What Frequency Was Mozarts Music Used to Heal in the Renaissance or Baroque Era?

Those who set on equal temperament, the tuning of our modern pianos - equally I practice on my But Intonation Explained page - seem to be attacking the keen European musical tradition itself. After all, the music of Bach, Mozart, Beethoven, et al, was written for 12 equally-spaced pitches to the octave, right? And if nosotros alter our tuning, that music would no longer be playable equally it was intended to exist heard, right?

Expressionless wrong.

Equal temperament - the banal, equal spacing of the 12 pitches of the octave - is pretty much a 20th-century miracle. It was known about in Europe equally early as the early on 17th century, and in China much before. But it wasn't used, considering the consensus was that information technology sounded awful: out of melody and characterless. During the 19th century (for reasons nosotros'll discuss later), keyboard tuning drifted closer and closer to equal temperament over the protestation of many of the more sensitive musicians. Non until 1917 was a method devised for tuning exact equal temperament.

So how was before European music tuned? What are we missing when we hear older music played in 20th-century equal temperament?

Let'due south start with Europe's near successful tuning, if endurance can be equated with success. Meantone tuning appeared old around the late 15th century, and was used widely through the early on 18th century. In fact, information technology survived in pockets of resistance, particularly in the tuning of English language organs, all the way through the 19th century. No other tuning has survived in the west for 400 years. Let'south run into what meantone offered.

Every elegant tuning has a generating principle. The generating principle behind meantone was that it was more of import to preserve the consonance of the major thirds (C to Eastward, F to A, G to B) than it was to preserve the purity of the perfect fifths (C to 1000, F to C, K to D). There are acoustical reasons for this, namely - though I wouldn't want to become into the math involved - that the notes in a slightly out-of-tune third, existence closer together than those in a 5th, create faster and more disturbing beats than those in a slightly out-of-melody fifth. (I tin confirm this from experience with my own Steinway grand, which I keep tuned to an 18th-century tuning.) The aesthetic motivation for meantone was that composers had fallen in honey with the sweetness of the major third, and were trying to get abroad from the medieval austerity of open perfect fifths.

In a purely consonant major third, the ii strings vibrate at a frequency ratio of 5 to 4. For example, if

Or if G vibrates at 100 cycles per 2nd, and then B vibrates at 125, and so on. (If yous'd like this explained in more detail, visit my Just Intonation Explained folio.) The size of a pure 5:iv major third is 386.three cents, a cent being one 1200th of an octave, or i 100th of a half-step. Since an octave is 1200 cents, by definition, it is easy to see that three pure major thirds (3 10 386.3 cents = 1158.9) do not equal an octave. That's the whole problem of keyboard tuning, where you're express to 12 steps per octave. Where do you lot put the gaps in your chains of perfect major thirds?

A pure perfect 5th is a 3 to ii frequency ratio; if

vibrates at
440 cycles per second,

then

Due east
vibrates at
660 cycles per 2d.

A pure perfect fifth should be 702 cents wide, which is just virtually 7/12 of an octave; our electric current equal-tempered tuning accomodates perfect fifths (at 700 cents) within 2 cents, which is closer than almost people can distinguish, merely the thirds (at 400 cents) are way off, and course audible beats that are ugly one time y'all're sensitized to hear them.

Let'southward wait at the meantone solution. There was no one changeless meantone tuning; before the 20th century, tuning was an art, not a science, and each tuner had his own method of tuning according to his own gustation. The post-obit is a chart of what was initially the about mutual form of meantone, called 1/4-comma meantone, kickoff documented by Pietro Aaron in 1523, though he didn't draw it out to all twelve pitches:

Pitch:	C	C#	D	Eb	Eastward	F	F#	G	Grand#	A	A#	B	C
Cents:	0	76.0	193.2	310.3	386.three	503.four	579.five	696.eight	772.six	889.7	1006.8	1082.ix	1200

(I adjust this nautical chart, and ones following below, from an invaluable book, the bible of historical keyboard tuning: Owen Jorgensen's Tuning: Containing the Perfection of Eighteenth-Century Temperament, the Lost Fine art of Nineteenth-Century Temperament, and the Science of Equal Temperament, Michigan State University Press, 1991.) Now let'due south look at the sizes of the major thirds and perfect fifths on each pitch:

Major 3rd	Cents	Perfect 5th	Cents
C - E	386.3	C - One thousand	696.8
Db - F	427.4	Db - Ab	696.half-dozen
D - F#	386.3	D - A	696.v
Eb - G	386.five	Eb - Bb	696.5
E - G#	386.3	Eastward - B	696.6
F - A	386.three	F - C	696.half dozen
F# - A#	427.iii	F# - C#	696.v
G - B	386.1	Grand - D	696.4
Ab - C	427.iv	Ab - Eb	737.7
A - C#	386.3	A - E	696.6
Bb - D	386.4	Bb - F	696.half-dozen
B - D#	427.4	B - F#	696.vi

A major third and perfect fifth on the aforementioned pitch, of course, make upward a major triad, the most common chord in European music from 1500 to 1900 - the meantone era. Let's await at what kind of major triads we take in meantone tuning.

The major thirds that are near 386 cents wide will be sweetness, consonant, bonny. 8 pitches have virtually perfect major thirds on them - all except Db, F#, Ab, and B, whose major thirds are all virtually 427 cents. A third of 427 cents sounds like this: WAWAWAWAWAWAWAWAWA...!!! and is unusable for normal musical purposes. (Trust me on this.) All of the fifths are about 696 cents except for one, that on Ab, which is 737 cents and sounds terrible. The fifths would audio better at 702 cents, merely at 696 or 697 you lot don't actually find the deviation, particularly if the chord is filled in with that perfect major third to smoothen over the discrepancy. This is where the practice originated in European music of never having an open fifth sounding by itself without a third filling it in: the spare perfect fifth isn't quite consonant, and that fact becomes obvious if the 3rd isn't there.

And then meantone tuning gives us eight usable major triads: on C, D, Eb, E, F, Yard, A, and Bb. If you're writing a piece in meantone, those are the major triads you have available. Look through some 16th-century keyboard music: how many F#-major and Ab-major triads do you run across? Probably none, and if you lot do see some, information technology ways the composer was counting on a meantone tuning centered around some pitch other than C. If y'all desire to use I, IV, and V chords in your piece, yous tin can write in the keys of C, D, F, Chiliad, A, or Bb major. If you're writing in A major, you tin can't go to the Five/V chord (B major), because it sounds atrocious. Renaissance and early Baroque music tends to be in a few keys grouped (in the circumvolve of fifths) around C, usually C, F, G, D, Bb, or A. Ever wonder why Palestrina and Orlando Gibbons and Heinrich Schutz didn't get effectually to composing in F# major or Ab major? They couldn't, it sounded terrible in their tuning. (At that place were a few purely vocal early on works that went through triads in various keys, such as Josquin'south motet Absalon fili mi and Di Lasso'southward Prophetiae sybyllarum, the tuning and even note of which have been subjects of much 20th-century controversy.)

Before nosotros leave the subject of meantone, lets look at the available minor triads:

Minor third	Cents	Perfect Fifth	Cents
C - Eb	310.3	C - One thousand	696.8
C# - E	310.3	C# - G#	696.six
D - F	310.2	D - A	696.5
Eb - Gb	269.2	Eb - Bb	696.5
E - G	310.v	E - B	696.6
F - Ab	269.ii	F - C	696.6
F# - A	310.2	F# - C#	696.5
G - Bb	310.0	Thousand - D	696.4
M# - B	310.3	G# - D#	737.seven
A - C	310.3	A - E	696.6
Bb - Db	269.2	Bb - F	696.6
B - D	310.3	B - F#	696.vi

A pure pocket-sized third is supposed to accept a frequency ratio of 6:five. For example, if C# vibrates at 550 cycles per 2d, E should vibrate at 660. A 6:v ratio interval is 315.64 cents wide. None of the minor thirds in this meantone are quite that wide, but virtually of them are 310 cents, which is, pardon the expression, shut enough for jazz. (Actually, a narrow 7/6 minor third, often used past La Monte Young, is 266.8 cents, invitingly close to that 269; simply seven/6 is an interval that was never recognized by European theory, though used in jazz and Arabic music among others.) Therefore the pocket-size triads on C, C#, D, E, F#, G, A, and B are acceptable. (Not the one on K#, despite its OK minor third, considering it has that wildly beating fifth.) If you lot think about information technology, these triads define the relative minor of the major keys implied by the major triads to a higher place:

Major:	C	D	Eb	E	F	Chiliad	A	Bb
Pocket-size:	A	B	C	C#	D	Due east	F#	Yard

These 16 triads, eight major and 8 minor, constitute the harmonic vocabulary of Renaissance and early Baroque music. Don't believe me? Wait through a 16th- or 17th-century keyboard collection, such every bit the Fitzwilliam Virginal Book.

One important keyboard piece of work from the early 17th century (a real masterpiece, in fact) is Orlando Gibbons's Lord Salisbury Pavane. It'southward in A minor. If yous look at information technology (it's in the Historical Album of Music), Gibbons several times goes to the major triads on F, Chiliad, and C (which are in A natural minor), E (in the harmonic major), and D (not in A minor). He never, nonetheless, uses an F# major (V/ii) triad, because it doesn't really be in the tuning of his harpsichord. He does utilise, quickly, a B major triad fifty-fifty though D# doesn't exist in his scale; but considering he never uses Eb it's entirely possible that he retuned the Eb strings to D#, which would have only taken a moment on his clavichord or virginal. Had Gibbons begun in the cardinal of C minor, he would have had to write a different piece, because instead of moving from A minor to F major, he would have had to motion from C minor to Ab major, and Ab major, strictly speaking, didn't be on his harpsichord.

Here is the beginning of Orlando Gibbons's Lord Salisbury Pavane played in ane/four-comma meantone temperament:
Here is the same passage played in 12-tone equal temperament:

Considering information technology determines what sounds expert, tuning has a pervasive influence on compositional tendencies. Every piece of pitched music is the expression of a tuning. Meantone encouraged composers to apply major and minor triads, to avert open perfect fifths without thirds, and to not stray more than iii or four steps in the circumvolve of fifths away from a central primal. Renaissance and early on Baroque music played in meantone sounds seductively sweet and attractive. By playing it in modern equal temperament, nosotros do violence to its essential nature. Perhaps that'south why this repertoire is no longer often heard. It's been painted over with the ugly gray of equal temperament.

Why is it called meantone? Because it splits the difference on where to identify certain pitches. If C and E are tuned as a perfect major third of 386 cents, D should be tuned at 204 cents (ix/8) for the key of C, just at 182 cents (10/9) for the primal of D. Tuned at 193, D is correct in the middle, halfway between C and E, and halfway between the two points it needs to exist in for the various common keys; 193 is the mean between 182 and 204. Meantone temperament sacrificed the seconds, which were mainly melodic intervals rather than harmonic ones anyway, to achieve beautiful thirds.

A last point of interest: 1/iv-comma meantone is very closely approximated (as was known in the 16th century) by a scale of 31 equal steps to the octave. Ten steps of such a scale equal 387.i cents, which is a very adept major third indeed. C to C# (the chromatic half-step) volition be two such steps, and C to Db (the diatonic half-step) will exist three; in meantone, C# and Db are not the same pitch, nor are F# and Gb, etc. Mozart favored 1/6-comma meantone, which is closely approximated by a 55-stride partitioning of the octave. His C to C# was 4/55 of an octave, and C to Db was five/55. (Come across "Mozart'south Educational activity of Intonation" by John Hind Chesnut, Journal of the American Musicological Guild Vol. 30, No. 2 (Summer, 1977), pp. 254-271.)

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iii. Werckmeister III and Bach'due south Westward.T.C.

If you are or were ever a higher music student, you probably read, or were told, that Johann Sebastian Bach wrote his collection of preludes and fugues The Well-Tempered Clavier in all 24 major and pocket-size keys in gild to demonstrate equal tempered tuning.

If so, yous were misinformed.

Bach did non use equal temperament. In fact, in his twenty-four hours there was no style to tune strings to equal temperament, because there were no devices to measure frequency. They had no scientific method to achieve real equal-ness; they could only approximate.

Bach was, yet, interested in a tuning that would let him the possibility of working in all 12 keys, that did non make certain triads off-limits. He was a master of counterpoint, and chafed and fumed when the music in his head demanded a triad on A-flat and the harpsichord in front end of him couldn't play it in tune. (In fact, he once tormented the famous organ tuner Silbermann by playing sour Ab-major triads while trying out one of his organs.) So he was glad to come across tuners develop a tuning that, today, is known besides temperament. Back then, they did call it equal temperament (or sometimes circulating temperament) - not considering the 12 pitches were equally spaced, only because you could play every bit well in all keys. Each key, still, was a little unlike, and Bach wrote The Well-Tempered Clavier in all 24 major and minor keys in order to capitalize on those differences, not because the differences didn't exist.

In any case (according to Jorgensen), the error that Bach wrote the West.T.C. in order to accept reward of what nosotros telephone call equal temperament crept into the 1893 Grove Dictionary, and has since been uncritically taught as fact to millions of budding musicians. Lord knows how long it will take to get that mistake out of the reference books.

The theorist who came up with the easiest way to tune the kind of well temperament Bach needed was the German organist Andreas Werckmeister (1645-1706), whose most famous tuning, dating from 1691, is known as Werckmeister III. A table for Werckmeister Three is as follows:

Pitch:	C	C#	D	Eb	E	F	F#	One thousand	G#	A	A#	B	C
Cents:	0	90.225	192.xviii	294.135	390.225	498.045	588.27	696.09	792.xviii	888.27	996.09	1092.eighteen	1200

Detect that we've moved considerably closer to equal temperament; no pitch is more than than 12 cents off. The following perfect fifths are 3/ii ratios of 701.955 cents each: Gb - Db - Ab - Eb - Bb - F - C, as well as A - East - B. The Pythagorean comma is distributed among the remaining fifths, C - Grand - D - A and B - F#, each of which is 696.09 cents. Let's look at the triads nosotros now have on each pitch, organized for clarity'southward sake following the circumvolve of fifths:

Major third	Cents	Perfect 5th	Cents	Minor 3rd	Cents
C - E	390.225	C - G	696.09	C - Eb	294.135
Thou - B	396.09	Grand - D	696.09	G - Bb	300.0
D - F#	396.09	D - A	696.09	D - F	305.865
A - C#	401.955	A - E	701.955	A - C	311.73
East - G#	401.955	E - B	701.955	E - G	305.865
B - D#	401.955	B - F#	696.09	B - D	300.0
F# - A#	407.82	F# - C#	701.955	F# - A	300.0
Db - F	407.82	Db - Ab	701.955	C# - Eastward	300.0
Ab - C	407.82	Ab - Eb	701.955	G# - B	300.0
Eb - G	401.955	Eb - Bb	701.955	Eb - Gb	294.135
Bb - D	396.09	Bb - F	701.955	Bb - Db	294.135
F - A	390.225	F - C	701.955	F - Ab	294.135

As you await down the columns, yous can get an thought of the quality of each triad. Note that no perfect fifth is narrower than 696 cents, nor wider than 702; this is what renders all 12 (or 24 keys) usable. The closest major thirds to perfect are C-E and F-A. Yard-B, D-F#, and Bb-D are each 396.09 cents, still sweeter than equal temperament. A-C#, E-G#, and Eb-G are around 401 cents, close to equal temperament; they therefore accept a rather banal, neutral quality. The major thirds on F#, Db, and Ab are 408 cents wide, the same size as in Pythagorean tuning (for which, encounter beneath), and not very attractive. Again, the all-time small-scale triads are grouped around A minor, with the modest third A-C, at 312 cents, coming closest to the optimum of 316 cents.

So what is the outcome of Werckmeister Iii? Can the ear really hear a difference from equal temperament?

I've done experiments with students at Bard and Bucknell, playing preludes from the West.T.C. in different keys on a sampled piano tuned to Werckmeister III; say, playing the C major prelude in B, C, and D (computer-sequenced, so that the quality of the transposed performances wasn't a gene). The students could often pick which was the appropriate key for each prelude, and even when they were mistaken they formed strong opinions nearly their preferences. In keys with poor consonances, similar F# major, Bach will laissez passer quickly by the major 3rd, and the slight touches of dissonance requite the prelude a brilliant, sparkly air. In more consonant keys, as in the C major prelude, the tonality is much more mellow, and Bach can afford to dwell on the tonic triad. Each cardinal has a unlike color (every bit opposed to the uniform color of all keys in equal temperament), and even (or especially!) the unpracticed ear can hear appropriate and inappropriate correspondences between the grapheme of each prelude and the color of each cardinal. Of course, there are preludes that sound fine in more than ane central; just it'due south disconcerting to move a prelude to a distant key, such equally from Bb to B, or C# minor to Eb minor.

Playing Bach's Well-Tempered Clavier in today'southward equal temperament is similar exhibiting Rembrandt paintings with wax paper taped over them.

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4. Young's Well Temperament and Classical-Era Music

I keep my own grand piano tuned to Thomas Young'southward first well temperament of 1799. Some synthesizers offering an alternating temperament called Vallotti-Young, which is Young'south second temperament; the Young referred to is Thomas Young (not, of class, La Monte). Jorgensen considers Immature'southward Well Temperament to be the most elegant well temperament, with a fluid variety of tonal colors and a symmetry that matches the piano keyboard: all intervals are symmetrical around D and K# - that is, D-F# and D-Bb are the same size, G#-F# and Ab-Bb the aforementioned size, and so on. The chart is as follows:

Pitch:	C	C#	D	Eb	E	F	F#	G	G#	A	A#	B	C
Cents:	0	93.9	195.8	297.eight	391.7	499.nine	591.9	697.nine	795.eight	893.eight	999.eight	1091.eight	1200

This is even closer to equal temperament; even and then, when I switched to it, my piano tuner had to return twice within two months before information technology began to stabilize. (You lot'd exist surprised how exactly your pianoforte's soundboard can remember a vi-cent divergence.) Let'south look at the quality of the triads:

Major tertiary	Cents	Perfect 5th	Cents	Minor third	Cents
C - E	391.7	C - G	697.nine	C - Eb	297.8
Thousand - B	393.nine	Grand - D	697.9	G - Bb	301.nine
D - F#	396.ane	D - A	698	D - F	304.one
A - C#	400.i	A - E	697.9	A - C	306.two
E - G#	404.1	E - B	700.1	E - G	310.3
B - D#	406	B - F#	700.i	B - D	304
F# - A#	407.nine	F# - C#	702	F# - A	301.9
Db - F	406	Db - Ab	701.9	C# - E	297.viii
Ab - C	404.2	Ab - Eb	702	G# - B	296
Eb - K	400.i	Eb - Bb	702	Eb - Gb	294.one
Bb - D	396	Bb - F	700.1	Bb - Db	294.1
F - A	393.9	F - C	700.1	F - Ab	295.9

This is a subtle tuning, quite usable in all keys, and the differences from equal temperament are more than evident to the pianist playing in it than to the listener. The best major thirds are grouped in the circumvolve of fifths around C-Eastward, whereas the perfect fifths become more perfect in the black keys, which all have fifths of 702 cents. This gives the keys related to C a sweet, gentle quality, the blackness-note keys an austere, noble quality (specially in pocket-size), and middle keys like Eb and A a neutral, ambiguous quality.

Certain keys are warmer than others; F# pocket-sized, for instance, imparts a lush quality to the slow movement of the Hammerklavier Sonata. Db major is surprising, well-nigh besides harsh, and if I happen to play Db and F alone on the keyboard the buzzy beats make me spring as though I had played a wrong notation. Information technology'southward interesting that Beethoven chose thise brilliant key for the mellow dull movement of his Appassionata Sonata, but I find that it loses energy when I play it in C; in fact, in C I have a visceral urge to play it faster considering the harmonies aren't interesting enough, but in Db I can take my time.

Nineteenth-century musicians used to argue nigh what colors the various keys represented; whether Eb major was gilded, for example, and D major red. Twentieth-century musicians have dismissed such arguments every bit sentimental nonsense, but when you play 19th-century music in well temperament, yous begin to hear the differences of color. Is it far-fetched to advise that Mozart and Beethoven wrote keyboard music with sure key-colors in mind, and that nosotros miss subtle just pervasive qualities in the music when nosotros homogenize it into equal temperament?

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5. A Word about Pythagorean Tuning

Before the advent of meantone tuning, French theorists associated with Notre Dame (13th and 14th centuries) followed a medieval tradition since Boethius (4th century) and Guido d'Arezzo (11th century) in decreeing that merely a series of perfect fifths could make upwardly a calibration; their ratio was 3/ii, and 3, later all, was the perfect number, connoting the Trinity amidst other things. Thus the Pythagorean scale is a just-intonation scale on a series of perfect fifths, all the ratio numbers powers of either iii or ii:

Pitch:	C	C#	D	Eb	E	F	F#	G	M#	A	A#	B	C
Ratios:	1/1	2187/2048	9/8	32/27	81/64	4/3	729/512	3/2	128/81	27/sixteen	16/9	243/128	two/i
Cents:	0	113.7	203.9	294.i	407.8	498	611.7	702	792.two	905.nine	996.ane	1109.eight	1200

This was an appropriate scale for a music in which perfect fifths and fourths were the overwhelmingly ascendant sonority. Though used, the thirds were theoretical dissonances, and therefore avoided at final cadences: the major third, 81/64, was 408 cents broad, and the minor third, 32/27, 294 cents. Equally Margo Schulter has convincingly written me, however, those wide thirds practise provide a compelling pull to the perfect fifths they usually resolve outward to; that is, in a cadence typical of Guillaume de Machaut (c. 1300-1377), a D and F# 408 cents apart will move outwardly to C and G. Gradually, especially under the English influence of John Dunstable and others, the thirds began to be redefined every bit v-related intervals, 5/iv and 6/5, precipitating the necessity of meantone tuning and a revolution in musical style that led to the Renaissance. Since equal temperament has close-to-perfect fifths (700 cents compared to a perfect 702), much music written in Pythogorean tuning doesn't fare besides badly in equal temperament. The Hilliard Ensemble observes Pythagorean tuning in its recordings of the Machaut Notre Dame Mass (Hyperion) and the organum of Perotin (ECM).

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6. Conclusion

I wish I could offering a wider disography of recordings in historical tunings. Luckily, a few recordings in documented historical temperaments have appeared in the terminal few years:

Enid Katahn, piano; Edward Foote, piano tuner: Beethoven in the Temperaments - Gasparo GSCD-332 (Moonlight Sonata, the Waldstein, and the Pathetique in tardily-18th-century temperaments)

Enid Katahn, piano; Edward Foote, piano tuner: Six Degrees of Tonality - Gasparo GSCD-344 (Scarlatti, Mozart, Haydn, Beethoven, Chopin, Grieg, in a diverseness of meantone and well temperaments)

J.S. Bach: Well Tempered Clavier, Robert Levin, keyboards - Hanssler 116 (in Werckmeister temperament)

Lou Harrison: Piano Concerto - Keith Jarrett, piano; Naoto Otomo conducting the New Nippon Philharmonic; New Globe NW 366-2. (Harrison tunes the solo piano to Kirnberger temperament.)

Guillaume de Machaut: Messe de Notre Matriarch, Hilliard Ensemble - Hyperion

It may be that some of the many original-practice harpsichord recordings and European organ recordings apply meantone without documenting their tuning. For those further interested, I highly recommend Owen Jorgensen'southward four-inch thick Tuning compendium (Michigan State University Printing, 1991). And I promise this volition spark some interest that will atomic number 82 to further experiments in reclaiming the original beauty of Europe's musical past.

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