88CET Ear Training CDs, Part 6

88CET Notation
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The other important aspect of 88CET to learn a bit about in order to
understand the exercises on my 88CET ear-training CDs, is my notation
system. This is one of those non-auditory, "intellectual", memorization
exercises I mentioned earlier. As powerful as it is to be able to conjure
up the sound of any 88CET interval in your mind at will, you still can't
tell what E up to G sounds like without the strictly intellectual knowledge
of what the interval between those two notes is!

Without an octave to organize a tuning around, you have to choose some
other interval. Two intervals stand out as most likely candidates, because
they are very intuitively basic to our ears, and because you can build
interesting scales around them. The first such interval is the 3:2 perfect
fifth, and the second is the 7:4 subminor seventh. As you can see from the
earlier-presented table of frequency-ratio approximations, in 88CET tuning
8 steps forms a perfect fifth, and 11 steps is a subminor seventh.

Since this series is not basically about 88CET Music Theory, I'll just
say that using those two intervals to form a cyclic scale pattern of
2122121 (2 meaning two 88CET steps - a small whole step, and 1 meaning a
half step). This 2122121 pattern repeat in 7:4s rather than octaves,
meaning that two As are not octaves 7:4s apart. The pattern also maps
nicely to a traditional piano (or MIDI) keyboard if you imagine away the
G#/Ab key, and leave a half-step between G and A:

A# C# Eb Gb
A B C D E F G A

On my 88CET MIDI instruments, I make the G#/Ab key play the same pitch as
the adjacent A key, but it doesn't really matter, since I never use that
key anyway.

Perhaps skipping over the G#/Ab key seems odd, but this structure is
really a lot more intutive, audibly speaking, than not skipping the G#/Ab
key. The entire 7+5 structure of a traditional keyboard corresponds to a
pattern of tritones wrapping within a span of a not-very-accurate neutral
seventh (11:6). Perhaps you'll just have to listen to those two pairs of
intervals for yourself, but when you do, I think you'll have to agree that
3:2 and 7:4 are much more intuitive, simple, and meaningful to our ears
than 10:7 and 11:6!

Once you establish this 2122121 pattern, you have a system of
"pseudokeys", each with its pseudokey signature. Then again, they're not
like any key signature you've probably ever seen before though! Although
these pseudokeys don't necessarily suggest a tonality (thus "pseudo"keys
rather than a true key), it turns out that they can be used that way. When
you do use them to suggest a tonality, you promptly find that related keys,
even in as such a nontraditional tuning, are still fifths apart. Since
this scale pattern is cyclic in fifths, you find that related keys have
pseudokey signatures that differ by a single sharp or flat.

By "higher bits" I meant the higher precision (finer tuning) bits, so
all but one of the remaining bits will always be correct after truncation.
It's that highest bit that can be compromised if you don't round off,
putting you one tuning step away from the closest possible value.

For example tuning an 8/5 calls for a pitch bend of +561 (13.7c) which
QT would truncate to +544 (32*17), giving you 13.28c. If you rounded
the bend up to +576 (32*18) yourself, you'd get 14.06c which is slightly
closer to your goal. So the worst that truncation can do to you is one
tuning step, or 0.78c in QT.

I don't bother rounding off bends in MIDI files since they're intended
to be played on different machines with different PB precisions, but
if you're targeting one particular synthesizer it might be useful.

Ken Wauchope


>>I'm not a MAX user, but assuming that xbendouts are the same as MIDI
>>pitch bend messages (+-8192 = 14 bits), you can round off to the
>>nearest multiple of 32 for best results since I imagine QT just
>>truncates the higher bits.

>You mean I have to recalculate by myself?
>Or does QT automatically take me to the nearest value?
>In other word ... does higher bit does the 'more' fine tunings?...I mean
>when 5 bits truncated ... the remaining 9 is not "not correct" but just
>robust?...hope so...