6 The system allows composers to utilize any key and expect the same amount of consonance and dissonance from each. This also means that pieces can be freely transposed and each interval will remain identical. Furthermore, like all standards, equal temperament becomes even more beneficial when widely used. Widespread usage of equal temperament allows ensembles of varied instrumentations to play together with relative ease. Despite this widespread usage, it is plagued with one fundamental flaw: the only interval that is acoustically in tune is the octave.

The concept of acoustically pure intervals is based on the harmonic series, a collection of pitches or harmonics produced simultaneously by a vibrating string or air column.^{7} The string or air column's ability to vibrate not only as a whole but in halves, thirds, fourths, and so on creates the various harmonics of the series.^{8} The harmonics extend infinitely and each is present in varying degrees in every musical note. Table 1 shows the first 9 harmonics of an A pitched at 110hz.

Table 1
**Partial Length Note Frequency Interval**
1 1 A 110hz Fundamental
2 1/2 A 220 Octave
3 1/3 E 330 Octave + Perfect fifth
4 1/4 A 440 Two octaves
5 1/5 C# 550 Two octaves + Major third
6 1/6 E 660 Two octaves + Perfect fifth
7 1/7 Gb 770 Two octaves + Minor seventh
8 1/8 A 880 Three octaves
9 1/9 B 990 Three octaves + Major second

When this A is sounded with an A pitched at 220hz, one octave above it, the first partial of the first A coincides perfectly with the second A and the interval is said to be pure. Other intervals have matching overtones and can be made pure by tuning until the frequency of the overtones match. For example, the partials of an A pitched at 110hz and of an E pitched at 165hz coincide at 330hz.^{9} If one of the pitches is not tuned correctly, however, and the overtones do not coincide perfectly, beating^{10} occurs and the interval sounds out of tune.

The era of equal temperament's dominance is merely the latest chapter in the tuning debate that has lasted over 2 millennia. The ancient Greek scholar Pythagoras, in his experiments with vibrating strings, discovered the frequency ratios of the first three overtones of the harmonic series. Pythagoras found that the ratio of a pitch to its octave is 1:2, the ratio of a pitch to its perfect fifth is 2:3, and the ratio of a pitch to its perfect fourth is 3:4. The Greeks considered this 1:2:3:4 series to be holy,^{11} and Pythagoras is believed to have put forth a tuning system based completely on perfect fifths. To achieve this system, now known as Pythagorean Tuning, one begins with a fundamental pitch and tunes a perfect fifth (702 cents) above it. Using this as the new fundamental, the process is repeated 12 times and a fundamental seven octaves above the original is attained.^{12} Twelve successive perfect fifths, however, are equal to 8424 cents and 7 successive octaves are equal to 8400 cents. Essentially, the perfect fifths do not fit into the octaves, and the second fundamental is 24 cents sharp to the original--nearly an eighth of a step off and noticeably out of tune. This 24 cent difference is known as the Pythagorean comma, or ditonic comma.

Clearly, something must be done about this discrepancy. If the pitch of the original fundamental is retained, the final fifth is equivalent to a perfect fifth minus the ditonic comma, or 488 cents. This fifth is unbearably flat and unusable, especially in a scale where the fifths are designed to sound perfectly consonant. In the time of Pythagoras, however, the holy 2:3 ratio of the perfect fifth was to be upheld. Thus, the troublesome fifth, known as "the wolf"^{13} or simply the wolf fifth, was placed between rarely used notes, usually G# and Eb, and composers merely avoided it.

In Pythagorean tuning, major thirds are particularly wide because of the purity of the fifths and fourths. To see this, find a major third by choosing a fundamental pitch, measuring four perfect fifths above it, and subtracting two octaves.^{14} Four perfect fifths is equal to 2808 cents, and two 1200 cent octaves subtracted leaves 408 cents, 22 cents sharp of a pure 386 cent major third. This discrepancy is known as the syntonic comma and causes Pythagorean thirds to beat noticeably. This comma can also be thought of as "the amount by which four pure fourths and a pure major third fall short of two octaves."^{15}

Despite their inaccuracy, Pythagorean thirds were not particularly troublesome in the medieval styles dominated by Pythagorean tuning. The Pythagorean scale is melodically pleasant and lends itself well to monophonic chant of the era. The system also held up to early experiments with simple organum, based mostly on fourths and fifths. Moreover, when thirds and sixths are used, their active nature creates an intriguing contrast with the perfection of the fourths and fifths. In "Pythagorean Tuning and Medieval Polyphony," Margo Schulter writes:

In addition to presenting fifths and fourths in their ideal just ratios, Pythagorean tuning makes mildly unstable major thirds (81:64) and minor thirds (32:27)^{16} somewhat more active or tense. In a style where these intervals represents points of instability and motion standing in contrast to stable fifths and fourths, this extra bit of tension may be seen not only as a tolerable compromise, but indeed as an expressive nuance.^{17}

When performing repertoire of this era, Pythagorean tuning is an obvious choice and will recreate the color of the original composition. Great care should be taken to achieve the holy perfection of fifths and fourths as well as wide, active major thirds when they appear.

By the 15th century, the purity of the third had overtaken that of the fourth and fifth in importance, mainly because composers began to use the third much more frequently and the wide major thirds of Pythagorean tuning finally became troublesome. Since four perfect fifths are equal to two octaves and a major third, it follows that altering the size of the fifth will alter the size of the third.^{18} This is the driving force behind the most common tuning systems of the Renaissance, mean-tone temperaments.^{19} Mean-tone temperaments became important to performance on keyboard instruments, which lacked flexibility of intonation during performance but on which composers desired sweeter thirds.^{20}

If four perfect fifths exceed an octave and a pure major third by the amount known as the syntonic comma, then the fifth can be decreased by 1/4 of the comma in order to make the major third pure.^{21} This method of tempering is known as 1/4 syntonic comma mean-tone temperament. In this system, 5.5 cents (1/4 of the syntonic comma) are subtracted from eleven of the perfect fifths and a wolf fifth is created to fill the gap left in the circle.^{22}

The downside to 1/4 comma mean-tone is that only two out of every three major thirds are pure; the other is unusable.^{23} Three consecutive major thirds should span an octave, but three pure major thirds fall short by 42 cents and one major third of 428 cents^{24} for every two pure major thirds must be used to compensate for the discrepancy.^{25} The eight pure thirds out of twelve are centered at the top of the circle of fifths, and composers did not stray far from them to avoid the "jarring dissonance"^{26} of the distant keys.

Another common mean-tone temperament is 1/6 comma mean-tone temperament, a "practical compromise between [1/4 comma mean-tone temperament] and [equal temperament]."^{27} It contains purer fifths, tempered by only 1/6 of the syntonic comma. This results in major thirds that are not pure, but remain more in tune than their equally tempered counterparts.^{28} Slightly impure thirds allow 1/6 comma mean-tone to sound good in a greater variety of keys; some distant keys, however, remain unusable.

When played in a suitable key, mean-tone temperaments sound more harmonious on keyboard instruments than today's standard of equal temperament.^{29} This fact, when combined with its usability in a greater variety of keys, makes 1/6 comma mean-tone the most common choice of early music groups who wish to perform or record with a historically accurate unequal temperament.^{30}

### Their Role in Modern Performance and Recording

Improvements made by newly invented tunings bring with them corresponding losses,^{31} and in the case of equal temperament, versatility of key and instrumentation was achieved at the cost of perfectly consonant intervals. This versatility, however, is not always needed. When performing or recording alone, one does not need to conform to a standard tuning system. The vast majority of, if not all, previous tuning systems attain perfection for a portion of their intervals, and this fact can be used to the performer's advantage when performing or recording specific repertoire. The use of historically accurate tuning systems adds a degree of authenticity not achievable with equal temperament and the use of such tunings should always be considered.

### Endnotes

^{1} David Dolata, "An Introduction to Tuning and Temperament," 2000, 1.

^{2} Owen H. Jorgenson, Tuning (East Lansing:Michigan State University Press, 1991), 15.

^{3} Jorgenson, 15.

^{4} Since the correct string length ratios of the equally tempered half step were then known, the correct fret positions could be calculated.

^{5} Jorgenson, 15.

^{6} Dolata, Abstract.

^{7} Instruments that utilize vibrating air columns for sound production include brass instruments, woodwinds, and pipe organs.

^{8} Guy Oldham, "Harmonics," The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie (London: Macmillan, 1980), vol. 8, 165.

^{9} Their partials also coincide at 660hz and elsewhere, but the coincidence at 330hz is the most obvious.

^{10} In Tuning, Jorgenson defines beats as "a phenomenon heard as waverings, flutterings, pulsations, or a vibrato... Opposite frequency phases of the non-coinciding (conflicting) harmonics cause periodic cancellations of the tones of the harmonics." The further the overtones are from coincidence, the more they beat and the more the interval sounds out of tune.

^{11} Paul Guy, "Tuning and Temperament," http://home.swipnet.se/~w-37192/eng/handbook/Tuning/history.html (1999), accessed 27 October 2002.

^{12} Using C as the fundamental, this yields C G D A E B F# C# G#/Ab Eb Bb F C

^{13} Named so because its beating was compared to the howling of a wolf. Wolf intervals plagued many early tunings.

^{14} C - G - D' - A' - E'' = C - C' - C'' - E''

^{15} Dolata, 5. C - F - Bb - Eb' - Ab' - C'' = C - C' - C''

^{16} A notation for naming the two string lengths of an interval. The first number is the string length of the fundamental, and the second number is the string length of the other note in the interval.
^{17} Margo Schulter, "Pythagorean Tuning and Medieval Polyphony," http://www.medieval.org/emfaq/harmony/pyth.html (10 June 1998), accessed 27 October 2002.

^{18} Altering the size of an interval to affect the size of another interval is known as "tempering" the interval. Since the size of the fifth remained perfect in Pythagorean tuning, the system is not considered a temperament.

^{19} Mean-tone temperaments get their name from their whole tone, which is exactly half of the pure major third.

^{20} Mark Lindley, "Mean-tone," The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie (London: Macmillan, 1980), vol. 11, 875.

^{21} (702 - 5.5) * 4 - 2400 cents = 386 cents

^{22} (702 - 5.5 cents) * 11 = 7661.5 cents. This falls short of 7 octaves by 738.5 cents, the wolf fifth. This fifth was usually placed where it would rarely be used, such as between G# and Eb, between C# and Ab, or between D# and Bb.

^{23} Edward Foote, notes to Enid Katahn, Beethoven In The Temperaments--Historical Tunings on the Modern Concert Grand (1997), CD, Gasparo GSCD-332.

^{24} 386 + 386 + 428 cents = 1200 cents

^{25} This discrepancy is known as the lesser diesis.

^{26} Foote.

^{27} Dolata, 15. Equal temperament is also known as 1/11 syntonic comma mean-tone temperament because the fifths are each tempered by 1/11 of the syntonic comma (2 cents).

^{28} (702 - 3+2/3) * 4 - 2400 cents = 393+1/3 cents. This third is only 7+1/3 cents sharp of pure, compared to 14 cents sharp of pure in equal temperament.

^{29} Foote.

^{30} Dolata, 15.

^{31} The basis of tuning systems is that not all intervals of a scale can be made pure, so the purity of certain intervals is gained through sacrificing the purity of others. In equal temperament, the purity of all intervals is sacrificed in order to gain the equal spacing of tones within the octave.

### Bibliography

Barbour, J. Murray. Tuning and Temperament: A Historical Survey. New York: Da Capo Press, 1972.

Dolata, David. "An Introduction to Tuning and Temperament." 2000.

Foote, Edward. Notes to Enid Katahn, Beethoven In The Temperaments--Historical Tunings on the Modern Concert Grand. CD, Gasparo GSCD-332, 1997.

Guy, Paul. "Tuning and Temperament." http://home.swipnet.se/~w-37192/eng/handbook/Tuning/history.html. 1999; accessed 27 October 2002.

Jorgenson, Owen H. Tuning: Containing the Perfection of Eighteenth-Century Temperament; The Lost Art of Nineteenth Century Temperament; and The Science of equal temperament. East Lansing: Michigan State University Press, 1991.

Lindey, Mark. "Mean-tone." The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. London: Macmillan, 1980. Vol. 11, 875.

Lindey, Mark. "Temperaments." The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. London: Macmillan, 1980. Vol. 18, 165-66.

Oldham, Guy. "Harmonics." The New Grove Dictionary of Music and Musicians, ed. Stanley Sadie. London: Macmillan, 1980. Vol. 8, 660-64.

Schulter, Margo. "Pythagorean Tuning and Medieval Polyphony." http://www.medieval.org/emfaq/harmony/pyth.html. 10 June 1998; accessed 27 October 2002.