 21/17 is a better description of this
flavor. While 16/13 (359 cents) is a large neutral third, it's
still for me at the high end of the central neutral region rather
distinct from major or minor. By the point we get to the
mid-360's, however, a friend of mine and I have both concluded
that were really in "submajor" territory: 367 is a like a small,
active, and beautiful major third or ditone! Thus if I were
describing this tuning, or looking for a JI version, I'd go with
21/17 (366 cents).

We could get more into tetrachords, but I wanted to let you know
that it was fun to confirm your tuning in cents -- corrections
are welcome, and I used a value for Phi of 1.618034.

It was neat how that small 236-273 cent step immediately reminded
me of Robert's septimal Dorian from a few years back. That,
alone, would be good reason to explore your scale further!

Best,

Margo Schulter
mschulter@...

Hi, Michael. For reference, why don't I include again the Scala
file for your scale (hoping again that I got it right):

! sheiman_michael-phi_section.scl
!
Michael Sheiman's Phi Section scale, from Tuning List
9
!
149.46400
235.77400
273.02400
366.91000
466.18100
560.56600
597.31600
683.62700
833.09000

> Of course...there's ongoing debate over whether fractions or
> cents are better...if I do either without the other someone
> will mind. :-D

Here I'd say that the most important guideline is that many of us
understand either cents or _integer ratios_ ("JI fractions" as
you call them below) such as 9/8 or 9:8 much better and faster
than a fraction expressed in decimal form. Thus either 9/8 or
203.91 cents is fine, and both would be ideal; but 1.125 is
harder for most of us to read. It's not the fraction, but the
decimal form, that causes the problem in comprehension.

Thus I'd say, "Use either cents or integer ratios, or often
ideally both" is a good guideline, along with "When in doubt, use
cents."

This is maybe a quirk of musical mathematics, because in lots of
areas of life, we tend to prefer decimal fractions like 1.6. But
in tuning math, 8/5 or 814 cents is instantly and effortlessly
recognized, while 1.6 requires (at least for me) a bit of
thought. This isn't to say that some people don't use decimal
forms extensively, or that there's anything wrong with them in
themselves, for example to show how a calculation is done. However, especially in a listing or file for a tuning, my rule
would be always to use either integer ratios or cents or both.

> But, fair enough, I shouldn't have been so lazy as to leave my
> values in decimal form and should have converted them to cents
> and not just "nearest fairly low-limit JI fractions"...and I
> do have Scala (only I used another program to convert decimal
> to cents when I do so).

Please let me say that the JI integer ratios that you suggested
were very interesting, and comparing views on how a given
irrational tuning might be approximated by such ratios is
something lots of us here enjoy. These views can and will differ,
of course, and that's part of the fun. As you now understand, the
only problem was getting a file or listing of your scale in cents
also.

Also, in fairness, I should confess that I often have a tendency
to mention some rather complex JI ratio such as 56/39 without
giving the size in cents (here about 626 cents). If people aren't
already familiar with the ratio, that isn't so helpful. Let me
resolve to do better from now on.

>> "You have a very interesting tetrachord 0-236-274-466, which
>> in JI terms might be rather like 1/1-8/7-7/6-21/16
>> (0-231-267-471). The very small "semitone" or diesis between
>> your large tone and small minor third at 236 and 274 cents"

> Interesting because when I composed with the scale in the past,
> I avoided having "8/7" and "7/6" in the same chord because I
> was afraid they were comma-tic, but now you're letting me know
> it is just far enough apart to be interpreted as otherwise.

This raises an interesting point on which I'll give my view with
a caution that I'm certainly ready to here that others see things
differently.

To me, the fact that two notes in a scale like 7/6 and 8/7 (or
your 235 and 273 cents) have between them a small interval that's
a fine melodic step, here around 38 cents, doesn't necessarily
mean that you'd want to use that interval simultaneously, in a
chord. They're two different questions: you might routinely use
it melodically, but consider using it simultaneously or
harmonically more as a "special effect."

While many people would think of something like 38 cents as a
comma, a term used very widely, I'd call it a small diesis: an
interval large enough to be felt distinctly as a melodic step,
and yet small enough to be felt as distinct from a usual
"semitone."

The upper limit for a "comma" in the narrow sense, something a
bit small to be felt as a distinct step (although a very slow
tempo can often help), might be around 30 cents or a bit more.

In general, while a small melodic step can be very refreshing,
using it as an interval in a chord should be carefully
considered. Often a result is high dissonance -- although at
times that's just what we might want.

In fact, when we get in the range of 50-70 cents, such an
interval might be very effective both melodically and vertically
in some appropriate harmonic or contrapuntal setting. George
Secor and I have both noted this, with his focus especially on
semitones of around 60-80 cents.

There are situations where small dieses and even commas can be
used effectively as harmonic intervals or in chords. One of them,
for example, is in the "complex unisons" and compressed or
stretched octaves of gamelan, where the beating is part of the
desired texture.

Another is a wonderful effect George Secor pulled off, playing a
13th-century European dance with voices a Pythagorean comma apart
creating a "chorusing" effect.

What I would want to caution against -- in least from my view --
is the idea that a good scale is meant to have all the steps, or
as many as possible, used together in a single chord. There are
pentatonic scales, for example, which nicely lend themselves to
this; but this isn't what I'd generally expect.

Then, again, given my own musical background, I often tend to
regard a single voice alone as beautiful melody (and the basis of
mnay world musical cultures); two as company; three as quite
rich; and four as very rich. So this business of stacking up
lots of notes at once, common of course in lots of musical
styles, is not what I would myself assume in looking at a scale,
although that doesn't mean that it's in any way wrong.

> A comment about your 367-cent interval: unless you're a spccial
> 16/13 fan, I'd say that 21/17 is a better description of this
> flavor."

> Could very well be, I just used 16/13 as I figured it would be
> easier for people to calculate JI-style relationships with
> being 13 limit and not 21 limit.

You're probably right that 16/13 might be better known that
21/17, but there is an interesting quirk in this that George
Secor pointed out to me around late 2001.

If our object is to divide a fifth into just two neutral thirds
with the simplest odd ratio for the division as a whole, then the
simplest division is actually 14:17:21, with our larger 21:17
third at around 366 cents and the smaller 17:14 at around 336
cents. While either 11:9 or 16:13 has a lower odd ratio than
either 17:14 or 21:17, their simplest divisions of the fifth are
18:22:27 (for 11:9 at 347 cents, with 27:22 at 355 cents); or
32:39:48 (for 16:13 at 359 cents, with 39:32 at 342 cents).

Actually, while this kind of calculation may (or may not) be
very interesting on paper, I'd say the various neutral shadings
(not necessarily just!) are all beautiful and worth exploring.

Here it's maybe a question of how fine a scale of mapping. Certainly 366 cents isn't far from 16/13, and it may be mainly
the "locals" to this part of the spectrum who will tell you:
"Yes, you're in the same general neighborhood as 16/13, but just
a bit wider, with 21/17 as almost a kind of GPS address."

In musical terms, one might say that 16/13 sounds more like a
clearly neutral third, and 21/17 more like a "submajor" third
leaning toward a major quality -- but impressions can definitely
vary. If you explore this region, you may find it interesting to
see how your perceptions develop (as I've been doing).

>> We could get more into tetrachords, but I wanted to let you
>> know that it was fun to confirm your tuning in cents -- >
>> corrections are welcome, and I used a value for Phi of >
>> 1.618034.

> Glad you enjoyed it. I'll have to double check but (for once)
> it looks like you (unlike many others) have finally converted
> this scale correctly. :-D

Since no one here is infallible, checking it out would be very
wise, and again I would apologize for any mistakes, since I know
that creating a scale like this is an effort that it's important
to honor by being careful that we get it right.

> That's the odd thing about this scale though: that Phi is
> indeed the period and neither PHI nor anything else in ideal
> form, is rounded to JI. In fact I've found (both in the past
> and in a few recent "re-tests") doing things like making the
> Wolf fifth into a pure fifth or using the low-limit 13/8 as PHI
> or using 2/1 as the period actually tends to upset the sense of
> balance of the scale for composing.

While 13/8 is a wonderful interval, it's no replacement for Phi,
which might better be approximated by 21/13 (830 cents), or
closer yet by 34/21 (834 cents). All these neutral sixths are
part of the Fibonacci series leading to Phi itself -- and here
the more complex the ratio along this series, the better the
approximation.

Maybe the operative rule is an axiom of a woman named Nicole
Lennon in Australia about dog training that can also apply to
humans and tunings alike: "Work with the personality of the dog
you've got, rather than trying to impose a boring one of your
own."

And a neat aspect I haven't looked at is that since Phi is the
periodicity, we'll get intervals repeating at this periodicity,
not the same as 2/1! I'd didn;t really look at that in Scala, but
it could give us all kinds of rich resources for this scale.

Best,

Margo