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Definitions of tuning terms
© 1998 by Joseph L. Monzo
All definitions by Joe Monzo unless otherwise cited
hexamu, 6mu AND 768-edo
The unit which i call a tetradekamu
is specified in the MIDI spec (1983)
as the smallest increment available
for the pitch-bend controller, and is the
frequency data format for MTS (1999).
However, as of 2003, almost all MIDI hardware, including
both keyboards and other "regular" musical instruments
and most computer soundcards, ignores the less
significant byte of the two MIDI pitch-bend data bytes,
thus limiting tuning resolution to only 7 bits.
Because the pitch-bend protocol uses the most significant
of these bits only to designate the frequency of the
12edo MIDI-note which lies
as close as possible to the center of the pitch-bend
range (namely, 1/2-unit above the exact center), in effect
making this bit merely a flag which indicates the sign
showing the direction of the pitch-bend, and
because the smallest total pitch-bend range from this
center is +/- 100 cents, the effective maximum resolution
for all of this hardware is 6 bits per Semitone.
At the setting for hexamu pitch-bend resolution, a
semitone is
divided into 26 = 64 pitch-bend units.
Thus there are 64 * 12 = 768 hexamus in an
"octave",
so the hexamu measurement system may be thought of
as 768-EDO
tuning, with a hexamu being one
degree
of 768-EDO.
Below is an illustration of exactly how this works.
Therefore the hexamu gives a range of possible values from
0 to +/- 63 [decimal] = 00 to +/- 3F [hex].
For practical use in tuning MIDI-files, an interval's
semitone value
must first be calculated. The nearest
integer
semitone is translated into a MIDI note-number
(which can generally also be described by letter-name
plus optional accidental: A, Bb, C#, etc., followed
by an "octave"
register-number). Then the remainder or deficit
is converted into hexamus plus or minus, respectively.
Below is a table showing the exact cents-values
for all hexamus in one Semitone.
See also:
[from Joe Monzo, JustMusic: A New Harmony]
updated:
I had previously suggested the name
'midipu' for the 14-bit resolution
of the MTS (MIDI tuning standard). In July 2003, Aaron Hunt suggested
the general name 'Mu' for 'MIDI unit', and suggested that
Greek prefixes be used to specify the number of bits resulting
in the unit. Prefixes were offered and agreed upon by Aaron,
myself, Gene Ward Smith and John Chalmers.
The internal data structure of the hexamu requires
one byte, with the first two bits reserved as flags,
one to indicate the byte's status as data, and one
to indicate the sign (+ or -) showing the direction of the
pitch-bend up or down, as follows:
let "d" designate the bits that cannot be used
because it is reserved for the SysEx flag, to
indicate that this is a byte of pitch-bend data.
let "s" designate the bit that represents the
sign of the pitch-bend data, + or - .
the hexamu spec thus uses a total of 2+4 = 6 bits.
thus, the maximum possible value is:
ds11 1111 [binary]
= +/- 3 F [hex]
= +/- 63 [decimal]
note that the data value of the first nibble can only be 0, 1, 2, or 3.
The "x" represents the status flag at the
beginning of the byte, and is not recognized
as part of the tuning resolution.
x 64 32 16 8 4 2 1 -- decimal value
x 1 0 0 0 0 0 0 -- bits
= 64 decimal = the plain MIDI-note, 0 cents deviation from 12edo.
x 64 32 16 8 4 2 1 -- decimal value
x 1 0 0 0 0 0 1 -- bits
= 65 decimal = one unit (1.5625 cents) above the 12edo MIDI-note.
x 64 32 16 8 4 2 1 -- decimal value
x 0 1 1 1 1 1 1 -- bits
= 63 decimal = one unit (1.5625 cents) below the 12edo MIDI-note.
A hexamu is calculated as the 768th root of 2, or 2(1/768),
with a ratio of approximately 1:1.000902943.
It is an irrational number, but is
extremely close to the ratio 2217:2215
( 31 5-1 443-1 7391 ):
the difference is ~ 1/70,000 of a
cent, which makes them for
all intents and purposes identical.
A hexamu is
The formula for calculating the hexamu-value of any ratio
is:
hexamus = log10(ratio) * [ (26 * 12) / log10(2)]
------- cents ------
6mus decimal fraction
0 0.0000 0
1 1.5625 1 & 9/16
2 3.1250 3 & 2/16
3 4.6875 4 & 11/16
4 6.2500 6 & 4/16
5 7.8125 7 & 13/16
6 9.3750 9 & 6/16
7 10.9375 10 & 15/16
8 12.5000 12 & 8/16
9 14.0625 14 & 1/16
10 15.6250 15 & 10/16
11 17.1875 17 & 3/16
12 18.7500 18 & 12/16
13 20.3125 20 & 5/16
14 21.8750 21 & 14/16
15 23.4375 23 & 7/16
16 25.0000 25
17 26.5625 26 & 9/16
18 28.1250 28 & 2/16
19 29.6875 29 & 11/16
20 31.2500 31 & 4/16
21 32.8125 32 & 13/16
22 34.3750 34 & 6/16
23 35.9375 35 & 15/16
24 37.5000 37 & 8/16
25 39.0625 39 & 1/16
26 40.6250 40 & 10/16
27 42.1875 42 & 3/16
28 43.7500 43 & 12/16
29 45.3125 45 & 5/16
30 46.8750 46 & 14/16
31 48.4375 48 & 7/16
32 50.0000 50
33 51.5625 51 & 9/16
34 53.1250 53 & 2/16
35 54.6875 54 & 11/16
36 56.2500 56 & 4/16
37 57.8125 57 & 13/16
38 59.3750 59 & 6/16
39 60.9375 60 & 15/16
40 62.5000 62 & 8/16
41 64.0625 64 & 1/16
42 65.6250 65 & 10/16
43 67.1875 67 & 3/16
44 68.7500 68 & 12/16
45 70.3125 70 & 5/16
46 71.8750 71 & 14/16
47 73.4375 73 & 7/16
48 75.0000 75
49 76.5625 76 & 9/16
50 78.1250 78 & 2/16
51 79.6875 79 & 11/16
52 81.2500 81 & 4/16
53 82.8125 82 & 13/16
54 84.3750 84 & 6/16
55 85.9375 85 & 15/16
56 87.5000 87 & 8/16
57 89.0625 89 & 1/16
58 90.6250 90 & 10/16
59 92.1875 92 & 3/16
60 93.7500 93 & 12/16
61 95.3125 95 & 5/16
62 96.8750 96 & 14/16
63 98.4375 98 & 7/16
64 100.0000 100
MIDI tuning units
enamu
doamu
triamu
tetramu
pentamu
heptamu
oktamu
enneamu
dekamu
endekamu
dodekamu
tridekamu
tetradekamu
cawapu
midipu
my Gentle
Introduction to the MIDI Tuning Specification
the Official
MIDI Tuning Specification.
2003.07.04 -- page created
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