Date: Sat, 13 Jun 1992 04:30:56 GMT From: "Neil E. Midkiff midkiff@netcom.com" Organization: BTR Public Access UNIX, MtnView CA. Contact: Customer Service cs@BTR.COM Subject: Tuning and Temperament, was Re: tuning information needed In article <1066@zogwarg.etl.army.mil> mike@zogwarg.etl.army.mil (Mike McDonnell) writes: >I would like to have the details needed to set a meantone bearing on >my harpsichord. What I specifically need is the intervals for both >quarter-comma and sixth-comma meantone tuning. > >I can accept information in any form such as Hz or cents difference >from equal temperament. The tuner I use takes its tables in the form >of cents deviation from equal temperament, so that is the best form >for me to get the information in. The most difficult information to >work with is "beats", though I think I can convert that to cents. Fortunately, I've been gathering just that information! Your article prompted me to prepare it for posting. Here it is. ------------------------------------------------------------------------- TUNING AND TEMPERAMENT - An Overview with Some Real Numbers Neil Midkiff (midkiff@netcom.com) 6/11/92 This document is intended for musicians who wish to understand some of the mathematical details involved in commonly-encountered systems of tuning keyboard instruments. It does not purport to give a complete historical picture of temperament, nor does it replace tuning recipes based on beats. Non-Western tunings and tunings with more than 12 notes per octave are also specifically excluded from consideration. This information should be especially useful with electronic tuning aids which are calibrated in cents of deviation from the equal-tempered scale. With some additional information it can be used to generate beat charts for tuning by ear, or to compute the data for microtuning of synthesizers such as the Yamaha TG77/SY77 and -99 which allow the user to specify individual note pitches. (Contact me at the address above if more information is desired.) As always, I'd appreciate comments and corrections if any of this is unclear or inaccurate. I'm sure that others will be able to help flesh out the bits of historical data I include. There may be a little repetition since this was assembled from a set of individual documents created from Microsoft Excel spreadsheets. (Spreadsheets turn out to be useful "calculators" for this sort of work!) I hope, however, that others will find this as useful as I would have found it several months ago before I began tracking all this down to put it into one place. --------------------------------------------------------------------------- Let's start with a few definitions. Engineers and scientists deal with musical tones in terms of frequency, with the units of hertz (Hz), equivalent to vibrations per second. Musicians also have agreed (well, mostly) on a standard frequency of 440 Hz for the A above middle C. But because our sense of pitch is related to the logarithm of frequency, it's easier to deal with these logarithmic numbers in discussing tuning. This allows us to add and subtract intervals, rather than multiplying and dividing ratios of frequencies, and has the advantage that musical intervals like octaves have the same size in pitch units, no matter where they are on the keyboard (how high or low the pitch, that is). Musicians usually talk about tuning in units of cents, or hundredths of an equal-temperament semitone. An octave is a 2:1 frequency ratio, and is divided in equal temperament into 12 semitones, or 1200 cents. Cents are computed by taking 1200 times the logarithm (base 2) of the frequency ratio. EQUAL TEMPERAMENT Equal temperament, the system used today for keyboard instruments, makes all semitones equal in size; the frequency ratio is the twelfth root of 2 (about 1.0594631), or 100 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 523.25 1200.00 0.00 B3 493.88 1100.00 0.00 A#3 466.16 1000.00 0.00 A3 440.00 900.00 0.00 G#3 415.30 800.00 0.00 G3 392.00 700.00 0.00 F#3 369.99 600.00 0.00 F3 349.23 500.00 0.00 Note: Because I originally computed these E3 329.63 400.00 0.00 tables for use on a MIDI synthesizer, the D#3 311.13 300.00 0.00 note names use the Yamaha octave numbers D3 293.66 200.00 0.00 so that C3 equals middle C, and C4 is one C#3 277.18 100.00 0.00 octave above it. C3 261.63 0.00 0.00 The "problem" with equal temperament is that it is only an approximation of the ideal intervals our ears hear as pure. We tend to hear consonances between frequencies whose ratios are equal to the ratio of small integers. Thus the octave is 2:1, the fifth is 3:2, the fourth is 4:3, the major third is 5:4, the minor third is 6:5 and so on. The trouble is that these intervals can't be combined in ways that "come out even" with the scales we see on our keyboards. The Pythagorean Comma To take the simplest example, start at the lowest C on the piano and go upward by fifths to G, D, A, and so forth. When you come back to C after twelve fifths, you'll have covered the seven complete octaves on the keyboard. But this is only possible because we've agreed to use equally- tempered intervals on the piano, where a fifth is 700 cents, or 7/12 of an octave. A pure 3:2 fifth turns out to be 1200*log2(3/2), or 701.96 cents. That is, we make fifths on the piano about 2 cents flatter (narrower) than pure, so that twelve of them equal seven octaves. The total discrepancy if we used pure fifths would be (3/2)**12/2**7, or about 23.46 cents. This interval is called the Pythagorean comma, and the amount by which equal- tempered fifths are flattened is one-twelfth of it. The following table of intervals may look a little redundant since all the lines are the same. But I'm using it here to prepare for the discussion of other tunings. The note names at the left refer to the lower note of an interval. So the first entry means that the minor second from B3 up to C4 is 100 cents, which is 4.96 cents flat with respect to a 17:16 frequency ratio. (This is one of several choices of integer ratios for the minor second, but I'll use it throughout for consistency). minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 100.00 -4.96 200.00 -3.91 300.00 -15.64 A#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 A3 100.00 -4.96 200.00 -3.91 300.00 -15.64 G#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 G3 100.00 -4.96 200.00 -3.91 300.00 -15.64 F#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 F3 100.00 -4.96 200.00 -3.91 300.00 -15.64 E3 100.00 -4.96 200.00 -3.91 300.00 -15.64 D#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 D3 100.00 -4.96 200.00 -3.91 300.00 -15.64 C#3 100.00 -4.96 200.00 -3.91 300.00 -15.64 C3 100.00 -4.96 200.00 -3.91 300.00 -15.64 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 400.00 13.69 500.00 1.96 700.00 -1.96 A#3 400.00 13.69 500.00 1.96 700.00 -1.96 A3 400.00 13.69 500.00 1.96 700.00 -1.96 G#3 400.00 13.69 500.00 1.96 700.00 -1.96 G3 400.00 13.69 500.00 1.96 700.00 -1.96 F#3 400.00 13.69 500.00 1.96 700.00 -1.96 F3 400.00 13.69 500.00 1.96 700.00 -1.96 E3 400.00 13.69 500.00 1.96 700.00 -1.96 D#3 400.00 13.69 500.00 1.96 700.00 -1.96 D3 400.00 13.69 500.00 1.96 700.00 -1.96 C#3 400.00 13.69 500.00 1.96 700.00 -1.96 C3 400.00 13.69 500.00 1.96 700.00 -1.96 As the table makes evident, fourths and fifths are not badly compromised by equal temperament, but major and minor thirds are much farther out of pure tuning. We have learned to accept these slightly dissonant thirds as the price we have to pay for a tuning system which allows modulating into different keys without encountering even worse out-of-tuneness. Historically, other tradeoffs have been made in an attempt to have greater consonance in the key of C and those "near" it (with few sharps or flats in the key signature), at the expense of unusable intervals in remote keys. The so-called "wolf" interval is found between G# and Eb in the tunings described below, though another place around the circle of fifths could equally well be chosen. The Syntonic Comma Before we describe some of the tunings, we need to define another kind of comma, the syntonic comma. It's most quickly defined in the key of C as the difference between the major 2nd between F and G and the major 2nd between Eb and F. That is, it's (fifth - fourth) minus (fourth - minor 3rd). In frequency terms it's ((3/2)/(4/3)) divided by ((4/3)/(6/5)), or 9/8 divided by 10/9. And this ratio is the key to its real definition, and to its importance. In the harmonic series based on the C three octaves below middle C (that is, C0 on my synthesizer), C1 (one octave up) is twice the fundamental frequency, G1 is close to three times the fundamental, C2 is four times, E2 is close to five times, G2 is close to 6 times, A#2 is about seven times, C3 is eight times, D3 is close to 9 times, and E3 is close to 10 times. If we tuned pure intervals based on C, then we'd make all the "close to" intervals exact. Then the major second from C3 to D3 would be 9:8, and the major second from D3 to E3 would be 10:9, and the difference is once again the syntonic comma. 9/8 divided by 10/9 is 81/80; in pitch this works out to about 21.51 cents. MEANTONE TUNINGS The various varieties of meantone tunings are attempts to average out the differences in these major seconds. In a quarter-comma meantone, all but one of the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fourth of the syntonic comma, or 5.38 cents. So fifths are tuned to 696.58 cents, except for the wolf which must be seven octaves minus eleven flattened fifths, or 737.64 cents. Here the wolf is shown inverted as the fourth of 462.36 cents, between G# and Eb. All other fourths are 503.42 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 526.36 1200.00 10.26 B3 491.93 1082.89 -6.84 A#3 470.79 1006.84 17.11 A3 440.00 889.74 0.00 G#3 411.22 772.63 -17.11 G3 393.55 696.58 6.84 F#3 367.81 579.47 -10.26 F3 352.00 503.42 13.69 E3 328.98 386.31 -3.42 D#3 314.84 310.26 20.53 D3 294.25 193.16 3.42 C#3 275.00 76.05 -13.69 C3 263.18 0.00 10.26 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 117.11 12.15 193.16 -10.75 310.26 -5.38 A#3 76.05 -28.91 193.16 -10.75 269.21 -46.43 A3 117.11 12.15 193.16 -10.75 310.26 -5.38 G#3 117.11 12.15 234.22 30.31 310.26 -5.38 G3 76.05 -28.91 193.16 -10.75 310.26 -5.38 F#3 117.11 12.15 193.16 -10.75 310.26 -5.38 F3 76.05 -28.91 193.16 -10.75 269.21 -46.43 E3 117.11 12.15 193.16 -10.75 310.26 -5.38 D#3 76.05 -28.91 193.16 -10.75 269.21 -46.43 D3 117.11 12.15 193.16 -10.75 310.26 -5.38 C#3 117.11 12.15 234.22 30.31 310.26 -5.38 C3 76.05 -28.91 193.16 -10.75 310.26 -5.38 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 427.37 41.06 503.42 5.38 696.58 -5.38 A#3 386.31 0.00 503.42 5.38 696.58 -5.38 A3 386.31 0.00 503.42 5.38 696.58 -5.38 G#3 427.37 41.06 503.42 5.38 737.64 35.68 G3 386.31 0.00 503.42 5.38 696.58 -5.38 F#3 427.37 41.06 503.42 5.38 696.58 -5.38 F3 386.31 0.00 503.42 5.38 696.58 -5.38 E3 386.31 0.00 503.42 5.38 696.58 -5.38 D#3 386.31 0.00 462.36 -35.68 696.58 -5.38 D3 386.31 0.00 503.42 5.38 696.58 -5.38 C#3 427.37 41.06 503.42 5.38 696.58 -5.38 C3 386.31 0.00 503.42 5.38 696.58 -5.38 In summary: Quarter-comma meantone gives pure major thirds in most cases (except for four wide ones) and evens out the major seconds (except for two wide ones from C# to D# and G# to A#) as half the pure major third. This is accomplished at the expense of minor seconds of two widely different sizes, a very sour "wolf" fifth, and even worse mistunings for the wide major thirds and three narrow minor thirds. Usually when "meantone" is mentioned without further specifics, this is the variety that is meant. In a fifth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-fifth of the syntonic comma, or 4.30 cents. So fifths are tuned to 697.65 cents, except for the wolf which must be 725.81 cents. Here the wolf is shown inverted as the fourth of 474.19 cents, between G# and Eb. All other fourths are 502.35 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 525.38 1200.00 7.04 B3 492.55 1088.27 -4.69 A#3 469.33 1004.69 11.73 A3 440.00 892.96 0.00 G#3 412.50 781.23 -11.73 G3 393.06 697.65 4.69 F#3 368.49 585.92 -7.04 F3 351.13 502.35 9.39 E3 329.18 390.61 -2.35 D#3 313.67 307.04 14.08 D3 294.06 195.31 2.35 C#3 275.68 83.58 -9.39 C3 262.69 0.00 7.04 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 111.73 6.78 195.31 -8.60 307.04 -8.60 A#3 83.58 -21.38 195.31 -8.60 278.88 -36.76 A3 111.73 6.78 195.31 -8.60 307.04 -8.60 G#3 111.73 6.78 223.46 19.55 307.04 -8.60 G3 83.58 -21.38 195.31 -8.60 307.04 -8.60 F#3 111.73 6.78 195.31 -8.60 307.04 -8.60 F3 83.58 -21.38 195.31 -8.60 278.88 -36.76 E3 111.73 6.78 195.31 -8.60 307.04 -8.60 D#3 83.58 -21.38 195.31 -8.60 278.88 -36.76 D3 111.73 6.78 195.31 -8.60 307.04 -8.60 C#3 111.73 6.78 223.46 19.55 307.04 -8.60 C3 83.58 -21.38 195.31 -8.60 307.04 -8.60 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 418.77 32.46 502.35 4.30 697.65 -4.30 A#3 390.61 4.30 502.35 4.30 697.65 -4.30 A3 390.61 4.30 502.35 4.30 697.65 -4.30 G#3 418.77 32.46 502.35 4.30 725.81 23.85 G3 390.61 4.30 502.35 4.30 697.65 -4.30 F#3 418.77 32.46 502.35 4.30 697.65 -4.30 F3 390.61 4.30 502.35 4.30 697.65 -4.30 E3 390.61 4.30 502.35 4.30 697.65 -4.30 D#3 390.61 4.30 474.19 -23.85 697.65 -4.30 D3 390.61 4.30 502.35 4.30 697.65 -4.30 C#3 418.77 32.46 502.35 4.30 697.65 -4.30 C3 390.61 4.30 502.35 4.30 697.65 -4.30 Once again, we have achieved major seconds that are half the size of the major thirds in most cases, so this is also a meantone tuning. The difference is that we've allowed the "nice" major thirds to expand from pure to one-fifth comma wider than pure. This reduces most of the other deviations from pure intervals; only the "nice" minor thirds are a little farther from pure than in quarter-comma. The wolves are howling less loudly! In a sixth-comma meantone, the fifths are flattened from the pure 3:2 ratio of 701.96 cents by one-sixth of the syntonic comma, or 3.58 cents. So fifths are tuned to 698.37 cents, except for the wolf which must be 717.92 cents. Here the wolf is shown inverted as the fourth of 482.08 cents between G# and Eb. All other fourths are 501.63 cents. Hz cents adjustments from A=440 equal temperament (cents) C4 524.73 1200.00 4.89 B3 492.95 1091.85 -3.26 A#3 468.36 1003.26 8.15 A3 440.00 895.11 0.00 G#3 413.35 786.96 -8.15 G3 392.73 698.37 3.26 F#3 368.95 590.22 -4.89 F3 350.55 501.63 6.52 E3 329.32 393.48 -1.63 D#3 312.89 304.89 9.78 D3 293.94 196.74 1.63 C#3 276.14 88.59 -6.52 C3 262.37 0.00 4.89 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 108.15 3.19 196.74 -7.17 304.89 -10.75 A#3 88.59 -16.36 196.74 -7.17 285.34 -30.30 A3 108.15 3.19 196.74 -7.17 304.89 -10.75 G#3 108.15 3.19 216.29 12.38 304.89 -10.75 G3 88.59 -16.36 196.74 -7.17 304.89 -10.75 F#3 108.15 3.19 196.74 -7.17 304.89 -10.75 F3 88.59 -16.36 196.74 -7.17 285.34 -30.30 E3 108.15 3.19 196.74 -7.17 304.89 -10.75 D#3 88.59 -16.36 196.74 -7.17 285.34 -30.30 D3 108.15 3.19 196.74 -7.17 304.89 -10.75 C#3 108.15 3.19 216.29 12.38 304.89 -10.75 C3 88.59 -16.36 196.74 -7.17 304.89 -10.75 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 413.04 26.73 501.63 3.58 698.37 -3.58 A#3 393.48 7.17 501.63 3.58 698.37 -3.58 A3 393.48 7.17 501.63 3.58 698.37 -3.58 G#3 413.04 26.73 501.63 3.58 717.92 15.97 G3 393.48 7.17 501.63 3.58 698.37 -3.58 F#3 413.04 26.73 501.63 3.58 698.37 -3.58 F3 393.48 7.17 501.63 3.58 698.37 -3.58 E3 393.48 7.17 501.63 3.58 698.37 -3.58 D#3 393.48 7.17 482.08 -15.97 698.37 -3.58 D3 393.48 7.17 501.63 3.58 698.37 -3.58 C#3 413.04 26.73 501.63 3.58 698.37 -3.58 C3 393.48 7.17 501.63 3.58 698.37 -3.58 By this point, I think the trend should be clear. The wolves are almost tamed; the major thirds aren't too wide, but they're getting farther off, and the minor thirds are going flat. The logical extension of this trend is a one-twelfth (Pythagorean, not syntonic) comma meantone, which is precisely equal temperament. WELL-TEMPERED TUNINGS Other systems of tuning, which get lumped under the heading of "well- tempered" tunings, don't follow the meantone pattern of having a single wolf fifth. Instead, they distribute the discrepancies more-or-less evenly through some of the remote-from-C-major intervals. In this way, they attempt to preserve the consonances of purer intervals in the usual keys, while smoothing over the consequences of modulating into remote keys. All keys are usable, but the sizes of the intervals between various degrees of the scale are not exactly the same in each key. This is often thought of as an advantage, since it lends a distinctive character to the different keys. The tuning Bach had in mind for Das Wohltemperierte Klavier was almost certainly *not* equal temperament (though it had been invented prior to Bach's time) but one of the many forms of well-tempered tunings. Werckmeister III A well-tempered tuning adopted for organs in the time of Bach is known as Werckmeister III. In this tuning, the fifths C-G-D-A and B-F# are each tempered by 1/4 of the Pythagorean comma, or 5.87 cents. Pure fifths of 3:2 are 701.96 cents; the tempered ones are then 696.09 cents. Pure fourths of 4:3 are 498.04 cents; the tempered ones are then 503.91 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 526.81 1200.00 11.73 B3 495.00 1092.18 3.91 A#3 468.27 996.09 7.82 A3 440.00 888.27 0.00 G#3 416.24 792.18 3.91 G3 393.77 696.09 7.82 F#3 369.99 588.27 0.00 F3 351.21 498.04 9.78 E3 330.00 390.22 1.96 D#3 312.18 294.13 5.87 (Yes, it's true: F# and A are the same D3 294.33 192.18 3.91 as in equal temperament, and the others C#3 277.50 90.22 1.96 are all *sharper*. This is a coincidence C3 263.40 0.00 11.73 due to using A=440 as a standard pitch.) minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 107.82 2.86 198.04 -5.87 300.00 -15.64 A#3 96.09 -8.87 203.91 0.00 294.13 -21.51 A3 107.82 2.86 203.91 0.00 311.73 -3.91 G#3 96.09 -8.87 203.91 0.00 300.00 -15.64 G3 96.09 -8.87 192.18 -11.73 300.00 -15.64 F#3 107.82 2.86 203.91 0.00 300.00 -15.64 F3 90.22 -14.73 198.04 -5.87 294.13 -21.51 E3 107.82 2.86 198.04 -5.87 305.87 -9.77 D#3 96.09 -8.87 203.91 0.00 294.13 -21.51 D3 101.96 -3.00 198.04 -5.87 305.87 -9.77 C#3 101.96 -3.00 203.91 0.00 300.00 -15.64 C3 90.22 -14.73 192.18 -11.73 294.13 -21.51 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 401.96 15.65 498.04 0.00 696.09 -5.87 A#3 396.09 9.78 498.04 0.00 701.96 0.00 A3 401.96 15.65 503.91 5.87 701.96 0.00 G#3 407.82 21.51 498.04 0.00 701.96 0.00 G3 396.09 9.78 503.91 5.87 696.09 -5.87 F#3 407.82 21.51 503.91 5.87 701.96 0.00 F3 390.22 3.91 498.04 0.00 701.96 0.00 E3 401.96 15.65 498.04 0.00 701.96 0.00 D#3 401.96 15.65 498.04 0.00 701.96 0.00 D3 396.09 9.78 503.91 5.87 696.09 -5.87 C#3 407.82 21.51 498.04 0.00 701.96 0.00 C3 390.22 3.91 498.04 0.00 696.09 -5.87 The result is that major thirds are stretched by 2, 5, 8, or 11/12 of the Pythagorean comma, and minor thirds flattened by 2, 5, 8, or 11/12 of it. 2 5 8 11 Pure major third is 386.31 cents 390.22 396.09 401.96 407.82 Pure minor third is 315.64 cents 311.73 305.87 300.00 294.13 Vallotti Another popular well-tempered tuning is the Vallotti tuning, historically accurate for music of Mozart's time. In the Vallotti tuning, the fifths F-C- G-D-A-E-B are each tempered by 1/6 of the Pythagorean comma, or 3.91 cents. Pure fifths of 3:2 are 701.96 cents; the tempered ones are then 698.04 cents. Pure fourths of 4:3 are 498.04 cents; the tempered ones are then 501.96 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.03 1200.00 5.87 B3 492.77 1090.22 -3.91 A#3 467.75 1000.00 5.87 A3 440.00 894.13 0.00 G#3 415.77 796.09 1.96 G3 392.88 698.04 3.91 F#3 369.58 592.18 -1.96 F3 350.81 501.96 7.82 E3 329.26 392.18 -1.96 D#3 311.83 298.04 3.91 D3 294.00 196.09 1.96 C#3 277.18 94.13 0.00 C3 262.51 0.00 5.87 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 109.78 4.82 203.91 0.00 305.87 -9.77 A#3 90.22 -14.73 200.00 -3.91 294.13 -21.51 A3 105.87 0.91 196.09 -7.82 305.87 -9.77 G#3 98.04 -6.91 203.91 0.00 294.13 -21.51 G3 98.04 -6.91 196.09 -7.82 301.96 -13.68 F#3 105.87 0.91 203.91 0.00 301.96 -13.68 F3 90.22 -14.73 196.09 -7.82 294.13 -21.51 E3 109.78 4.82 200.00 -3.91 305.87 -9.77 D#3 94.13 -10.82 203.91 0.00 294.13 -21.51 D3 101.96 -3.00 196.09 -7.82 305.87 -9.77 C#3 101.96 -3.00 203.91 0.00 298.04 -17.60 C3 94.13 -10.82 196.09 -7.82 298.04 -17.60 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 407.82 21.51 501.96 3.91 701.96 0.00 A#3 396.09 9.78 498.04 0.00 701.96 0.00 A3 400.00 13.69 501.96 3.91 698.04 -3.91 G#3 403.91 17.60 498.04 0.00 701.96 0.00 G3 392.18 5.87 501.96 3.91 698.04 -3.91 F#3 407.82 21.51 498.04 0.00 701.96 0.00 F3 392.18 5.87 498.04 0.00 698.04 -3.91 E3 403.91 17.60 501.96 3.91 698.04 -3.91 D#3 400.00 13.69 498.04 0.00 701.96 0.00 D3 396.09 9.78 501.96 3.91 698.04 -3.91 C#3 407.82 21.51 498.04 0.00 701.96 0.00 C3 392.18 5.87 501.96 3.91 698.04 -3.91 The result is that major thirds are stretched by 3, 5, 7, 9, or 11/12 of the Pythagorean comma, and minor thirds flattened by 5, 7, 9, or 11/12 of it. 3 5 7 9 11 Pure major third is 386.31 cents 392.18 396.09 400.00 403.91 407.82 Pure minor third is 315.64 cents 305.87 301.96 298.05 294.14 The Fisk-Vogel tunings at Stanford Finally, a pair of tunings which have gotten some discussion on the net: the tunings devised by Charles Fisk and Harald Vogel for the Fisk organ in Memorial Church at Stanford University, which can be switched by a lever above the music desk from mean-tone to well-tempered; the lever activates an alternate set of tracker mechanisms for the sharp keys, so that there are 17 pipes per octave. Stanford's Fisk organ uses a modified meantone tuning which shares the natural keys with the well-tempered tuning. This means that the six intervals F-C-G-D-A-E-B are identical in both systems; they're flattened by one-fifth the Pythagorean comma, or 4.69 cents. The five intervals F-Bb-Eb and G#-C#-F#-B are flattened by one-fourth the syntonic comma, or 5.38 cents. So natural fifths are tuned to 697.26 cents and "other" fifths are tuned to 696.58 cents, except the wolf which must be seven octaves minus six natural fifths minus five accidental fifths, or 733.53 cents. Here the wolf is shown inverted as the fourth of 466.47 cents, between G# and Eb. All natural fourths are 502.74 cents, and "other" fourths are 503.42 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.74 1200.00 8.21 B3 492.32 1086.31 -5.47 A#3 470.05 1006.16 14.37 A3 440.00 891.79 0.00 G#3 411.55 776.05 -15.74 G3 393.24 697.26 5.47 F#3 368.10 582.89 -8.90 F3 351.44 502.74 10.95 E3 329.11 389.05 -2.74 D#3 314.34 309.58 17.79 D3 294.13 194.53 2.74 C#3 275.22 79.47 -12.32 C3 262.87 0.00 8.21 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 113.69 8.73 193.16 -10.75 308.21 -7.43 A#3 80.16 -24.80 193.84 -10.07 273.31 -42.33 A3 114.37 9.41 194.53 -9.38 308.21 -7.43 G#3 115.74 10.78 230.11 26.20 310.26 -5.38 G3 78.79 -26.17 194.53 -9.38 308.90 -6.74 F#3 114.37 9.41 193.16 -10.75 308.90 -6.74 F3 80.16 -24.80 194.53 -9.38 273.31 -42.33 E3 113.69 8.73 193.84 -10.07 308.21 -7.43 D#3 79.47 -25.48 193.16 -10.75 273.31 -42.33 D3 115.05 10.10 194.53 -9.38 308.21 -7.43 C#3 115.05 10.10 230.11 26.20 309.58 -6.06 C3 79.47 -25.48 194.53 -9.38 309.58 -6.06 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 423.27 36.96 502.74 4.69 696.58 -5.38 A#3 388.37 2.06 503.42 5.38 696.58 -5.38 A3 387.68 1.37 502.74 4.69 697.26 -4.69 G#3 423.95 37.64 503.42 5.38 733.53 31.57 G3 389.05 2.74 502.74 4.69 697.26 -4.69 F#3 423.27 36.96 503.42 5.38 696.58 -5.38 F3 389.05 2.74 503.42 5.38 697.26 -4.69 E3 387.00 0.69 502.74 4.69 697.26 -4.69 D#3 387.68 1.37 466.47 -31.57 696.58 -5.38 D3 388.37 2.06 502.74 4.69 697.26 -4.69 C#3 423.27 36.96 503.42 5.38 696.58 -5.38 C3 389.05 2.74 502.74 4.69 697.26 -4.69 These should be compared with quarter-comma meantone; the differences are hardly ever more than a couple of cents. The nifty thing is that by switching only the sharps, you get the well-tempered tuning described next. Stanford's Fisk organ uses a well-tempered tuning which shares the natural keys with the modified meantone tuning. This means that the six intervals F- C-G-D-A-E-B are identical in both systems; they're flattened by one-fifth the Pythagorean comma. G#-C# is similarly flattened in the well-tempered tuning. The three intervals Bb-Eb-G# and F#-B are pure; the two intervals F-Bb and C#-F# are sharpened (widened) by one-fifth Pythagorean comma. So natural fifths are tuned to 697.26 cents, and wide fifths are tuned to 706.65 cents, compared with pure fifths of 701.96 cents. All natural fourths are 502.74 cents, and wide fourths are 493.35 cents, compared with pure fourths of 498.04 cents. Hz cents adjustment from A=440 equal temperament (cents) C4 525.74 1200.00 8.21 B3 492.32 1086.31 -5.47 A#3 467.32 996.09 4.30 A3 440.00 891.79 0.00 G#3 415.40 792.18 0.39 G3 393.24 697.26 5.47 F#3 369.24 588.27 -3.52 F3 351.44 502.74 10.95 E3 329.11 389.05 -2.74 D#3 311.55 294.13 2.35 D3 294.13 194.53 2.74 C#3 277.68 94.92 3.13 C3 262.87 0.00 8.21 minor 2nd wrt 17:16 major 2nd wrt 9:8 minor 3rd wrt 6:5 B3 113.69 8.73 208.60 4.69 308.21 -7.43 A#3 90.22 -14.73 203.91 0.00 298.83 -16.81 A3 104.30 -0.65 194.53 -9.38 308.21 -7.43 G#3 99.61 -5.35 203.91 0.00 294.13 -21.51 G3 94.92 -10.04 194.53 -9.38 298.83 -16.81 F#3 108.99 4.04 203.91 0.00 303.52 -12.12 F3 85.53 -19.42 194.53 -9.38 289.44 -26.20 E3 113.69 8.73 199.22 -4.69 308.21 -7.43 D#3 94.92 -10.04 208.60 4.69 294.13 -21.51 D3 99.61 -5.35 194.53 -9.38 308.21 -7.43 C#3 99.61 -5.35 199.22 -4.69 294.13 -21.51 C3 94.92 -10.04 194.53 -9.38 294.13 -21.51 major 3rd wrt 5:4 fourth wrt 4:3 fifth wrt 3:2 B3 407.82 21.51 502.74 4.69 701.96 0.00 A#3 398.44 12.13 498.04 0.00 706.65 4.69 A3 403.13 16.82 502.74 4.69 697.26 -4.69 G#3 407.82 21.51 502.74 4.69 701.96 0.00 G3 389.05 2.74 502.74 4.69 697.26 -4.69 F#3 407.82 21.51 498.04 0.00 706.65 4.69 F3 389.05 2.74 493.35 -4.69 697.26 -4.69 E3 403.13 16.82 502.74 4.69 697.26 -4.69 D#3 403.13 16.82 498.04 0.00 701.96 0.00 D3 393.74 7.43 502.74 4.69 697.26 -4.69 C#3 407.82 21.51 493.35 -4.69 697.26 -4.69 C3 389.05 2.74 502.74 4.69 697.26 -4.69 ------------------------------------------------------------------------- version 1.0 (C) Copyright 1992 by Neil Midkiff. This document may be freely distributed for educational or other non-profit use; please retain this notice and do not distribute it in altered form. Commercial use or publication in whole or part requires my consent. In other words, it's freeware, but not public domain.