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Just intonation reference


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Hello peoples. (yes, with an s)

I'm trying to program a nifty little mac application (to use my old and otherwise useless powerbook 170) that could assist me with providing me some drone notes while I play some passages. The precision of the sound chip is rather good in terms of frequency. Also, programming a little software using this would mean that I'm in command over how these values are calculated.

Of course, I can start from the orchestral A at 440Hz, which is a standard reference. But from there, what are all the aspects which must be taken into account? Tell me if my plan is reasonable:

I think I'd let the program require the user to specify a scale, and use the well-defined factors from that base note to all others of the scale. The thing is, how do you do it in a nifty way while still holding A=440Hz true? What about the other open string notes for the violin, do they also have to be held as hard references? The problem with doing this, is that if you have many "hard reference notes", you cannot define exact frequencies for all notes of your scale and still keep these open string notes intact.

I'm starting to think there's something from the math aspect of music theory that I've forgotten.

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There is a way suggested by my Inside Macintosh documentation, but I have an inkling it's going to be subpar for violin practice. They provide a huge list of the frequency values used in an equal-tempered scale across 7 octaves. They also provide the just-tempered values (providing, for example, different values for C# and Db) for the C key. Then they tell you a way to construct the values for another just-tempered key: take your pick of value from the equal-tempered list and construct your scale from there. That will change the value of A=440Hz in general, and might leave it unchanged in only a few cases.

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I don’t really pay attention to the just temperament versus equal temperament. In real life you have to play in equal temperament with other equal temperament instruments, notably the piano. When playing by yourself, you tend to play in just temperament because it sounds better. I’d stick to equal temperament using a computer and treat it as playing along with a tempered scale instrument. Tempered scale midi is handy for practice because it’s easy to slow it down with software.

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This process may be more useful than its final product. Have fun proving again how awesome it is to have a good pair of ears. You'll enjoy your findings in string quartet rehearsal, provided that at least one of your colleagues has as much on the ball as you have, and that all four of you hear well.

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Okey doke.

Well-tempered tuning.

n = number of half steps from A

Frequency = 440 x 2^(n/12)

To go up a fifth: 440*2^(7/12)

To go down a fifth: 440*2^(-7/12)

(Here I use '^' to mean raised to the power of.

In a spreadsheet, write 440*2^(n/12) . For n, you have to insert a cell reference.

For perfect fifths:

An interval of a fifth is 3/2 or 2/3.

I.e., to go down, take 2/3 of the frequency; to go up, take 3/2.

Here's how it works out:

----------Tempered--Perfect fifths





Fifths work out pretty well, as you can see. Work out major and minor thirds. They clash pretty badly.

Other "perfect" intervals:

Fifth 3/2

Forth 4/3

Major 3rd 5/4

Minor 3rd 6/5 ? sort of. Not sure. I have to think that one through. Thirds and more complicated intervals get kinda fuzzy, hard to hear because you are listening for higher harmonics. Clear as mud. Read on....

The idea of the perfect intervals is that higher harmonics match. Example: 1st harmonic of E string is and E with twice the frequency.

Second harmonic of A string is an E with 3 times the frequency. You've probably tried this with your instrument. So when you tune to "perfect" fifths, you tune to 2/3 or 3/2 the frequency. Get it? I thought not. Just joking.

Don't ask me any more complicated tuning systems. Durned if I know.

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There's a guy who sells a cd called "The Tuning CD" that does pretty much what you describe- I think his drones have the tonic and the fifth of the scale. I have one and it can be handy sometimes, for scales and arpeggios especially. (Most pieces switch harmony too fast too make a drone useful, but if you're practicing just one bar or so it handy there too.) I forget the guy's name but he advertises in the International Musician all the time- he has some kind of circular breathing cd out too, which I've never tried, being a string player.

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I think my qualms can become clear if I give an example of a problem situation:

Let's say the key is F. There is no quick relation between F and any open string (no direct octave, nor fifth relationship between it and any open string). The fifth above F is C (not an open string). Do you set that fifth relation in stone in a just temperament setting, even though that might give it a less than stellar relation to open D for example (haven't checked it or anything). OR, do you still give priority to the open strings (and the octaves they can form with 3rd finger in 1st position, for instance) because they so often come into minor sympathetic resonance. This might happen if your fingers are playing a certain note that might share a higher resonance mode with any untouched open string.

I know all about equal-temperament and even already have sheets of frequency-related values for them. I just wanted to figure out the next step (i.e. perhaps a system based on just intonation, but that takes care of the unique intricacies of the violin's open strings). I will check out the link that was provided in this thread a bit later (practicing now!)

Thanks for your input.

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There is a wonderful world out there to explore, that I have barely begun to explore. Mean tuning or just tuning on an instrument such as a harpsichord can sound absolutely heavenly, while well-tempered tuning can sound pretty crude, depending on what you are playing. I can personally vouch for this. "Fast and loud is best," to quote PDQ Bach. It covers up all beats and other awful sounds.

For a start, I searched Google for just tuning. The fifth hit is wonderful, and goes on to give you a very nice explanation of historical tunings. The first hit gives you some nice sound samples, which unfortunately are damped, not sustained, and contain an unknown mixture of overtones.

Fortunately, as string players, we can sometimes dispense with tempered tuning, and get heavenly results. It doesn't get any better than a string quartet, except maybe for a gamalan. The good news is that the ear is just as good as math -- maybe better.

Moving on, there's this nice explanation of bells . Bells don't work well for major chords. They sound minor.

Hey, Maestronetters, this would be a great time to tell us what you think of intonation. And MuOn, let us know what you find discover.

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In F major, the third is an A. For a start, consonance of the thirds is very important, and just thirds or mean-tuned thirds don't sound much like tempered thirds. Once you have the third, the triad is also in tune. I'll let you or someone else fill in the other important chords, depending on what you have in mind musically.

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