Tartini tones, consonance, temperament, harmony and scales
Consonance and temperament in music are related to Tartini tones. In this background page to the multimedia chapter Interference, we look at consonance and the Tartini tones it produces. Simple consonances pose difficulties for keyboard instruments and lead to problems in tuning thirds and in the circle of fourths or circle of fifths. The approximations that deal with these problems are called temperaments.
We saw in
the chapter Interference that simple musical consonance is produced by notes whose frequencies are in the ratios of small integers. Here, using sine waves, we illustrate musical intervals called the just major third (ratio 5:4), the perfect fourth (4:3) and the perfect fifth (3:2).
Nomenclature and details. To make the arithmetic simple, I've tuned the note G to 400 Hz and used it to introduce these topics n
the chapter Interference. In the key of G, a major scale has the notes G, A, B, C, D, E, F#, G. The third note in that scale is B, so the major third is the musical interval G-B, the fourth G-C and so on. So, while the bottom note remains fixed on 400 Hz, the higher note goes from 500 to 533 to 600 Hz and back again. Most people agree that these are harmonious or consonant intervals: they go together well. Try it and see whether you agree.
The next step really requires either headphones or a good speaker system. We want you to turn the sound up loud and listen to them again. If you play it loud enough, you will probably hear the Tartini tone – a faint buzzing tone at low pitch, with a frequency equal to that of the difference between the two higher tones. We've shown the pitch of these Tartini tones in small notes in parentheses on the bass clef. If you don't hear them, it may be that your sound isn't loud enough, or it may be that you are not susceptible to this auditory illusion*. If you do hear them, we hasten to point out that these tones are not present in the sound file. If your sound system is good, they are not present in the sound, either. They are created in your head by nonlinear interference, either in your ear or in your auditory processing centre. (See Interference beats and Tartini tones and Linear and nonlinear superposition for more examples and background.)
* I use the word 'illusion' to stress that, for example, when you hear listen to pure tones at 200 Hz and 300 Hz together producing a Tartini tone with pitch corresponding to a pure tone at 100 Hz, there is no acoustic power at 100 Hz. It is true that three cycles of 300 Hz takes the same time as two cycles of 200 Hz, so the pattern repeats every 10 ms. What is interesting is that, from those two pure tones, your ear (or suitable software or hardware) can create the impression of a tone at 100 Hz. In a nice analogy that also involves non-existent frequency components, it's interesting to refer to the colour yellow, as produced on a computer monitor or television, as a visual illusion, in the sense that no yellow light is ever emitted from those screens. See this site for details.) If you would prefer to call a Tartini tone a real note (or computer yellow a real colour), that's fine.
Consonance example: recorder notes
In this example, I use two descant recorders to provide notes. The lower note is the G an octave higher than that of the sine wave example above, and on the other I play B, C and D, also an octave higher. The intervals are no longer exactly
in the simple ratios 5:4, 4:3 and 3:2, because I can't control the pitch of both of them independently. Consequently, you may hear the Tartini tones to be somewhat out of tune.
Again you'll need to play it loudly to hear the Tartini tones.
These instruments are cheap to buy, so it's an easy experiment to do yourself. In this case, it's very obvious that the low sound of the Tartini tone cannot possibly be produced by these very high pitched instruments.
Tuning with Tartini tones
To return to our first example: if we have a major third to play, such as G-B, the Tartini tone will fall exactly two octaves below the G — provided that the frequency ratio is 5:4 or 1.25. In the example below, we begin with 504 and 400 Hz, a ratio of 1.26 and then reduce it to 500 and 400 Hz, a ratio of 1.25.
If you play this softly, it might sound like a chord that changes slightly half way through, both of them reasonably pleasant. If you played it loud, however, you have probably heard the Tartini tone go from two thirds of a semitone sharp (104 Hz) down to being exactly two octaves below the G (100 Hz) — aah. Musicians often use Tartini tones to tune chords: if the two flutes or oboes in an orchestra are playing a sustained high G and B, then they need to tune it so that the Tartini tone is a G (in tune with that of a lower instrument, such as a clarinet or bassoon sitting behind them).
Problems with just intervals
thirds, with ratio 5:4, are very pure and their Tartini tones are exactly in tune. But they can pose a problem in music that involves modulation into different keys. Let's play the first three notes in a major scale in G major: G A B. Then we use that B to play the first three notes in the scale of B major: B C# D#. Then we use that D# to play the first three notes in the scale of D# major: D# E# Fx, where Fx means F double sharp.
The sound file will convince you that, with this just intonation, F double sharp is not the same note as G. Alternatively, if you said (quite reasonably) that there is no key of D# major, then we could have used E flat, F and G as the final third, and proved that E flat and D# are not the same note in just tuning.
Equal temperament and keyboards
The observation we have just made is fine for instruments like flutes, violins etc that can make fine adustments to pitches as they go. However, it poses a big problem for keyboards: on most keyboard instruments, we don't have a choice between D sharp and E flat, or between F double sharp and G, they must be played by pushing the same
The interval between adjacent keys (white-white for B-C and E-F, white-black and black-white for all others) is called a semitone. On nearly all pianos, the semitones are tuned with an equal ratio – let's call it r. After ascending 12 semitones, we have covered an octave, which in pure tuning has a frequency ratio of 2:1 exactly * . So, for twelve equal semitones to make one octave, we require that r multiplied by itself twelve times gives two, i.e. that
r * r * r * r * r * r * r * r * r * r * r * r = r12 = 2 so
r = 21/12 = 1.059
So the equal tempered semitone has a frequency ratio of 1.059 or a frequency increase of 5.9%.
Incidentally, guitar makers sometimes use the 'rule of 18' to position the frets on an equal tempered fingerboard. If you reduce the length of an ideal string by 1/18, you increase its frequency by 18/17, which is 1.059. However, guitar strings are not ideal in the physical sense, because of their finite bending stiffness, especially if they are made of solid steel. For this reason, while the rule of 18 may be used to make the fingerboard, the bridge is usually set further from the nut than twice the length to the octave fret.
Fifths and fourths in equal temperament are not a big problem: From the keyboard, we see that there are five semitones in a fourth and seven in a fifth. Using a calculator, we see that
25/12 = 1.335 and 27/12 = 1.498
These values are very close to those of the just intervals: 4:3 = 1.333 and 3:2 = 1.500. So in equal temperament, the fourth is very slightly wider than a just fourth, and the fifth is very slightly narrow.
This doesn't pose serious problems for harmony in itself, but it does pose problems for the
Circle of fourths or circle of fifths
Let's ascend in fourths, starting from C: To start, we have C F Bb Eb Ab Db Gb. But on a piano, Gb = F#. So we continue F# B E A D G C. So, on a piano
C-F-Bb-Eb-Ab-Db-F#-B-E-A-D-G-C = five octaves = 25 = 32
So twelve just fourths are not quite the circle, by 1.3% or a quarter of a semitone. Similarly, twelve just fifths is larger than 7 octaves.
Usually, this is not a problem, because one rarely uses the complete circle in any given musical context. Some composers do, of course, just for the challenge. In fact I've written an orchestral piece called Circle of Fourths that uses this chord as a central idea.
* On pianos, the octave is very slightly more than 2:1 at the very low and very high ends of the keyboard, but near the middle of the keyboard it is usually very close to 2:1. But that is another story, and has to do with non-ideal strings and their inharmonic resonances.
Thirds in equal temperament (ET)
The difference between the just and ET fourth (or fifth) is small enough (0.1%) not to cause much problem. The
ET major third is 0.8% wide (an eighth of a semitone), which is more serious, but can be disguised. Here, for example, is a set of ET major thirds, in which A flat is exactly equal to G#.
In the familiar
piano timbre, this seems quite harmonious. One of the reasons for this is that a piano uses three strings for each note (over much of its range). The three strings are tuned to slightly different frequencies, which has two purposes. One purpose is to extend the sustain of the note: if the three strings were tuned to the same frequency, they would be struck and remain in phase, exerting a large force on the bridge and thus transferring their energy quickly to the soundboard: a loud rapidly decaying note. Tuned to three slightly different frequencies, their relative phases change and they exert a smaller force on the bridge, so energy is not lost so quickly: a note that begins loudly but decays quickly to a softer sound with a long sustain.
The second purpose of the unequal tuning of the string triplets is that the beating within a note disguises the beating due to the ET tuning compromise. On a piano, an ET third sounds fine – not to mention being very familiar!
Just because it works on a piano, however, doesn't mean that it works in general. For wind and bowed string players, tuning to just intervals will sound purer than ET. Try this example, first softly, then loudly through headphones.
Listening to Tartini tones has made us aware of the musical importance of the frequency ratios 3/2, 4/3, 5/4 and others. The relationship between ratios of small integers and musically important intervals was known to Pythagoras in the sixth century BC, and probably earlier, and is one of the earliest examples of a quantitative science, or at least of a relation between numbers and the natural world. It has thus had enormous influence on culture and metaphor – and has an enormous literature, from which we only show some simple examples.
Now to the big question: how are these important in harmony? The Greek word harmonia (αρμονια) means fitting together, in the geometrical and practical sense as well as the musical one. Why do some combinations of notes fit together? Getting the Tartini tone in tune is part of the answer, but there is more.
Let's start with the octave, an interval whose importance is recognised by our note names: A3 is an octave above A2, but we call both A (specialists would say they have the same pitch chroma). The octave occurs in nearly all human singing: if a man sings a phrase, then a woman is asked to repeat it, she will usually sing it an octave higher, and we wouldn't normally call this transposing. When men and women sing a song together (say a national anthem or Happy Birthday), we usually sing in octaves. Paradoxically, the octave is so harmonious that we don't regard it as harmony: we think we are singing the same note. Similar observations can be made of many different musical cultures.
To see why the octave is so special, remember that most notes (including those of singers) are not pure sine waves but have a series of harmonics (See Spectrum, Harmonics and Timbre). To continue our example, consider the note A2 and the octave above it, A3, and look at the frequencies of their harmonics:
The harmonics of the higher note are a subset of the note an octave lower. So, when we add the higher note, we are adding no new frequencies. No wonder the octave fits in so neatly.
In many cultures, the fifth (ratio 3:2) is regarded as the next most consonant interval. If we add E3 at 110 Hz * 3/2, their harmonics are:
Here, every second harmonic of the higher note coincides with one of those of the note a fifth below, and the other harmonics are well separated, which avoids beats and another effect called interference roughness (see Beats and interference). Further, the Tartini tones are in tune: 55 Hz
is A1, an octave below A2.
Where did we get 12 semitones to the octave and 7 steps to the diatonic scale? Let's discuss this in the key of C major. Regarding the perfect fifths as very consonant, the notes G (a fifth above C) and F (a fifth below) are important notes. Calling C our tonic, G is called the dominant and F the subdominant, and these have important harmonic roles in very many simple tunes. (All of those beginning guitarists who know only three chords know a tonic, a subdominant and a dominant.)
In Western music, major and minor thirds (5:4 and 6:5 in just intonation) and sixths (about 5:3 and 8:5) have, for the last several centuries, been regarded as consonant. Thirds can be combined to give major and minor triads (three note chords) in which all notes are consonant. So, in the key of C, the triads on tonic, subdominant and dominant are C-E-G, F-A-C and G-B-D. That collection of notes gives us the C major scale: C-D-E-F-G-A-B-C.
To maintain those pure consonances, we stay with just intonation. The major scale then has notes with these frequency ratios:
As required above, the tonic, subdominant and dominant chords (the triads on do, fa and so) include the ratios 5/4 and 3/2. These chords sound very pure and consonant. The minor triad on re, however, sounds very rough in just intonation.
Strictly, this scale has three different sized steps: 9/8, 10/9 and 16/15. However, 9/8 and 10/9 differ by only 1%, so this scale is approximately diatonic: it has five big steps and two small ones. If we call the larger step a tone and the smaller a semitone, then, dividing each larger interval into two gives 12 semitone steps to the octave.
Of course, in the equal temperament approximation, all these semitones are set equal. As noted above, this makes little difference to the fourths and fifths, but a substantial difference to the thirds.
Dependence on loudness and ...
If you've followed the suggestions above, you will have noticed that the Tartini tone is much easier to hear when the interfering tones are loud. In fact, if you've been listening at low levels, you may not have heard the Tartini tone.
A consequence is that the effects listed above are much more noticeable to closely spaced singers and players than they are to a more distant audience. Does the audience in a concert of barbershop singing or original instrument performance care very much about the precision of the intonation? If they hear a mainly linear superposition, would equal temperament be good enough? Interesting questions, and I don't know the answers.
But the answers are also complicated by other effects. The performers have listened carefully and trained, and they have been long exposed to these tunings. The audience often less so. Many people say they don't care about temperament and don't notice differences. And of course people from different cultures may be familiar with very different tunings, some of which (e.g. Arabic musics) use harmonic ratios and some of which (e.g. Indonesian gamelan music) do not.
A distinction between Tartini tones and heterodyne components
It is sometimes useful to make the distinction between a heterodyne component, which is a difference or sum frequency produced from two excitation frequencies by a physical nonlinear component and which produces a frequency component in the sound wave and a pure Tartini tone, which one hears but which has no corresponding component in the sound wave. (If you like, a pure Tartini tone is a heterodyne component produced only by nonlinearities in the ear.) The difference can be experimentally subtle, but fundamentally important. Here are some examples:
Two violinists playing notes a fifth apart (one note each) will produce a pure Tartini tone that they will hear if standing close enough. So if one plays D4 on the open D string and the other plays A4 on the open A string, they will hear D3 even though the microphone and spectrum analyser show no component at the frequency of D3 (about 147 Hz),
One violinist playing a double stop fifth (e.g. again the open D4 and A4) will however produce a heterodyne difference tone: this time the sound produced will have a frequency component at about 147 Hz.
The (strong) nonlinearity in the double stop is in the stick-slip interaction between bow and string. We often say that the bridge end of a bowed string is stationary, but that is not quite true. The whole point of the bridge and body is to transmit power from the string into the air, so that end of the string is not quite stationary but is moving. In a double stop, it is moving with component frequencies from the other string.
Another example is in the sound examples on this site. For the pure Tartini examples, the signals on our site are made from two pure sine waves. If your headphones are high quality and the level is not too high, the sound wave produced will have two pure sine waves and there will be no heterodyne component in the sound, even though you will be able to hear a pure Tartini tone. If, however, you play back via computer speakers, the (considerable) nonlinear distortion in the speakers will produce a heterodyne component, and you’ll hear a stronger component at the difference frequency.
There is more on the multimedia chapter Interference and the supporting pages.