I’m not the piano tuner or music theory expert, but I am pretty good at physics. I think the idea is that with equal temperament, each half-step note is the twelfth root of two times the previous lower note. And you know that the twelfth root of two is an irrational number.
Two notes sound “in tune” when they’re a nice ratio with each other, and that can never happen on a system that’s based on the twelfth root of two spacing. For example, two notes that are a major fifth apart, on an equal temperament tuning piano, would have a frequency ratio of 1.4983. A ratio of 1.5000 would sound perfectly in-tune, so it’s close but not exact.
The equal temperament is an approximation that works equally well for all the major and minor keys, but if you wanted to play a piano in a specific key it would be possible to tune it so that the notes are nice exact ratios of the key note. But try to play on that piano in a different key and it would probably sound worse than ET.
Yes again. Very generally speaking, the temperament’s tonic key will have the “purest” intervals and sound “best”; the closely related keys will be less pure, but still acceptable; the most distant keys will be unusable. For example, using C as tonic note, F and G will be second best; B and F#/Gb will be worst.
It’s worth mentioning that in this context, “best” and “worst” are to some extent in the ear of the beholder. Within reasonable limits, it’s not a question of right, wrong, bad, or good.
For what it’s worth, I find non-ET’s interesting to hear, but the novelty wears off quickly, and the pure thirds and sixths (both major and minor) become dull and lackluster. Of course, ET is what I’m used to, having heard it all my life.
.
Can you explain how you would do this? You can change the tuning of the strings (and string length at the bridge, but I don’t know if that’s what you’re thinking of), but you are stuck with the fret layout, so I don’t see how you can tune to any temperament.
Just for the record, that would be “perfect fifth”
I believe that keyboard instruments were sometimes tuned to specific keys using systems like just intonation before the widespread adoption of equal temperament (and other temper approaches). But I’m not a scholar on the subject.
Yep, that’s what I meant. My point was that changing the tuning on a guitar is quick and easy, while changing the tuning on a piano is not. I should have left it at that, while I still knew what I was talking about. Instead,
I made an assumption about matters outside my field of expertise, a mistake I try to avoid. Sorry.
Sorry to further hijack this thread, but let’s say we go with standard tuning of A=440 and other As would be 110, 220, and 880. And follow it theoretically down to middle C=261.63. Isn’t that how a piano is tuned? Is a synthesizer keyboard done to the precise tones?
Yes, it’s usual for pianos to be tuned to that value, which is the equal temperament one.
What do you mean by “precise”? I’d think that most synthesizers would get tuned to E.T. as well, so it would be precisely 261.6256. However, a synthesizer theoretically should be much more easily tuned than a piano, so I would think the high-end models would let you select non-E.T. tunings, but you’d have to be able to select the tuning based on the key you’re playing in.
For example, from A to C is a major third, which should be an exact ratio of 1.2. But the E.T. tuning gives a value of 1.1892. If you were tuning just to play in the key of A, you’d set C to be 264 Hz instead of 261.6256. However, that value for A and C wouldn’t be so good for playing in another key like F. If you want to set it once and play in any key without re-tuning, that’s exactly why E.T. was invented. It’s already precise, it’s just not optimized for each key separately.
I just got this book and am about halfway through. I feared that it might be deadly dull, but no, it’s quite good.
No, not really. That’s equal temperament, and while it’s the starting point for how pianos are tuned, something called stretch tuning is applied (as mentioned by ** TreacherousCretin**) that is meant to correct for a problem called inharmonicity. The “error” is greatest in the lowest and highest notes.
Middle A (A4) will be the only note that is tuned precisely to its theoretically correct frequency (440Hz in your example), and becomes the reference point for the rest of the tuning; every other note within the temperament octave is tuned in relation to A4, and on any given piano the ideal frequency for a specific note won’t necessarily be the theoretically correct frequency.
Once the temperament area has been tuned, the rest of the notes are then tuned in relation to the temperament as octaves, double octaves, triple octaves, and various other intervals. None of these octaves will be tuned to their theoretically correct frequencies.
Octaves above the temperament will be tuned sharper (higher) than TCF, while octaves below the temperament will be tuned flatter (lower) than TCF. This is “octave stretching”, and just to make things more interesting, as the notes get farther away from the temp octave, the degree or amount of octave stretching increases dramatically.
All of this is because pianos exhibit an interesting phenomenon called “inharmonicity.” Unlike those of any other instrument (that I can think of, anyway) a piano string is in fact a stiff wire under extremely high tension, which causes the overtones for any given string to be skewed sharp of their theoretical pitches. Furthermore, the amount or degree of inharmonicity increases from each overtone to the next highest.
For example, if the string’s fundamental frequency is 440Hz, the first overtone (one octave) won’t be 880Hz, but instead will be a little higher, let’s say x% higher. The next overtone (octave plus fifth) will skewed even sharper, say x+2%. And so on.
So in a nutshell, if A4 has a fundamental frequency of 440Hz, A5 must be tuned at least x% sharper than 880Hz. And so on and on as you go farther up the keyboard.
Same principle applies to octaves as you move downward. If A4= 440Hz, then A3 has a TCF of 220Hz. However: the A3 string is also inharmonic, so it’s first overtone will be x% higher than 440Hz, and the A3 string must be tuned at least far enough below 220Hz to bring that first overtone down to 440Hz. And so on.
Inharmonicity, with all of its ramifications, is what makes a piano sound like a piano.
Last but not least: in any given piano, the amount of inharmonicity will vary from one area of the keyboard to another; and, except for identical pianos in identical condition, no two pianos are likely to have identical inharmonic profiles. Which means the degree of divergence from TCF will vary from piano to piano.
Confusing? You betcha. But if you haven’t had enough, go here:
I assume that any synth starts out with tones at their TCF. But to make a synth sound like a piano, inharmonicity will have to be programmed into it. I think.
.
NB: I am not a specialist in tuning (barring all the well-temperaments I have screwed around with on my harpsichord), nor in acoustics. I have futzed around at one time in what follows:
For one thing, perceived frequency is nonlinear. One thing the OP is asking (or the trend of the discussion) is on “how narrow can you slice the pitch pie.” For this, psychoacoustics is the field where they now play the game. First, “octaves” is a theoretical construct of the history of the West. And, as has been admirably discussed above by TreacherousCretin, as pure modern physics they are not simple redoubles in frequencies; ( Auditory Perception (Stanford Encyclopedia of Philosophy)]; (/url] Mel scale - Wikipedia] has the math).
The scientific gauge of frequency difference perception (simply known as the just noticeable difference) has two parts. One is the measure of any two pure frequencies (the “frequency difference limen”) or the just noticeable difference in pure waveforms. The other is the just noticeable difference of the fundamental produced by any instrument, which is what the OP asked–the fundamental pitch being the frequency “beneath which” the other waveforms are, and which makes each instrument sound unique–unsurprisingly known as the “fundamental frequency limen.”
Depending on loudness and the general area of the output frequency–important factors–any change lower than 2 Hz cannot be perceived. However, the interference of two even closer pitches can be heard as beating, or the wha-wha when you’re so close you can taste it. I never count the beats between different intervals–except for listenening for none of them–in some of them in the various well temperaments.
As a side note, some people say the beat-count for every interval difference while tuning, but I could never figure it, or I don’t believe them, because the number of those beats changes as you march up the scale. I’m happy to be told off on this subject by other tuners out there…
I’m reviving this posting because I’ve done a little bit of reading about guitar tunings since my first response, which was based on three assumptions:
While the frets on a standard guitar are positioned to accomodate Equal Temperament,
a guitar can easily be tuned to a different temperament, because
changing the temperament involves nothing more than changing the various string tensions as needed.
I haven’t exactly done extensive research on the question, but so far everything I have read confirms all of those assumptions. I haven’t found any mention of having to change string lengths at the bridge, much less reposition any frets.
Re my remarks posted above, I gave the wrong URL. A nice one-pager with a graph and a sound file of the sounds of what what pure frequency division produces w/o non-linear scaling (a cool scale) is at http://www.sfu.ca/sonic-studio/handbook/Mel.html. The math for the non linear basis of perceived equidistant intervals is Wiki-ed at Mel scale - Wikipedia.
–Leo
I don’t think that #3 is correct. You can retune a guitar such the the intervals between the notes sounded by the open strings are in any particular temperament that you choose, but the moment you start to move up the neck, the intervals are strictly equal temperament because of the fixed, unchanging relationship of the frets. Except now you have the problem that a given note on one string will now no longer be the same pitch as the same note fretted on an adjacent string.
Believe in them, because comparing the frequency of interference beats is how pianos have always been tuned. At some stages of tuning, the beat rate (beats per second) of a few selected intervals will be counted, but for the most part it’s more a process of comparing the beat rates of two or more related intervals-- “Faster - Slower”, rather than “x.6 beats per second.”
The bps for any interval will increase as one ascends the scale (from one major third to the next, e.g.), because although the frequencies of the interval’s two notes are increasing, the ratio between them remains the same. That progression of beat rates is what a tuner listens to while setting temperament, octaves, and some other intervals.
Beat rates that are too slow to be discernable are obviously of no use for tuning, and beat rates higher than about 16bps are too fast for us to hear as discrete pulses. But in any section of the scale, including extreme low bass and high treble, for most intervals being tuned there will be some combination of overtones/fundamentals that will beat within the useful range of 7-14bps.
Needless to say, this is much easier said than done.
Well, I did a little more searching, and found the source of your quote, as well as a couple of other sources that agree with it.
Then I rechecked the contrary sources that I’d already consulted earlier.
My conclusion:
You’re right.
Fortunately for me, I’m not too proud (or stubborn) to learn something.
Regards,
Cretin (Treacherous)
.
From Leo, a correction:
Gave the wrong URL. The math for psychoacoustics of listeners actually perceiving equivalent distances is at Mel scale - Wikipedia.
