I have been studying temperament and I have a few questions about just temperament that are not quite answered by what I have been able to find on the wired. I am hoping someone here could direct me to a bit more detailed (math oriented) discussion of the topic or maybe just answer these questions off the top of their head if they know.
Okay, first question. (nomenclature)
Suppose I start by tuning the open A-string to 440. And then tune the E string to that A string by playing both strings together and adjusting the E string until the ratio of frequencies is exactly 1.5:1 (or 3:2) which is 660 Hz. Is this 3:2 ratio on fifths all that is required to establish a "just" temperament? I think so, therefor I am gonig to use that terminology in the next question, but if not, some of the questions below might server to elucidate where I have gone wrong in my understanding of just temperament.
Question 2 (playing octaves on justly tuned instruments)
First, suppose a break out a hypothetical perfectly tuned, equal temperamnet piano. If we play that with our justly tuned violin, then that "E" at 660 Hz, will be slightly sharp (about 1.955 cents sharp) compared to the piano's 659.2255 Hz. For that matter, every note we play on that E string will be likewise sharp (from the point of view of the piano) by exactly the same amount. So far, so good. From what I understand, that is just the consequence of playing a justly tuned instrument alongside an instrument tuned with temperament. So put the piano away and stick with justly tuned instruments for now. Suppose I crawl up the E string to the A above E. That note should be exactly one octave above the open A-string. But that E-string is sharp... and if you play it simultaneously with the open A there will be dissonance (about 2 cents worth). Is it the responsibility of the violinist to play that note on the E-string just slightly flat so that dissonance goes away? What if your are playing the higher A (on the E-string) by itself, without the open-A would it still need to be flattened? (if you don't you won't get the harmonic resonance with the A string.)
This brings us to question 3 (the notes before the 5th)
Suppose that one the A's on my hypothetical piano is tuned to the same 440 Hz as the open A on my justly tuned violin. Now if the Piano were present I would want to climb the A string in such a way that the 2nd (B), the 3rd (C#), and 4th (D) were in tune with my equal temperament piano. In that case I would have 100 cents between each equally spaced tone. But now consider that I have only justly tuned instruments nearby. Do I need to adjust my 2nd, 3rd and 4th so that when I get to the 5th (E) it is tune with my (sharp - 660Hz) open E? That is to say, do I use evenly spaced, but wider, steps between each note on the A-string so that 100 "just-cents" are equal to about 100.2793 "equal-temperament-cents"? Or do I play the 2nd, 3rd, and 4th on the A string as if I were playing with the piano and then sharpen only the 5th to make it match my open E (to catch the harmonic resonance off the open E string)? Perhaps I do neither of those things! Perhaps in a "just temperament" I play 3rds and 4th slightly flat (relative to equally tempered piano) to achieve precise 4:3 and 5:4 frequency ratios. Then I could then play the 2nd (B) slightly sharp so that it is an exact octave below the (justly tuned) fifth over open E (also a B but one octave higher). The point is, that the spacing between on a just instrument tones is not constant at 100 cents or some other value but dependent whatever you might have played recently or simultaneously
This discussion has been archived and is no longer accepting responses.
Violinist.com is made possible by...
Discover the best of Violinist.com in these collections of editor Laurie Niles' exclusive interviews.