4.6 Questions for the reader

4.6 Questions for the reader#

Exercise 15

Reading a score. Consider the score pq.Score([(0.0, {"pitch": "C4", "duration": 2.0}), (1.0, {"pitch": "E4", "duration": 2.0})]). At time \(t = 1.5\) seconds, how many notes are sounding, and which? Explain why, referring to each event’s onset time and duration.

Reveal solution

Two notes: C4 (sounding on \([0, 2)\)) and E4 (sounding on \([1, 3)\)).

Exercise 16

Pitches and frequencies. Using the relationship that raising a pitch by \(k\) steps multiplies its frequency by \(2^{k/12}\), and the anchor A4 \(= 440\) Hz \(=\) MIDI pitch \(69\):

  1. What is the frequency of A5, one octave above A4?

  2. A4 and E5 are \(7\) steps apart. What is E5’s frequency (to the nearest Hz)?

  3. Two tones are \(19\) steps apart. By what factor do their frequencies differ?

Reveal solution

  1. \(880\) Hz.

  2. \(440 \cdot 2^{7/12} \approx 659\) Hz.

  3. \(2^{19/12} \approx 3.0\).

Exercise 17

Timbre or score? You synthesize four simultaneous sustained sinusoids at \(200\), \(400\), \(600\), and \(800\) Hz. Are you more likely to hear a single fused tone or four separate tones? What if the frequencies were \(200\), \(283\), \(327\), and \(412\) Hz instead? Justify your answers in terms of harmonic relationships.

Reveal solution

\(200, 400, 600, 800\) Hz are all harmonics of \(200\) Hz and fuse into one tone. \(200, 283, 327, 412\) Hz are inharmonic and are heard as four separate tones.

Exercise 18

Envelope values. An attack/decay envelope has control points \((0, 0)\), \((0.2, 1)\), and \((0.5, 0)\), where each control point is time / amplitude pairs \((t_i, a_i)\). What is the envelope’s value at \(t = 0.1\) s? At \(t = 0.35\) s? At \(t = 0.8\) s?

Reveal solution

\(0.5\) at \(t = 0.1\) s; \(0.5\) at \(t = 0.35\) s; \(0\) at \(t = 0.8\) s.

Exercise 19

Designing an ADSR envelope. Many synthesizers use a four-parameter attack-decay-sustain-release (ADSR) envelope, where attack \(A\), decay \(D\), and release \(R\) are durations (in seconds) and sustain \(S\) is a level (in \([0, 1]\)). The envelope rises from 0 to a peak of 1.0 over \(A\), falls from 1.0 to the level \(S\) over \(D\), holds at \(S\) for some sustain duration, then falls from \(S\) to 0 over \(R\). Write down a set of control points \((t_i, a_i)\) that implement an ADSR envelope with \(A = 0.05\) s, \(D = 0.1\) s, \(S = 0.7\), a 0.5 s sustain, and \(R = 0.2\) s.

Reveal solution

\((0, 0),\ (0.05, 1),\ (0.15, 0.7),\ (0.65, 0.7),\ (0.85, 0)\).

Exercise 20

Reading a topology. A synthesis patch is built from four unit generators with the following “spec”:

  • Generator \(A\) takes a single input \(X\).

  • Generator \(B\) takes two inputs: \(Y\) and the output of \(A\).

  • Generator \(C\) takes two inputs: the output of \(A\) and \(Z\).

  • Generator \(D\) takes two inputs, the outputs of \(B\) and \(C\), and produces the final output.

  1. Write this topology as a single nested function-call expression of the form \(D(\ldots)\).

  2. Why does the output of \(A\) appear twice in your expression, and what does that tell you about how many times \(A\) must be computed?

Reveal solution

  1. \(D\big(B(Y, A(X)),\ C(A(X), Z)\big)\).

  2. \(A(X)\) appears twice, so \(A\) must be computed twice unless its output is computed once and reused.

Exercise 21

Block-based bookkeeping. You synthesize 5 seconds of audio at \(f_s = 44{,}100\) Hz using a network of \(M = 3\) unit generators, processed block-by-block and ugen-by-ugen with a block size of \(B = 512\) samples.

  1. How many blocks are processed?

  2. How many total unit-generator calls are made?

  3. Compute the call count for the sample-by-sample strategy instead of ugen-by-ugen.

Reveal solution

  1. \(431\) blocks (or \(430\) if you drop incomplete blocks)

  2. \(1293\) calls (ugen-by-ugen)

  3. \(661{,}500\) calls (sample-by-sample)

Exercise 22

Choosing a block size. Suppose you halve the block size \(B\). What happens to (1) the peak memory used and (2) the total function-call overhead? Though we haven’t yet discussed real-time computer music systems, why might such systems benefit from block-based computing, and why might a small block size be preferred there despite the overhead?

Reveal solution

Peak memory roughly halves; total function-call overhead roughly doubles.