Physics of Music - Notes

Why G# and Ab are (not) the same: a summary

When comparing two notes, the ear most easily picks up "beats." These are easiest to hear for approximate unisons, relatively easily for octaves, and after that, for fifths. For pure tones (sine waves, which have no overtones) the comparison for all but the unison can be difficult. For sounds with significant overtone content, such as from the violin or oboe, the comparisons are quite easy, even for most non-musicians.

For any tone which is constant in time there will be an underlying "fundamental frequency" (the reciprocal of the repeat time) and overtones which will be harmonic -- that is integer multiples of the fundamental frequency. These harmonic overtones repeat exactly 2, 3, 4, or more times during the repeat time of the fundamental. When you hear beats you are generally hearing the slowest beats found when comparing all of the harmonics of each tone. That is, if you compare a violin playing a middle C at 262.5 Hz and an oboe playing a G at 393.0 Hz, you can compute the beat frequency you will hear by looking for the smallest differences between any of the harmonics (up to about 8 harmonics, that is). In this case:
     Violin harmonics: 262.5, 525.0, 787.5, 1050.0, ...
     Oboe harmonics: 393.0, 786.0, 1179.0, ...
and so you will hear slow beats at (787.5-786.0) = 1.5 Hz.

In general, it seems, people do not like to hear these beats. Also, if you need to use harmonics larger than about the 8th to find a close match, most people have considerable trouble hearing those beats even for instruments with significant overtone content. Hence, only smaller integer multiples are relevant. (It is somewhat interesting that we often hear performers using vibrato, sometimes to great extent, which puts beats back into the music as ornamentation, but that is beyond the presentation here).

When considering what musical notes to include in a scale it is then quite natural to first include notes related by intervals which do not have beats. The intervals which do not have beats will have fundamental frequencies related by rational numbers (one integer divided by another). Of particular interest will be frequencies related by ratios of relatively small integers. In the example above, the fundamental frequency of the oboe is close to 3/2 times that of the violin. If the oboe (or violin) adjusts so that the 1.5 Hz beats move toward 0 Hz, then the frequency ratio moves toward exactly 3/2.

Some "nice" intervals have frequency ratios as given in the following table with their usual name.

IntervalFrequency Ratio
Unison1:1
Octave2:1
Fifth3:2
Fourth4:3
Major Third5:4
Minor Third6:5
Major Sixth5:3
The names for the intervals are historical and are derived from the position of those notes in our major and minor scales. Additional notes can be derived from these by considering intervals on top of these intervals (e.g. a fifth above a third, for example).

If we start by defining a single note, say middle-C (an arbitrary choice), by specifying its frequency (say 262 Hz), then all the other notes of the C scale (where we are using the usual naming scheme) are found using the ratios above. This is called "just" or "harmonic" tuning. (More commonly one defines the A as 440 Hz, and works backwards, but that is not important here. See here for more info on scales.) The C major (just) scale would then include the following notes:

Note NameInterval from CFrequency
CUnison262.0 Hz (defined)
EMajor Third327.5 Hz
FFourth349.33 Hz
GFifth393.00 Hz
AMajor Sixth436.67
C'Octave524.0 Hz

The C major scale has (by definition from history) no sharps or flats. The naming convention usually used and as illustrated by the "circle of fifths" is that, starting from C major, going "up" by fifths you should add sharps to the major scale and going "down" by fifths you add flats. Thus, for example, the E major scale has as its major third the note G#. On the other hand, the Ab major scale has as its major third, the note C' (or C, etc.). Thus, starting with the C major scale defined above, one concludes that one should have
     G# = (5/4) x 325.0 Hz = 406.25 Hz
     Ab = (4/5) x 524.0 Hz = 419.2 Hz
which are quite noticeably different. The question one should really ask is, why do we make them be the same?

The argument above should be taken only as an argument showing that G# and Ab should be different, but the specific difference in the example above rests on several specific assumptions and definitions. Any changes in those assumptions will change the specific results, though not the general conclusion. While the details will vary, one will generally find that to satisfy all the possible intervals, for all scales using a small number of notes, is just plain impossible.

In addition to the numerical arguments above there are also aesthetic arguments. Even the issue of which note should be higher in frequency, G# or Ab, has been argued both ways. To review those arguments, please see the references below.

In order to keep the number of notes on a keyboard manageable, one resorts to tempering -- that is making "small" adjustments and compromises to all these note frequencies for the sake of convenience. A scheme which results is called a "temperament."

Equal temperament has been known for centuries though was not (nearly) universally adopted until the early 1900's. Just prior to that most used some form of "mean tone" or "well-tempered" scale, which provided for less of a compromise for scales with a smaller number of sharps and flats, and more of a compromise for the others. Equal temperament divides the octave into "equal intervals." Here by "equal" we mean equal multipliers (which are equally spaced on a logarithmic scale).

If you divide the octave into N notes, the equal tempered interval between notes is the number which when raised to the N-th power equals 2. Mathematically this would be called "the N-th root of 2." Our normal chromatic scale has N = 12 and the 12-th root of 2 is 1.059463..., an irrational number. Since rational numbers and irrational numbers can never be equal, equal temperament will always be out of tune (there will be beats). If you want to use an equal tempered scale which closely matches the just intervals above labeled 3rd, 4th, and especially the 5th, then N = 12 happens to work reasonably well. The next higher value of N which works much better is above 30. Having over 30 keys on a keyboard per octave is just too many so we compromise and use 12. A comparison between the equal temperated scale and the just scale can be found here.

One other equal tempered-like tuning scheme suggested in the past effectively had N > 50 but only a fraction of the notes (frequencies) would actually be used. In that scheme each whole tone is spaced as we do with our 12 note chromatic scale (two equal tempered half-steps or a factor of 1.122462... in frequency). The whole tone is then divided into 9 equally spaced intervals with notes placed at 4/9 and 5/9 of the way (again, in the multiplicative sense). This was actually implemented on keyboards at one point in history using split black keys (divided forward and back). While more difficult to manage, this was a better compromise in terms of minimizing the beats while allowing most musical keys to be used equally well. It was apparently not very convenient to play and has not survived.

Hence we use only one whole tone and one division within the whole tone, forcing G# and Ab to be the same, because (1) the convenience of an equal tempered scale is attractive, (2) N = 12 happens to work well for the most important intervals and other choices are unmanageable, and (3) N = 12 includes only one interval between whole tones.

It should be understood that any scheme which uses a manageable number of fixed value frequencies (such as is found on a keyboard) to attempt to produce beat-free intervals in all musical keys, will be a compromise. Having a sharp and flat be different may be less of a compromise than having them the same, but it will still be a compromise.

The scheme we use is certainly not universal. While influenced by the scales of Greek's "Western music," Turkish music has a virtual continuum of notes within the whole step, with 6 included in the notation (3 kinds of flats and 3 kinds of sharps, all of which are different).

Recommended References:

"How Equal Temperament Ruined Harmony," R. W. Duffin (W. W. Norton & Co., NY, 2007).

J. Wild, "The computation behind consonance and dissonance," Interdisciplinary Science Reviews, Vol 27, No. 4, p 299 (2002).

H. Helmholtz, "On the sensation of tone as a physiological basis for the theory of music," translated by A. J. Ellis, (Dover, NY, 1954).

"Tuning: containing the perfection of eighteenth-century temperament, the lost art of nineteenth-century temperament, and the science of equal temperament, complete with instructions for aural and electronic tuning," by Owen H. Jorgensen, Michigan State University Press, East Lansing, MI 1991.


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