When comparing two notes, the ear will mostly pick up "beats." Beats 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 can be 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 and the interval is perceived as being "in tune."

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

Interval | Frequency Ratio |
---|---|

Unison | 1:1 |

Octave | 2:1 |

Fifth | 3:2 |

Fourth | 4:3 |

Major Third | 5:4 |

Minor Third | 6:5 |

Major Sixth | 5:3 |

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 based on this choice would then include the following notes:

Note Name | Interval from C | Frequency |
---|---|---|

C | Unison | 262.0 Hz (defined) |

E | Major Third | 327.5 Hz |

F | Fourth | 349.33 Hz |

G | Fifth | 393.00 Hz |

A | Major Sixth | 436.67 |

C' | Octave | 524.0 Hz |

The C major scale has, by definition from history, no sharps or flats.
The naming convention usually used 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 A^{b} 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

A^{b} = (4/5) x 524.0 Hz = 419.2 Hz

which are quite noticeably different. The question one really might
ask is, why do we make them be the same?

The argument above should be taken only as an argument showing that G^{#} and A^{b}
should be different, but the specific difference found in that example rest
on several specific assumptions and definitions. Any changes in those assumptions will
change the specific results, though not the general conclusion that the notes should
be different. 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.

The basic problem is much messier than just how many sub-divisions of the whole tone there should be. For example, in the above "no beat" scheme there are two different whole tone intervals: F to G is a ratio of 9/8 = 1.125 while G to A is 10/9 = 1.11111... . So, for example, the first two notes of the F major scale, F and A, would have a different spacing than the first two notes of the G major scale. That different (1.2%) is musically quite significant -- a half-step is about 6% -- and yet one does not generally see concern over the fact we commonly use equal whole steps. A small difference for a flat or sharp is actually minor in comparison.

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 A^{b}, has been argued both ways. To review
those arguments, please see the references below. There is not one correct answer.

For singers and for many instrumentalists, making small tuning adjustments while playing is normal. Those adjustments are made to make the music sound better and there is probably no mathematical calculation involved. Keyboards, however, are another matter. It is not easy to adjust the tuning of the notes as you play.

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, that is, when it is multiplied by itself N times, 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 (i.e. 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 per octave on a keyboard 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. That idea was apparently not very convenient to play and has not survived.

Hence we use only one whole tone spacing and one division within the whole tone,
forcing G^{#} and
A^{b} 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 in some specified way may be less of a compromise than having them the same, but it will still be a compromise.

The scheme described above is certainly not universal. As an example, while also influenced by the scales of Greek's "Western music," some 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). These choices for temperaments are human choices, perhaps initially motivated by the mathematics, but are certainly not defined by the mathematics.

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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