I noticed some time ago that the frequencies listed by GT Curle do not match exactly the current ones
quoted by today's authorities but some research attached shows Curle's freqs were recommended by
the British Society of Arts in 1860. However, I couldn't find the reference Curle gives for the Society
actually choosing his freqs in 1869.The attachement also explains how 'equal-tempered" tuning was
used to get the current freqs.
Journal of the Society of Arts, June 8, 1860
UNIFORM MUSICAL PITCH
Part II - Report of the Committee
A practical compromise
It is well known, that neither by the committee called together by the Society of Arts, nor by the Commission appointed by the French government, has the attempt to deal with the now intolerable evil of an extravagantly high pitch, been made for the first time. Among other attempts, that of a Congress of Musicians at Stuttgart in 1834, has attracted the most attention. This body recommended a pitch of 528 for C, = 440 for A , basing their calculation on 33 vibrations per second instead of 32. The following would be the scale at this pitch – the only one yet proposed which gives all the sounds in whole numbers:
C D E F G A B C
264 297 330 352 396 440 495 528
This pitch, of which the C is 16 vibrations per second higher than that of C512, and 18 vibrations lower than the C at the present pitch (of 546), is as near as possible half-way between the two latter, and, therefore, a quarter of a tone above the one and a quarter of a tone below the other.
To lower the stringed instruments to this pitch would obviously be attended with little difficulty. Depression to the extent of a quarter of a tone is said to be easy with the brass instruments and possible with the wooden wind instruments – the flutes, oboes, clarionets, and bassoons – now in use. (Many of these were originally tuned somewhat lower, and have been raised in pitch by modifications which do not of necessity permanently affect them. The standard adopted by one of the most eminent makers of instruments for military bands is somewhat lower than that proposed above.). Few organs exist of higher pitch than the Stuttgart, and the raising of those which have been tuned to C512 would not be attended with serious difficulty.
The Stuttgart pitch, then, if not the very best that could be conceived, may be regarded as the one which, with many recommendations, would have the best chance of attaining the general assent of contemporary musicians.
Though higher than the pitch of 512, the [previous] Philharmonic pitch, or the diapason normal, the Stuttgart pitch is but a few vibrations higher than the two last of these – one of which experience has proved to be a good pitch for instrumental music.
It is a quarter of a tone below the present pitch, by general consent voted intolerably high. (About forty letters have been received in answer to a circular addressed to various directors of musical societies, vocal and instrumental performers, musical instrument makers and others, soliciting information and opinions respecting the objects of the sub-committee. Of the writers of these, none advocate an elevation of the present pitch, three think no change called for, and twenty-nine recommended a depression of the present pitch to different amounts varying from ¼ to ¾ of a tone. Of these twenty-nine eleven specify the pitch of C512.).
Its adoption would involve little, if any, inconvenience, or pecuniary loss to instrumental performers or makers of musical instruments. It would, therefore, be likely to meet with the support of the majority of those interested in the question of pitch.http://www.angelfire.com/in2/yala/t4scales.htm
We can summarise the relationship between octaves and frequency as follows:Tube Length Note Octave FrequencyOriginal C Original 264 Hz = 264 HzHalf C Up 1 264 x 2 = 528 HzQuarter C Up 2 264 x 4 = 1,056 HzDouble C Down 1 264 / 2 = 132 Hz
For simplicity, let's call 132 Hz = "C1", 264 Hz = "C2", 528 Hz = "C3" and 1,056 Hz = "C4". By convention, the first note in a numbered octave is "A" (ie G#3 is followed by A4).
Let's look at the hollow tube length again. Halving it gives us an octave higher. What happens for lengths in between? Well, for lengths in between, we get the notes in between.
If we use fractions where the numerator and denominator are whole numbers, we are creating the "just intonation" sysem of tuning. The fractions are listed in the table below and are referenced to "C".Tube Length Frequency NoteOriginal 264 x 1 = 264 Hz C33 / 4 264 x 4 / 3 = 352 Hz F32 / 3 264 x 3 / 2 = 396 Hz G33 / 5 264 x 5 / 3 = 440 Hz A44 / 5 264 x 5 / 4 = 330 Hz E3
For most cultures, the "just intonation" tuning has been in use for thousands of years. This makes sense because we are using multiples of the original length (and then normalising them to the octave) to create notes.
The just-intonation tuning system works fine and sounds beautiful. However, it has only one drawback... you cannot transpose a song (ie you can only play songs in any key but "C"). When you play in another key (eg "D"), the tuning sounds wrong.
The "equal-tempered" tuning was developed to overcome this problem.
How does it work? Well, if you think about it, tuning is not linear. You can double the frequency to get the next octave up but you have to quadruple it to get the next octave after that. Consequently, the notes within a scale are not equally distributed in frequency (nor in length).
This is how it's worked out! "A4" (the note "A" at the fourth octave) is deemed to be at 440 Hz and, therefore, "A5" will be at 880 Hz. We then take logarithms of A4 and A5 frequencies. Next, we mark in 11 equally spaced points between log(A4) and log(A5). On the logarithmic scale, this is the same as having 12 equally spaced notes per octave. We then apply arc-logarithms to those points and arrive the equal-tempered tuning.Calculation for Equal-Tempered tuning [A4 = 440Hz]Hertz Octave=1 Octave=2 Octave=3 Octave=4 Octave=5 Octave=6
0 A 55.000 110.000 220.000 440.000 880.000 1,760.0001 A#/Bb 58.270 116.541 233.082 466.164 932.328 1,864.6552 B 61.735 123.471 246.942 493.883 987.767 1,975.5333 C 65.406 130.813 261.626 523.251 1,046.502 2,093.0054 C#/Db 69.296 138.591 277.183 554.365 1,108.731 2,217.4615 D 73.416 146.832 293.665 587.330 1,174.659 2,349.3186 D#/Eb 77.782 155.563 311.127 622.254 1,244.508 2,489.0167 E 82.407 164.814 329.628 659.255 1,318.510 2,637.0208 F 87.307 174.614 349.228 698.456 1,396.913 2,793.8269 F#/Gb 92.499 184.997 369.994 739.989 1,479.978 2,959.95510 G 97.999 195.998 391.995 783.991 1,567.982 3,135.96311 G#/Ab 103.826 207.652 415.305 830.609 1,661.219 3,322.43812 A 110.000 220.000 440.000 880.000 1,760.000 3,520.000
Since this tuning is mathematically derived, a song will sound "correct" when played in a different key.
Special note - The decision to use A4 = 440 Hz, 12 notes per octave and naming them A to G was due to historical circumstances. Any other combination would also be valid. However, the equal-tempered tuning is now the de facto system.