5 - Fundamentals of Data Representation

Musical Instrument Digital Interface

• A MIDI file is a more compact representation than an equivalent sampled recording. • The performance data be easily manipulated, for example, it’s easy to change instruments or change the pitch or duration of a note. • MIDI supports a very wide range of instruments providing more choices for music production.

Music is represented as a sequence of MIDI event messages. Examples of data that might be contained in a message are channel, volume and note envelope. MIDI messages are usually two or three bytes long and 16 channels are supported. Advantages: • More compact representation • Easy to modify the music, such as change the instrument • Musical score can be generated directly from a MIDI file

Align the binary numbers and perform column addition. eg.0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 0 + ────────── 1 0 1 1 0 1 1 1 ────────── ¹¹

Align the binary numbers and perform long multiplication. eg. 110 11 11x ─────── 110 11 110 11 0+ ───────── 1 0 1 0 0 0 1 ───────── ¹¹¹¹¹

A method of working with signed numbers using fixed-width binary where the most significant (left most) bit is the sign bit.

If the number is negative, place a 1 under the MSB then add a 1 under the column headings which, when added to the value of the MSB, give the number. eg. -128 + 16 = -112 - 112 + 8 = -104 - 104 + 4 = -100 - 100 + 2 = -98 - 100 + 1 = -97 -128 64 32 16 8 4 2 1 1 0 0 1 1 1 1 1 -97₁₀ → 10011111₂

Add up the column heading with a 1 under them (where the MSB is negative). eg. -128 64 32 16 8 4 2 1 1 0 0 1 1 0 1 0 -128 + 16 + 8 + 2 = -102 10011010₂ → -102₁₀

Turn the value that is being subtracted into a negative, then add the two values. eg. 0101₂ - 0011₂ → 0101₂ + (-0011₂) → 0101₂ + 1101₂ → (1)0010₂ a carry is generated which is ignored 0101₂ - 0011₂ → (1)0010₂

the range is -2ⁿ⁻¹ to 2ⁿ⁻¹ - 1

Fixed point binary is a method of representing numbers with a fractional part, where the binary point is in a fixed position.

Set up the correct column headings according to where the point is positioned. Then add up the values of the column headings with a 1 under them. eg. 8 4 2 1 . ½ ¼ ⅛ ¹⁄₁₆ 0 1 1 0 . 0 1 1 0 4 + 2 + ¼ + ⅛ = 6.375₁₀ 0110.0110₂ → 6.375₁₀

eg. 10.6875 - 8 = 2.6875 2. 6875 - 2 = 0.6875 0. 8675 - ½ = 0.1875 0. 1875 - ⅛ = 0.0625 0. 0625 - ¹⁄₁₆ = 0 8 4 2 1 . ½ ¼ ⅛ ¹⁄₁₆ 1 0 1 0 . 1 0 1 1 10.6875₁₀ → 1010.1011₂

Convert the exponent and move the binary point accordingly (adding preceding 1s for negative mantissa and 0s for positive mantissa for negative exponents). Then convert to decimal. eg. exponent → 4 0. 100100100 → 01001.00100 → 9.125

Convert to fixed point binary move the binary point so that it’s normalised (when the first two bits are different) and convert the exponent to binary. eg. -16 8 4 2 1 . ½ ¼ ⅛ 0 1 1 0 0 . 0 0 1 01100.001 → 0.1100001 exponent = 4 → 000100 → 01100001 000100

approximate value - original value |

Absolute error ÷ original value Then converted to a percentage

• Maximises precision for a given number of bits * There is a unique representation of each number

Move the binary point so that the first two bits are different. Then adjust the exponent to reflect the movement of the binary point. eg. 0.0001011 → 0.1011 old exponent = 5 new exponent = 5 - 3 = 2 → 0010 → 01011000 0010

storage requirement (in bits) = sampling rate × seconds × sample resolution

Binary represents numbers using 2 digits, 0 and 1. Binary is base 2.

Hexadecimal represents numbers using 16 digits, the numbers 0 - 9 and the letters A - F. Hexadecimal is base 16.

Decimal represents numbers using 10 digits, 0 - 9. Decimal is base 10.

Set up the column headings and move from left to right putting a 1 under the combination of numbers that make up the decimal value. eg. 119 - 64 = 55 55 - 32 = 23 23 - 16 = 7 7 - 4 = 3 3 - 2 = 1 1 - 1 = 0 1286432168421 01110111 119₁₀→01110111₂