So, I was thinking that since there are an infinite number of points on a line, how could a musician playing a fretless string instrument select the correct, singular, discreet position to play a desired note?
Are they always off by some minute fraction that is indiscernible to the naked ear?
They are always slightly off, but it’s the same with a fretted instrument. Changing the tension of a string affects the frequency it resonates at, and tuning is never perfect. Also, the act of holding a string against a fret changes the tension.
I know a guy who plays fretless upright bass in a jazz band in his free time. His IRL job is with the symphony. We have had discussions about the question in the OP. He says that (at least in jazz) “perfect” notes are not always what you want.
Not necessarily. They likely don’t produce precisely the same frequency every time, but it’s not impossible to hit some given frequency exactly some of the time. However, human hearing cannot discern differences smaller than approximately five cents, so practically speaking even if it’s off by some minute fraction it’s still the right note.
That’s probably true in many forms of music. We analyze and categorize notes and frequencies to help us comprehend them, but it comes down to what sounds good, and that isn’t necessarily the same as a textbook prediction of what it “ought” to be.
This really extends to all instruments. There are variables in tuning for any instrument such that the lack of frets is not necessarily the most important variable. Brass and woodwind players have a hell of a time in those marching bands where they’re outdoors in sub-freezing temperatures. The string quartet that played at Obama’s inauguration had a similar problem, so much so, in fact, that what the audience heard was prerecorded. Cold temps raise the pitch of a horn or string. As mentioned earlier, the amount of pressure on a fretted string can change the pitch. The amount of humidity can dramatically affect the tuning of a wooden instrument as the wood swells or contracts.
The OP seems to be asking a more philosophical question about infinite precision. So a slightly more philosophical answer. In the face of noise, in order to discriminate frequency you need to listen to it for enough time to be able to resolve the frequency. If you wish to be able to resolve the frequency to infinite precision you will need to listen for infinite time. Since we don’t live in a noise free universe, you have your answer. There is a fundamental limit after which it is meaningless to ask if the note is out of tune. So the same answer applies to the finger position on the fingerboard. Even measuring the position is subject to essentially the same limits. Infinite precision requires infinite time.
The notion of in and out of tune is at best pretty thorny anyway. A fretless instrument has a vastly better chance of playing the corect frequency than pretty well any fretted instrument. An instrument (rare as they are) that is tuned to equal temperamant is (being simplistic) just equally out of tune. Pianos and guitars are not tuned to equal temperament. You can, but most people don’t. Everything is a compromise.
Not sure exactly what you mean. A guitar’s frets are placed based on equal temperament proportions, although it’s true that guitarists often do not tune the strings themselves to correspond precisely to a true equal temperament. I think the proprietary Buzz Feiten system uses something different, for example. First time I ever heard that pianos are not tuned to true equal temperament, though. Some tuning issues on the guitar are due to playing 20-24 notes on the same string, and having overlap so that a single note can be played on 3 or 4 different strings, and differences between gauge and action on the strings, etc.
But a piano does away with that by having each note sounded on a dedicated string. Why would a piano not have equal temperament tuning?
You have the same situation with violins and the human voice that have no discrete stops.
As someone already mentioned, no instrument produces a perfectly correct pitch.
A digital sine wave generator theoretically produces a perfect note but ultimately you have to output that sine wave signal to speakers moving air particles to actually hear it. The mechanical limitations of the speaker’s voice coil introduces its own distortions so even that simple sine wave is no longer “pure.”
I’m curious about that. We may be disagreeing on terminology. Surely you stretch tuned pianos, and used one of the well known temperaments - like the Werckmeisters, quarter mean tone, or something similar? Did everyone prefer equal temperament?
The Buzz Fieten system for guitars was clearly based upon an approximation to a piano’s tempered tuning. If you read Buzz’s initial writings on how he developed it he describes how he realised his guitar was tempered differently to the piaono, and how he empirically worked out how to get the guitar to a closer approximation. His tuning involves a mixture of mechanical changes (moving the nut,) a specific intonation of the bridge and a set of tuning offsets for each string. The desire is a tuning that gets you a wide range of chords in harmony. If you talk to many luthiers they will tell you that they have used similar ideas for centuries. Most guitars are set up so that one part of the fretboard is more useful harmonically than the rest. Where you put that area depends upon the player.
Where as the guitar fretboard is laid out in principle to create the equal temperament - the rule of 18 for fret placement gets you close enough to the 12th root of 2, the reality is that the mechanics of the system plus the manner in which you tune each string relative to one another - and the exact location of the bridge, means that it isn’t equal tempered.
Yes, I think we may be misunderstanding each other.
Temperament and stretch tuning are two separate aspects of piano tuning.
I was a full-time, professional Piano Tech for 32 years, worked for everyone from “hands together” beginners, to concert pianists. I also did occasional work for a few concert harpsichordists. In all that time, I had exactly one (1) client request a non-equal temperament-- a classical pianist who wanted to work with selected baroque pieces in their historical temperament.
There have always been pianists and Piano Techs who use non-equal temperaments, but as far as I was ever aware they are VERY few and far between. I don’t mean that to sound dismissive or critical-- I’m not knocking them. Just saying that Equal Temperament has been the standard tuning for pianos since the mid 1800’s.
On a piano, the beauty of ET is that one can move freely from any major or minor key to any other, because no one key signature sounds better or worse than any other.
With a guitar, one can tune to any temperament one wishes, to suit a particular piece of music, or one’s personal preferences. Changing to a different temperament can easily be accomplished in a minute or two.
Not so with a piano. Changing the temperament requires a complete retuning of the entire instrument, which is extremely time-consuming and expensive, if done properly.
Regardless of which temperament is used, any properly tuned piano has stretched octaves. The variable factor is how much the octaves are stretched; it can depend on the characteristics of the specific piano being tuned, the judgement of the particular Technician doing the tuning, and the preferences of the pianist.