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Exploring the Psychoacoustics of Intonation

The Tone Circle Drone is a nice tool for exploring the psychophysics of tones and tuning. This page has three topics, each with demonstrations. The first topic discusses beating and roughness, which can occur with tones that are tuned closely but not exactly together. Tuning system generally have the goal of minimizing or controlling the degree of both. The second topic explores our ability to judge a true octave. We provide a test that you can take and discover how accurate your own octave sense is. Don't be surprised if your octaves are a little bit wide! This implies an important and somewhat neglected aspect of tuning to one another: for best results we should try to match pitches in the same octave. The third topic is the illustration of how complex tones can result from the merging of harmonic sine waves, and includes exercises to help develop the ability to pick out overtones. In addition to enhancing sensitivity to timbre, this skill can be very helpful for tuning when one is able to pick out the overtone that matches the octave of the reference note.

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Nearly in Tune: Beating and Roughness

Our ear is generally very good at discriminating between two different tones that are playing at the same time, at least as far as real world circumstances are concerned. But, there are limits. If two tones are close enough, our hearing mechanism and brain will merge or fuse them into the perception of a single tone.

Beating is the term that refers to a pulsing of volume (tremelo) that occurs when two tones are close but not exactly the same frequency. The combined sound waves alternately go in and out of phase. When in phase, the volumes add together to get louder. As tones go progressively out of phase, the increase in air pressure on the positive side of one wave will cancel out the decrease in pressure on the negative side of the other wave, and each other out and the result is a decrease in volume or even silence. The beating effect can range from a calm, slow pulsing to a rapid tremelousness.

Sometimes, a little beating is a desireable quality. For example, in an orchestra's string section, each player produces a slightly different frequency, and the multiple, microtonal differences and resulting complex of beating tones will contribute to a rich, lush texture. Similarly, piano notes (in the main) have three strings which are intentionally detuned a slight amount to create a richer sound. Too much beating, though, can be quite an irritating distraction.

As tones get further apart in frequency, they eventually become distinct as two separate pitches. However, there is a region of transition that has been given the term roughness. The beating becomes quite fast, climbing to a rate where the beating rate itself is reaches the low end of human frequency perception (around 20 Hz).

The rate of beating is determined by the difference in Hertz of the two tones. For example if two pitches are 1 Hz apart, they will cycle in and out of phase once per second. If they are 20 Hz apart, the volume fluctuations will occur 20 times per second. A more detailed explanation of how the two waves can combine into one beating wave can be found at HyperPhysics.

In the context of intonation systems, the usual goal is to minimize or manage the beating that occurs, and avoid roughness as much as possible. Chords that have a fast beatings are often referred to as being active and are generally not considered as desireable as chords with slower beating.

Demonstration One: Beating based on Hertz differences

Image of Drone settings for Beating Demo One

For this demonstration, open two copies of Tone Circle Drone. [You can also open the premade dronepatch: singleA.tcd in the file folder dronesets\psychoacoustics.] For each copy, set the Root to A4, and set the number of notes in the cycle to 12 in the Tones dropdown on the Control Panel. Click on the tone bring up the Note Editing Panel and set the number of beats (here, the term refers to the length of time until the note is released) to 11. The Note setting should be the root (e.g., set to I or A4, if the nomenclature is FUNCTION or ABSOLUTE, respectively). The Tuning can be any that leaves the note at 0 cents. Set the timbre to Pure Sine.

When playing the two Drones at the same time, their combined volume may be anywhere from near silent to quite loud. The volume will depend on the phase relationship between the two waves which may or may not be in phase depending on when the second tone is started.

Now, select the Options panel, and go to the Reference A field and set one Tone Cirle Drone to 440 and the other to 441. Now, when the two drones are played at the same time, we will hear the volume swing from loud to soft and back, once per second.

If you set the difference to be 2 Hz (e.g., 440 and 442) the rate of the beating will also be at 2 Hz. Try setting the differences to progressively larger numbers. At 20 Hz, the two tone should be entering the roughness stage. How far do you have to separate the two pitches before the notes become two distinct tones playing at the same time instead of a fused tone? For me, by the time we reach a difference of 50 Hz (e.g., 430 and 480) the pair have been distinct for a while and begun to resemble an emergency test tone pair--"this is only a test."

Demonstration Two: Beating based on Cents differences

For this demonstration, start out by making the same two copies of Tone Center Drone as in Demonstration One, or use two copies of the droneset: singleA.tcd

In this demo, instead of changing the pitch by using the master tuning control, we edit the cents control on the Note Editing Panel. A cent is defined as 1/100th of a semitone, and a semitone is in turn defined as 1/12th of an octave. An octave is not a fixed number of Hertz, but instead is defined by the ratio of 2:1. For example, an octave above A4 (440 Hz) is A5 equal to 2 * 440, or 880 Hz.

Now, in order to get a semitone, we might be tempted to divide the 440 Hz difference between A4 and A5 by 12, since twelve semitones make an octave. However the lower semitones would be wide (sharp) and the upper semitones would be narrow (flat). The true way to get the ratio that defines a semitone is to come up with a factor that when multiplied against any number twelve times, would result in a number twice as large as the starting number. The number that gives this result is the 12th root of 2, or approximately 1.05946309... If you have a calculator, try multiplying this number against itself 12 times. I get 1.999999901248959.

Similarly, one might be tempted to approximate a cent by dividing the difference between 1.0 and 1.05946309 by 100, but a better approximation is gotten by taking the 100th root of this number, or, the 1200th root of 2! The value I get from my PC is the approximation 1.0005777895065548. When multiplied against A4 (440) we get approximately 440.2542 Hz. As this is close to 1/4 of a Hertz difference, we would expect that when we play this note with a perfect A4, the beating would take 4 seconds for a single beat to complete. You can test this by setting the Cents setting on one of the A's to 1 and leaving the other at zero, and playing both at the same time. Does it work as expected?

Now, bump up the Root of each drone one octave to A5. Mutliplying our one cent factor against 880, we get 880.5084..., which should give us a beat length of about 2 seconds, or twice the rate of the once cent difference applied to 440 Hz. The higher the pitch, the faster the beating rate for a given number of cents difference between two tones.

Now, we have been discussing pure tones. What happens with tones that have harmonic content? As we go up the overtone series, each harmonic will have a progressively faster beating rate. The beating could well reach the roughness zone for higher harmonics. Usually though, since the harmonics are quieter than the main tone, the beating will be less audible.

Another experiment that you can do is to increase the number of cents between the two test tones. Again, if you double the difference in cents, the time it takes for one beat will halve, and vice versa.

One of the most notorious intervals in equal temperament is the Major Third. What is the rate of beating we would expect between an equal-tempered major third and a perfect ratio (5/4) major third? The integer ratio third is about 14 cents flatter than the equal tempered third. If we take our one cent value to the 14th power, the result is about 1.00811950. If we divide 440 Hz by this factor (to get the pitch that is flatter than the starting note), the result is about 435.45, or close to 4.5 beats per second. For the third comprising F4 and A4, where the clash is between the fourth harmonic of the A, and the fifth harmonic of the F, the beating rate is well into the roughness zone.

Pitch Stretching: When is an octave not an octave?

The octave is one of the most basic intervals, not only in music, but in all harmonic sounds. You might think, then, that our sense an octave would be biologically hardwired, but I have not read of any specific neuronal connections having been identified for this function, at least at the auditory cortex or prior. When tones with harmonic content are played, we have internal references (the overtones) that can be used as clues. But what happens when we are given pure tones with which to judge this interval?

Demonstration Three: Judging Pure-tone Octaves

For this demonstration, bring up five copies of the Tone Circle Drone, and load them with five presets: octavesa, octavesb, octavesc, octavesd, and octavese. Each plays a loop of two non-overlapping pure sines tuned approximately an octave apart. The octave intervals are arranged from narrowest to widest. Your task, if you'd like to take this test, is to determine which of the five is the true octave. The results may vary depending upon the root that you choose. I recommend starting with A4, 440 Hz. Once you have made your determination, open up the Note Editing Panel for the higher pitched note and check its cents setting. The true octave is the one set at 0 cents. The four decoys are set at multiples of 5 cents either sharp or flat. The octaves are all close to the perfect 2:1 ratio, so it may require a quiet environment and some close concentration to distinguish between the true and the mistuned octaves. The test can be repeated with roots at different pitch levels, though for lower roots, headphones or external speakers may be necessary as pure, sine low notes are often not audible via built-in speakers.

How did you do? If you are like most, the octaves either all sounded equally correct, or an octave was selected that was on the sharp side. It turns out that the subjective sense of an octave is often larger than a true 2:1 ratio octave. In fact, Diane Deutsch, in The Psychology of Music (2013) writes that this seems to be "universal across musical cultures" and that it reflects "a general tendency to stretch all the wider intervals." The underlying mechanism that causes this is still subject to theoretical speculation.

Still, there is an important take-away here, which is that when we tune to a reference note, it's best to match the exact frequency, not an octave of the frequency. For example, in an orchestral setting where the reference note is either the First Chair violin's or oboist's A, it's probably more reliable for a low-pitched string instrument to start by matching a harmonic. For example, a cellist might match to the octave harmonic of the A string. Alternatively, the instrumentalist might focus on mentally picking out the the matching overtone from the timbre of their instrument, and using that as a starting point for matching the target pitch.

For an instrument with a very rich timbre, particularly with strong overtones relative to a weaker (in volume) fundamental, the player may fall prey to another problematic situation if they let the overtones overly influence their pitch judgment. For example, let's say an oboist is monitoring intonation during a performance, and allows the upper harmonics of their tone to be used when matching to reference pitches. There is a danger that because the overtones are large intervals above the other instruments, that they will err on the sharp side in order to conform to the larger subjective octave, nd this will result in driving their pitch up.

Learning to hear individual overtones

In progress.