fachverwirrt clarifies this very nicely. I only wanted to add that the 3:2 ratio is an artifact of the Ancient Greeks, most notably Pythagoras who studied the tones of a plucked string. He noticed that strings of equal tension but lengths that were in a low ratio (2:1, 3:2, etc.) made a pleasing harmony when plucked at the same time. He then explored these ratios by what we today would call “taking the fifth” of each note (i.e. make a string 3/2 the length of the original), then dragging it back into the octave when necessary (i.e. cutting it in half if it exceeded twice the length of you baseline).
If you do this mathematically–taking one fifth, then the fifth of that, then the fifth of that…–you will end up on the 12th try with a length very close to the original length you started with. If, like Pythagoras, you used a compass and straightedge to compute these lengths, you probably wouldn’t even notice the slight discrepancy. The Pythagoreans were mystics, and 12 is a very “harmonious” number, so they naturally thought they had stumbled on something here.
Later Greek thinkers tried to line this scale of notes up with the seven known planets. They reasoned that, since the spheres holding the planets were immense yet obviously were in motion, they must make a tremendous noise, yet we don’t seem to hear any noise. They concluded the planetary spheres must be making sounds in such perfect harmony that we are unaware of it (I think this is in Aristotle somewhere, or buried in Cicero’s Somnium Scipionis). They then matched up the order in which each length is derived by the compass/straightedge process with the presumed order of the planets. IIRC, the first six notes derived, plus the one right before the end, make up the major scale; other choices give us the minor scale, etc. So if you’ve ever wondered why there are seven different notes (plus the matching octave) in a scale, it’s because the Greeks identified seven planets in the sky.
I think he meant that using the perfect fifth as the basis of a musical tuning system is an artifact of Pythagorean numerology.
To show where the system goes wrong, if we start at A=440 and we recursively multiply by 1.5 (perfect fifth), we get the following series:
A 440
E 660
B 990
F# 1,485
C# 2,227.5
G# 3,341.25
D# 5,011.875
A# 7,517.8125
F 11,276.71875
C 16,915.078125
G 25,372.6171875
D 38,058.92578125
A 57,088.388671875
However, if we multiply 440 by 2 (octave), we obtain the following values for A in various octaves:
440
880
1,760
3,520
7,040
14,080
28,160
56,320
56,320 is close, but not equal to 57,088.388671875.
The difference is known as the Pythagorean comma and it’s the reason why instruments had to be specifically tuned to a particular key before equal temperament was invented. (And that’s the origin of the weird artifacts that we have in the 12ET system, like sharps and flats and enharmonic equivalences which have no mathematical purpose.)
This link gives a good explanation of what I was trying to say; to wit:
But I feel I should qualify my earlier statement. The Greeks did not study music in order to develop what we would today call “music theory”, but rather as a mathematical/philosophical exploration of the universe. It is likely that music conforming to the Pythagorean rules was played long before Pythagoras came along. He merely codified a disorganized body of knowledge–perhaps even filling in some minor gaps along the way–and “sanctified” it (according to his own philosophical ideas) by association with the workings of the known universe, specifically the known heavenly bodies.
Thus, it was inaccurate for me to say Pythagoras invented the diatonic scale of seven notes to match the seven planets. He merely associated the already-in-existence scales with the seven planets in an effort to link music with mathematics and the working of the universe. One might then argue this work gave what would later become Western music theory a “sacred” foundation that ensured its survival intact thru the ancient world into the Middle Ages.
Best I can do is discussion groups - try searching “high pitch” “low pitch” group:music at Google Groups.
IIRC, High Pitch was more common during the late 19th century and began giving way to Low Pitch early in the 20th. You could, however, still get HP instruments on special order as late as 1930.
Another tidbit from the '30s: when the U.S. Bureau of Standards decided to start broadcasting a standard pitch note in 1936 at the request of “several musical organizations”, A-440 was the one chosen. Cite
The Pythagorean scale, which is based entirely on the fifth, is not the same as what is usually meant by the “just scale” (though it might be included under a broader meaning of “just”). The Pythagorean scale, for example, makes the major third a ratio of (3/2)^4 / 4, or 81/64, whereas the just scale makes it 5/4=80/64 (the difference is the syntonic comma).
And that C# would be 5*440=2200 in a just scale starting at A.
The other point is that none of this is Pythagoras’ fault (or the Pythagoreans’). The justly tuned fifth is what it is, and its being considered consonant has nothing to do with Pythagoras. Not all fifths can be justly tuned if octaves are justly tuned. People would have struggled with tempering and avoiding “wolf fifths” regardless of Pythagoras.
Indeed. Good orchestral string players will very obviously ignore him…
Violins tune a perfect fifth up from A440 to their E. Violas and cellos tune three perfect fifths down to their C. These are genuinely tuned 3:2, as the only way it’s possibly to quickly and aurally tune two strings to this interval, by avoiding interference ‘beating’. The basses have to be awkward, of course, going down to their E by a fourth :rolleyes: .
But play that E against the violins. It’s wrong. Something’s wrong, very obviously wrong.
Decent players will not only tune their own instruments within themselves, but within the ensemble, which means all string instruments tuning at once, and a common medium is found where the Cs and Es are reconciled. Quite what is reached, and whether it has actually been studied, I do not know.
Relatedly, this is a simple experiment that I think all violin players are faced with early on. Tune your A string at 440 Hz, if you tune your other strings accordingly, D = 293.333 Hz and E = 660 Hz. Now, place a finger on the A string to play a B and play the open D string along with that B. That’s a major sixth, if you place your finger so that it sounds best (i.e. minimal beating) you should be playing a B at around 488.888 Hz, or 293.333 * 5/3. Holding your finger still now play B together with the open E string: it’s going to sound terribly out of tune. Why? B-E is a perfect fourth – in just intonation a 4/3 ratio. 660 Hz * 3/4 = 495 Hz.
The lesson that is learned is that for a string player, there isn’t a single “B” and that in practice you must continually be making small adjustments to sound in tune.
I desperately need to tackle some cellists and question them about this. But they don’t take kindly to such things at this hour. You’re right, I’ve never liked electronic tuners beyond an A (I’ve had guitarists ask me ‘play a B’ - play it yourselves, you halfwits), but how the harmonics didn’t coincide between strings wasn’t something I’d paid much attention, perhaps, because violinists never encounter it. At least, until when dealing either with Stravinsky who knows exactly what he wants, or with composers who know what they want and will explore to find it. So what on earth are cellists actually doing when they’re fiddling about with those harmonics when ‘tuning’?
Just to throw a little more light on the subject (in case anyone here didn’t know), the just-intoned intervals are what they are because they line up with the various partials (overtones). That is, the relationship between the third partial and the second partial is 3:2, so a note that is 3/2 above another note (or, even better, 3/1 (an octave and a fifth) lines up nicely with the overtones of the lower note. This is what causes the “ring” of well tuned ensemble chords (that ring is what barbershopers are constantly striving for).
This same phenomenon has been what has caused the steady rise in the Bagpipe A up the scale. In competitions, judges seem to favour the sharper sound, so naturally bands at the upper level competitions tune upwards, and then the rest of the bagpipe world follows suit. Most chanters being made nowadays play much sharper than those made 30 or 40 years ago.
Some pipers don’t approve of the steady increase in the A note (and the rest of the scale accordingly), and some have been trying to go back to older chanters for a more mellow sound.
'Cellist checking in. I know very little about the topic of this thread, but I was (basically) told that the best way to tune a 'cello is by playing two adjacent strings and adjusting one of them until a pefect fifth is produced. For inexperienced musicians, finding this perfect fifth is not always easy, so by playing harmonics (an octave above on one string versus a fifth above on the string below), you can produce the ‘same’ note, which is more easily comparable. However, for the reasons previously mentioned in this thread, this will not necessarily be ‘accurate’. Make sense?
Well, part of the answer is here. It explains how other forks were tuned against a standard fork using a rather clever series of mirrors and a beam of light. However, even though the article asks “So how did we create that particular pitch before we had electronic measurements?” it doesn’t quite answer it to my satisfaction. Apparently, it involves organ pipes. Perhaps there’s some math you can do given the length and diameter of an organ pipe that will give you the frequency of the tone it produces. I don’t know.
Well, sortof. But wind pressure can have some effect of pitch. There may be other factors I can’t recall. (And, of course, we’re talking about flue pipes; the “pipes” on organ reed ranks are actually resonators and have little impact on pitch.)
This does suggest to me how the first “standard pitch” may have originated: it was the natural lowest pitch of some specific instrument of standard design.
Sure. What it doesn’t quite answer, though, is how they knew and measured exactly 435 Hz, which they apparently had the capability to do since at least 1859. We know how they tuned other tuning forks to a standard fork, but how did they get that first fork at 435 Hz and know it was that not, say, 436 Hz or 432 Hz?
