Snoman R. Dance Music Manual: Tools, Toys, and Techniques
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485
485
Binary and Hex
All hardware circuits, even computers, are based on a series of inter-
connected switches that can be in one of two states: either on or off. By con-
tinually switching these on and off in different confi gurations it’s possible for
the hardware to count and perform complex calculations. This switching is
accomplished by sending a series of bytes down an electrical cable. If it is equal
to 1 then the switch is turned on while if it’s a 0 then the switch is turned off.
Because of this, all computers must count using base 2, or ‘binary’ rather than
our usual method of counting which is base 10.
COUNTING IN BINARY
Our normal method of counting was developed because we have ten fi gures.
Using this system, any one digit in a number has a value that is based upon the
position of the digit in the number. Thus, when we are working to the power of
10, every time a digit moves one to the left its result is to the power of 10, then
by 100, then by 1000 and so forth. Consequently, the number 17 593 could be
seen as follows.
10000000 1000000 100000 10000 1000 100 10 1
0 0 0 1 7 5 9 3
Or (1 ⴛ 10000) ⴙ (7 ⴛ 1000) ⴙ (5 ⴛ 100) ⴙ (9 ⴛ 10) ⴙ (3 ⴛ 1) ⴝ 17 360
From this above example we can see that each position in a number essentially
adds a 0 to the meaning of a digit. Or, the value of each position is ten times
the value of the previous position (moving from right to left). This method
of counting is also implemented in binary, but rather than use a base 10 sys-
tem, we use a base 2 system. So, rather than saying that 10 to the power of
0 ⫽ 1, 10 to the power of 1 ⫽ 10, 10 to the power of 2 ⫽ 100, 10 to the power
of 3 ⫽ 1000, it’s 2 to the power of 0, 2 to the power of 1, 2 to the power of 2, 2
to the power of 4 and so forth.
128 64 32 16 8 4 2 1
0 0 0 0 0 0 0 0
Appendix A
APPENDIX A
Binary and Hex
486
Looking at the above table we are assuming that there are eight hardware switches
grouped together producing what is essentially an 8-bit byte. Also, as they are all
switched off, the sum produced will be zero. However, if we were to introduce
some positive bits we could sum them together to produce a decimal number.
128 64 32 16 8 4 2 1
0 1 1 0 1 1 0 1
(0 ⴛ 128) ⴙ (1 ⴛ 64) ⴙ (1 ⴛ 32) ⴙ (0 ⴛ 16) ⴙ (1 ⴛ 8) ⴙ
(1 ⴛ 4) ⴙ (0 ⴛ 2) ⴙ (1 ⴛ 1) ⴝ 109
Thus the decimal equivalent of the binary 01101101 would be 109. Additionally,
we could also determine that the maximum number that could be calculated
via binary would be 255 (11111111).
As MIDI uses this same 8-bit system when communicating with any devices,
it would seem sensible to assume that it should be able to offer this same
maximum parameter (of 255) but this isn’t the case. Similar to CC messages,
two forms of information need to be transmitted; a status bit and a data byte
(which is composed of 7 bits). The status bit informs the synth of an incoming
message that is arriving, while the following data byte informs it by how much
the parameter should be adjusted. Because of this initial status bit, only 7 other
bits are left to provide the information, resulting in a maximum decimal value
of 127; hence, the maximum number of any CC message can only be 127. To
transmit numbers larger than this, an 8-bit byte has to be split into two halves
and then converted into another numeration format, hexadecimal.
When we split a byte into two halves, both halves are commonly referred to
as ‘nibbles’. If the previous example were to be broken down into two nibbles,
it would become 0110 and 1101. These could then be individually summed
together as 0110 ⫽ 96 and 1101 ⫽ 13 (96 ⫹ 13 ⫽ 109) to produce the result
again. However, by splitting a byte into two and then converting it into a hexa-
decimal value, it’s possible to produce much higher values. The reason behind
this is that hexadecimal works to base-16, meaning that it is possible to access
up to 16 383 parameters in a synth, much more than the standard 127 offered
through CC messages.
COUNTING IN HEXADECIMAL
Hex uses a base-16 numbering system, but as there are not enough symbols to
represent 16 different digits, as soon as the number 10 is reached it has to be
converted into letters. Thus to represent number 10–15 the letters A–F are used.
Recall how we count in decimal. Counting upwards, we count from 1 through
9 and then we place a 1 to the left of this and go back to 0, making the number
10 (which actually means 1 to the power of 10 plus 0 to the power of 1). With
Binary and Hex
APPENDIX A
487
hexadecimal we count beyond 9, but because our entire numeric system is only
based on 9 digits, with hexadecimal, letters are used instead as below so that
we can count up to 15.
F E D C B A 9 8 7 6 5 4 3 2 1 0
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
When we count the zero, just as we do with the decimal counting system, we
have used a total of 16 numbers. You can see from the decimal to hexadecimal
conversion table that whenever we count up to F, increase the number to the
left by 1. This is also the same as the more common decimal counting system,
for example, if you count from 11 through 19, there are no more digits to use
to you change the 9 to 0 and increase the number to the left by 1, giving us 20.
You can see that hexadecimal counting is exactly the same as decimal, except
that in hexadecimal we count up to F before increasing the number to the left
by a factor of 1.
It follows that in decimal, the number 24 actually means ‘2 ⫻ 10 to the power
of 1 ⫹ 4 ⫻ 10 to the power of 0 ⫽ 24’, which is what you or I already under-
stand it to be because we use this every day like counting on our fi ngers.
In hexadecimal, the number 24 actually means ‘2 ⫻ 16 to the power of
1 ⫹ 4 ⫻ 16 to the power of 0 ⫽ 36’.
Logically it follows then that the number CE in hexadecimal means ‘12 ⫻ 16
to the power of 1 ⫹ 14 ⫻ 16 to the power of 0 ⫽ 206 ’ in decimal.
Using this principle, it’s easy to convert hexadecimal into decimal.
Using this method, it’s possible to count until we reach FF, which essentially
means 15 ⫻ 16 ⫹ 15 units or 255 in decimal. Creating a new position gives
100 or 16 to the power of 2 which is 256 in decimal. Continuing to count
using this base system it would be possible to produce two nibbles, both with
a value of 7F resulting in a total of 16 383. (127 ⫻ 128 ⫹ 127 ⫽ 16383!).
For instance, suppose we wished to convert the decimal number 12720 to
hexadecimal:
■ The largest power of 16 that fi ts into 12720 is 16
3
⫽ 4096
It fi ts 3 times and gives a remainder of 432. This number is derived from
the fact that
(3 ⴛ 4096) ⴝ 12288
(12720 ⴚ 12288) ⴝ 432
The fi rst hex number is 3
APPENDIX A
Binary and Hex
488
■ The largest power of 16 that fi ts into 432 is 16
2
⫽ 256
It fi ts 1 time and gives a remainder of 176. This number is derived from
the fact that
(1 ⴛ 256) ⴝ 256
(432 ⴚ 256) ⴝ 176
The second hex number is 1
■ The largest power of 16 that fi ts into 176 is 16
1
⫽ 16
It fi ts 11 times with no remainder (this remainder equates to 1 to the
power of nothing which is nothing and will make 0 the last digit in the
hex number). The third hex number is therefore B
■ The hex equivalent of the decimal number 12720 is therefore: 31B0
To clarify, to convert 31B0 hexadecimal into decimal the maths is as follows:
0 ⫻ (16 to the power of nothing) ⫽ 0
B (which is equivalent to 11 in decimal) ⫻ (16 to the power of 1) ⫽ 176
1 ⫻ (16 to the power of 2) ⫽ 256
3 ⫻ (16 to the power of 3) ⫽ 12288
0 ⫹ 176 ⫹ 256 ⫹ 12288 ⫽ 12720 in decimal
As another example, suppose we wanted to convert the decimal number 14683
■ The largest power of 16 that fi ts into 14683 is 16
3
⫽ 4096
It fi ts 3 times and gives a remainder of 2395
■ This remainder is derived from the fact that
(3 ⴛ 4096) ⴝ 12288
(14683 ⴚ 12288) ⴝ 2395
The fi rst hex number is 3
■ The largest power of 16 that fi ts into 2395 is 16
2
⫽ 256
It fi ts 9 times and gives a remainder of 91
■ This remainder is derived from the fact that
(9 ⴛ 256) ⴝ 2304
(2395 ⴚ 2304) ⴝ 91
■ The second hex number is 9
■ The largest power of 16 that fi ts into 91 is 16
1
⫽ 1
It fi ts 5 times and gives the remainder of 11
■ This remainder is derived from the fact that
(5 ⴛ 16) ⴝ 80
(91 – 80) ⴝ 11
■ The third hex number is 5
Binary and Hex
APPENDIX A
489
Again we can check this by converting 295B hexadecimal into decimal. The math is as
follows:
B ⫻ (16 to the power of nothing) ⫽ 11
5 ⫻ (16 to the power of 1) ⫽ 80
9 ⫻ (16 to the power of 2) ⫽ 2304
3 ⫻ (16 to the power of 3) ⫽ 12288
11 ⫹ 80 ⫹ 2304 ⫹ 12288 ⫽ 14683 in decimal
■ The remainder of 11 has no power of 16 that will divide into it as a
whole number so the last digit of the hex will be the hex equivalent of
11 which is B. This gives the fi nal hex number which is therefore 395B.
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491
491
Decimal to Hexadecimal
Conversion Table
Dec Hex Dec Hex Dec Hex Dec Hex Dec Hex Dec Hex
00 19 13 38 26 57 39 76 4C 95 5F
11 20 14 39 27 58 3A 77 4D 96 60
22 21 15 40 28 59 3B 78 4E 97 61
33 22 16 41 29 60 3C 79 4F 98 62
44 23 17 42 2A 61 3D 80 50 99 63
55 24 18 43 2B 62 3E 81 51 100 64
66 25 19 44 2C 63 3F 82 52 101 65
77 26 1A 45 2D 64 40 83 53 102 66
88 27 1B 46 2E 65 41 84 54 103 67
99 28 1C 47 2F 66 42 85 55 104 68
10 A 29 1D 48 30 67 43 86 56 105 69
11 B 30 1E 49 31 68 44 87 57 106 6A
12 C 31 1F 50 32 69 45 88 58 107 6B
13 D 32 20 51 33 70 46 89 59 108 6C
14 E 33 21 52 34 71 47 90 5A 109 6D
15 F 34 22 53 35 72 48 91 5B 110 6E
16 10 35 23 54 36 73 49 92 5C 111 6F
17 11 36 24 55 37 74 4A 93 5D 112 70
18 12 37 25 56 38 75 4B 94 5E 113 71
Appendix B
APPENDIX B
Decimal to Hexadecimal Conversion Table
492
Dec Hex Dec Hex Dec Hex Dec Hex Dec Hex Dec Hex
114 72 139 8B 164 A4 189 BD 214 D6 239 EF
115 73 140 8C 165 A5 190 BE 215 D7 240 F0
116 74 141 8D 166 A6 191 BF 216 D8 241 F1
117 75 142 8E 167 A7 192 C0 217 D9 242 F2
118 76 143 8F 168 A8 193 C1 218 DA 243 F3
119 77 144 90 169 A9 194 C2 219 DB 244 F4
120 78 145 91 170 AA 195 C3 220 DC 245 F5
121 79 146 92 171 AB 196 C4 221 DD 246 F6
122 7A 147 93 172 AC 197 C5 222 DE 247 F7
123 7B 148 94 173 AD 198 C6 223 DF 248 F8
124 7C 149 95 174 AE 199 C7 224 E0 249 F9
125 7D 150 96 175 AF 200 C8 225 E1 250 FA
126 7E 151 97 176 B0 201 C9 226 E2 251 FB
127 7F 152 98 177 B1 202 CA 227 E3 252 FC
128 80 153 99 178 B2 203 CB 228 E4 253 FD
129 81 154 9A 179 B3 204 CC 229 E5 254 FE
130 82 155 9B 180 B4 205 CD 230 E6 255 FF
131 83 156 9C 181 B5 206 CE 231 E7
132 84 157 9D 182 B6 207 CF 232 E8
133 85 158 9E 183 B7 208 D0 233 E9
134 86 159 9F 184 B8 209 D1 234 EA
135 87 160 A0 185 B9 210 D2 235 EB
136 88 161 A1 186 BA 211 D3 236 EC
137 89 162 A2 187 BB 212 D4 237 ED
138 8A 163 A3 188 BC 213 D5 238 EE
493
493
General MIDI Instrument
Patch Maps
Appendix C
Programme
Nos.
Instrument Set
(Piano)
Programme
Nos.
Instrument
(Chromatic Perc)
1 Acoustic Grand 9 Celesta
2 Bright Acoustic 10 Glockenspiel
3 Electric Grand 11 Music Box
4 Honky Tonk 12 Vibraphone
5 Electric Piano 1 13 Marimba
6 Electric Piano 2 14 Xylophone
7 Harpsichord 15 Tubular Bells
8 Clav 16 Dulcimer
Programme
Nos.
Instrument (Organ) Programme
Nos.
Instrument (Guitar)
17 Drawbar Organ 25 Nylon Acoustic Guitar
18 Percussive Organ 26 Steel Acoustic Guitar
19 Rock Organ 27 Jazz Electric Guitar
20 Church Organ 28 Clean Electric Guitar
21 Reed Organ 29 Muted Electric Guitar
22 Accordian 30 Overdrive Guitar
23 Harmonica 31 Distortion Guitar
24 Tango Accordian 32 Guitar Harmonics