88CET Ear-Training CDs, part 5

88CET? Wuzzat?
---------------

Before I go into more detail about the sorts of exercises that make
sense on such a CD, I need to explain a bit more about 88CET tuning itself.
Some of the exercises won't make a lot of sense without this background
information. Indeed, some of the exercises on these 88CET CDs aren't
likely to make much sense on, for example, a 34TET ear-training CD.

I did a series of tuning-list postings on 88CET tuning about 3 years ago
or so. Even though it's been that long, I'll paraphrase the qualities of
88CET tuning a lot here, since that's not the main thrust of this series.
I am working on some really killer web pages about 88CET tuning, but
they're only about 1/3 done now. So in the meantime (not to be confused
with "meantone"!), if you'd like to learn more, I can send any of you who
might be interested in more details about 88CET, the text of those postings
(assuming that I can find them all!).

Most basically, "88CET" stands for "88-Cent [per step] Equal
Temperament". It's an equal-temperament whose primitive step size is 88
cents. The 1200 cents in an octave's span clearly don't divide evenly by
88, so 88CET is a nonoctave tuning. No two notes on an 88CET instrument
are exactly an octave apart, nor exactly two octaves apart.

But the fact that it does not approximate octaves is not, in itself,
quite as critically important as several other attributes. For the
purposes of this series, here's what's most important about 88CET tuning:
1. 88CET has no traditional major or minor thirds. Instead it
has three nontraditional thirds: subminor (7:6), neutral (11:9),
and supramajor (9:7).
2. It has no approximation to a traditional major or minor scale.
3. Since it has no octave, 88CET has no octave-compound intervals.
That means that, rather than getting "octave-equivalent" intervals
in each octave's span, you instead get an different set of
harmonic resources.
4. 88CET has a pretty good approximation to a 4:6:7:9:10:11:15
harmonic-series fragment chord, or (obviously) any fragment of
that chord.
5. Since it's an equal temperament, it can also play that same
structure inverted, meaning a that same subharmonic-series
fragment.
6. Also because it's an equal-temperament, you can build other
chords as stacks of any one of those intervals, such as a neutral
triad, which is a stack of two 11:9 neutral thirds. That in the
same sense that a major triad is a minor third atop a major third.
7. Since 88CET has no approximation to the octave, it has two
intervals that come close to an octave. They are far enough to
sound "out of tune", and close enough not to sound like anything
in their own right. They are, in short, "off octaves" and I find
that they pretty much have to be avoided in most forms of harmony
(for approximately-harmonic timbres that is).
8. In addition to "off-octaves", 88CET also has a similarly dreadful-
sounding off twelfth (3:1 ratio), and off double-octave (4:1).
Other harmonic frequencies (5:1, 6:1, 7:1, etc.) seem to be either
sufficiently well-represented, or are harmonically remote enough
to be tolerable.
Again I'm brutally paraphrasing here, so I'm not coming even close to doing
justice to the importance of any of these qualities of 88CET tuning.

Here is a table of frequency-ratio approximations in 88CET music:

------------------------------------------
Number
of 88CET
Ratio Steps Interval Name
------------------------------------------
1:1 0 Perfect Unison
10:9 2 "Minor" Wholetone
7:6 3 (Septimal) Subminor Third
11:9 4 Neutral Third
9:7 5 Supramajor Third
10:7 7 (Large Septimal) Tritone
3:2 8 Perfect Fifth
5:3 10 Major Sixth
7:4 11 Subminor Seventh
15:7 15 Neutral Ninth
9:4 16 Major Ninth
5:2 18 Major Tenth
------------------------------------------

Paul,

I don't have any comments about these tunings, but I do have a question
about this:

>>In MT, the two German augmented sixth chords (usually Bb D F G# and Eb G Bb
>>C#) have a particularly concordant quality, resembling the seventh chords
>>sometimes encountered in barbershop singing. The two incomplete French
>>augmented sixth chords (Bb E G# and Eb A C#) also have a unique resonance.
>>Huygens, in advocating meantone tempermant, pointed out (back in the 17th
>>century, I believe) that these latter chords are used in compositions and in
>>meantone are tuned very nearly 1/7:1/5:1/4. This may have been the first
>>attempt to use the number 7 in explaining musical practice.

I haven't read Huygens (apart from what Barbour says about him), but this
seems a curious justification. Given that the minor seventh of the dominant
seventh chord resolves inward and the augmented sixth of the German
augmented sixth chord resolves outward, wouldn't one expect that, ideally,
the augmented sixth to be a larger interval than the minor seventh? (If,
indeed, one considers the seventh harmonic to be in need of resolution at
all.)

Bill

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^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
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