Pitch of complex tones: reproducing Houtsma, Rossing & Wagenaars

Listen for the moment a click turns into a tone. Each of the three series plays bursts of one tone lasting 1, 2, 4 and on up to 128 periods: first at 300 Hz, then 1000 Hz, then 3000 Hz. The booklet asks how many periods it takes to establish a sense of pitch, and describes the shortest bursts as clicks that take on a pitch as they lengthen.

Houtsma, Rossing & Wagenaars, Demonstration 13, after Doughty & Garner (1948). Bursts of 1, 2, 4, 8, 16, 32, 64 and 128 periods of 300, 1000 and 3000 Hz, as in the booklet, each starting and ending at a zero crossing of a sine with no ramps. Ours: the bursts start 0.6 s apart, with 1 s between the series, which are played once where the disc plays them twice.

Listen for the pair that sounds like a true octave. A 500 Hz tone alternates with a second tone that steps up from 985 to 1035 Hz, 5 Hz at a time, so that the exact octave, 1000 Hz, is the fourth pair. The booklet expects most listeners to choose a tone somewhere around 1010 Hz, an octave a little wider than 2 to 1.

Houtsma, Rossing & Wagenaars, Demonstration 15, after Ward (1954). A 500 Hz tone of 1 s alternating with a 1 s tone from 985 to 1035 Hz in 5 Hz steps, as in the booklet. Ours: equal levels, 20 ms ramps, 0.2 s between the tones of a pair and 0.5 s between pairs, and one pass where the disc plays two.

Listen for whether the pitch changes as the tone loses its lowest harmonics, one at a time. The timbre gets thinner and brighter at each step; the booklet asks whether the pitch changes too. A soft low rumble plays throughout.

Houtsma, Rossing & Wagenaars, Demonstration 20. Ten harmonics of 200 Hz at equal amplitude, first all ten, then without the fundamental, then without the lowest two, and so on, as in the booklet, which adds noise below 300 Hz at -10 dB to mask any 200 Hz difference tone that distortion in the playback might make. Ours: the steps go on until harmonics 7 to 10 are left; the tones are 0.8 s long, 1.2 s apart, in sine phase with 20 ms ramps; and the noise's -10 dB is taken relative to the full ten-harmonic tone.

Listen for a low pitch that rises a little at each step. Three partials 200 Hz apart, 800, 1000 and 1200 Hz, all move up 20 Hz at a time until they are 1000, 1200 and 1400 Hz. Each partial rises by 200 Hz in all, yet the booklet expects the low pitch heard to rise only from 200 Hz to about 241 Hz, a nearly common factor of the three, and notes that a second or even third rising pitch may be heard as well. A difference tone made by distortion would stay at 200 Hz, the spacing.

Houtsma, Rossing & Wagenaars, Demonstration 21, after Schouten, Ritsma & Cardozo (1962). Three partials, equal in level, 0.5 s long, shifted up in ten 20 Hz steps, as in the booklet. Ours: 20 ms ramps, a pause of 0.25 s between tones, and one pass where the disc plays two.

Listen for the pitch of each three-partial tone against the harmonic tone after it. First 800, 1000 and 1200 Hz, then a harmonic tone on 200 Hz; then 850, 1050 and 1250 Hz, then a harmonic tone on 210 Hz. The booklet reports the low pitches of the two three-partial tones well matched by the two harmonic tones, although the second set is not harmonic and its spacing is still 200 Hz.

Houtsma, Rossing & Wagenaars, Demonstration 21, its second part. The partials and fundamentals are the booklet's. Ours: the matching tones hold harmonics 1 to 10 at equal amplitude, every tone is 0.5 s long, and the pauses are 0.25 s within a pair and 0.75 s between pairs.

Listen for which way the pitch goes from the first tone of each pair to the second: up or down? In each pair, 800 and 1000 Hz are followed by 750 and 1000 Hz. The booklet explains that listening to the separate partials, you hear one go down, from 800 to 750 Hz, while listening to the whole, you hear the missing fundamental go up a major third, from 200 to 250 Hz. A soft noise plays behind them.

Houtsma, Rossing & Wagenaars, Demonstration 25, after Smoorenburg (1970). Two-tone complexes of 800 and 1000 Hz, then 750 and 1000 Hz, the pair played four times against a background of noise, as in the booklet. Ours: tones 0.5 s long with 20 ms ramps, 0.1 s apart, a pair every 1.5 s, and the noise, from 100 to 5000 Hz at 20 dB below one tone.

Listen for the tune of the Westminster chime twice over, in pairs: each note first as a pure tone, then as a tone of three high harmonics with the same pitch. The two versions carry the same melody, though the second has no energy at any of the notes' frequencies.

Houtsma, Rossing & Wagenaars, Demonstration 22. Each note is a pure tone, then a complex of its 4th, 5th and 6th harmonics, as in the booklet. The notes are ours: the first two changes of the Westminster Quarters, G#4 F#4 E4 B3 and E4 G#4 F#4 B3, in equal temperament (247 to 415 Hz), each tone 0.4 s long with 20 ms ramps, 0.1 s apart within a pair, a pair every 1.1 s. The complex tones have the same RMS as the pure tones.

Listen for the same pairs under a low noise, which covers the pure tones. The booklet reports that the pitches of the complex tones are still heard, so the melody is still there, although the noise covers the frequencies where those pitches lie.

Houtsma, Rossing & Wagenaars, Demonstration 22, its second part: low-pass noise added. The booklet does not give the noise; here it runs from 20 to 600 Hz, below the lowest harmonic of the complex tones (988 Hz), at 15 dB above a pure tone's RMS, and is ours.

Listen for the same pairs under a high noise, which covers the harmonics of the complex tones. The booklet reports that the pure-tone melody is still heard.

Houtsma, Rossing & Wagenaars, Demonstration 22, its third part: high-pass noise added. Ours: the noise runs from 800 to 5000 Hz, above the highest pure tone, at 20 dB above a pure tone's RMS.

Listen for the same tune, now with each note made of three successive harmonics chosen at random from the second to the sixth. The timbre changes from note to note; the booklet reports that the melody stays clearly recognizable, which shows that it is not followed by tracking any one harmonic from note to note.

Houtsma, Rossing & Wagenaars, Demonstration 23, after Houtsma (1984). Three successive harmonics of each note, with harmonic numbers between 2 and 6, as in the booklet. Ours: the tune and timing (0.4 s notes, one every 0.5 s), the random draws (seed 231), equal amplitudes and 20 ms ramps.

The three figures share one frequency axis. Above each note are its harmonic numbers, and the dashed line is its fundamental, which is not in the sound.

Listen for the tune again, with harmonics drawn from the fifth to the ninth.

Houtsma, Rossing & Wagenaars, Demonstration 23, its second part: harmonic numbers between 5 and 9. Ours as before (seed 232).

Listen for the tune once more, with harmonics drawn from the eighth to the twelfth: the highest and most crowded together of the three.

Houtsma, Rossing & Wagenaars, Demonstration 23, its third part: harmonic numbers between 8 and 12. Ours as before (seed 233).

Listen for a scale rising five octaves, played by pairs of clicks. Within each pair the second click repeats the first after a delay, and the delay shrinks from note to note; the booklet gives the pitch heard as one over the delay.

Houtsma, Rossing & Wagenaars, Demonstration 26, its first part. A diatonic scale of five octaves played with pairs of identical pulses whose delay goes from 15 ms (a pitch of 67 Hz) down. The booklet gives the last delay as 0.48 ms; this scale, in equal temperament, ends at 15/32 = 0.47 ms. Ours: the scale's tuning; the pairs repeat regularly, 25 a second, which is too slow to give a pitch of its own; each note lasts 0.25 s; and the clicks are band-limited, so that delays between samples are exact.

Listen for a scale rising four octaves, from click pairs that now come at random times. The booklet reports that the pitch at one over the delay is heard as before, though the pairs no longer come regularly.

Houtsma, Rossing & Wagenaars, Demonstration 26, its second part. A diatonic scale of four octaves; the pairs start at the times of a Poisson process, and the delay goes from 15 ms to 15/16 = 0.94 ms (the booklet gives 0.95 ms). Ours: 100 pairs a second on average, each note 0.25 s.

Listen for the same four-octave scale played by bursts of hiss, each added to a copy of itself delayed by the same amounts.

Houtsma, Rossing & Wagenaars, Demonstration 26, its third part: bursts of white noise echoed 15 ms to 0.94 ms later. Ours: 0.25 s bursts with 10 ms ramps, and the delay made exact by a phase shift in the frequency domain.

Listen for a steady tone, a second steady tone, then the two together, rising and falling in loudness four times a second: beats.

Houtsma, Rossing & Wagenaars, Demonstration 32, its first part. Pure tones of 1000 and 1004 Hz, first separately and then together, as in the booklet, which gives beats at 4 Hz. Ours: equal levels, 2 s each alone and 4 s together, 20 ms ramps and pauses of 0.5 s; one pass where the disc plays two.

Listen for beats again, four a second, in three pairs of tones that are far apart: a little more than an octave, a fifth and a fourth. The booklet invites you to compare how strong the beats are with those of the first part, where the two tones were 4 Hz apart.

Houtsma, Rossing & Wagenaars, Demonstration 32, its second part. A 1000 Hz tone with 2004, 1502 and 1334.67 Hz in turn, each mistuned from 2/1, 3/2 and 4/3 so that the second-order beats come 4 times a second, as in the booklet. Ours: equal levels, 2.7 s each, 20 ms ramps, pauses of 0.5 s.

Listen for a third, softer tone joining two steady ones, and beating slowly, four times a second. The two tones are 1000 and 1200 Hz; the third is 804 Hz. Nothing in the sound is at 800 Hz, but the booklet reports a combination tone made in the ear at 2 x 1000 - 1200 = 800 Hz, which the 804 Hz probe beats with. Combination tones are louder at higher levels; the booklet notes that at about 80 dB they are heard over a wide range.

Houtsma, Rossing & Wagenaars, Demonstration 34, after Goldstein (1967). Tones of 1000 and 1200 Hz, then an 804 Hz probe added, as in the booklet. Its text calls 800 Hz the quadratic difference tone; it is the cubic one, 2f1 - f2 (the quadratic one, f2 - f1, is at 200 Hz). Ours: equal levels for the two tones, the probe 20 dB below them, 2 s of the two tones before the probe joins for 4 s, and 20 ms ramps.

Listen for a low tone moving down while the upper of the two tones rises, from 1200 to 1600 Hz and back. The booklet describes the cubic difference tone, 2f1 - f2, moving opposite to the upper tone, soon joined by the quadratic one, f2 - f1, which rises with it; the two should cross when the upper tone is at 1500 Hz, where both are at 500 Hz. Neither is in the sound.

Houtsma, Rossing & Wagenaars, Demonstration 34, its second part. A steady 1000 Hz tone with a second tone gliding from 1200 to 1600 Hz and back, as in the booklet, which says only that it moves slowly. Ours: 8 s up and 8 s down in a straight line, equal levels, 20 ms ramps.