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Voltage values for 1v/oct just intonation.

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Eichburger
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Voltage values for 1v/oct just intonation.

Post by Eichburger »

Does anyone know how to calculate voltages to obtain a particular pitch ratio, assuming that the root note is at 0v?

So for example if I want to set a voltage on my sequencer to obtain a justly tuned minor third with a ratio of 6:5 (again assuming the root note is at 0v) would it be 0.2v?

Similarly would a perfect fifth be 0.5v?

Is it that simple or am I missing something? Thanks for any help.
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mskala
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Post by mskala »

The voltage needs to be the logarithm to base 2 of the frequency ratio. For rational frequency ratios (except powers of 2, that is, octaves), this will always be an irrational number.

If your desired frequency ratio is A/B, you want to compute voltage = (log(A)-log(B))/log(2). Your calculator will probably offer a choice of "natural" or "base 10" logarithms and it doesn't matter which one you use as long as you use the same one for all three.

So for instance, if you want frequency ratio 6/5, you compute (using natural logarithms):

log(6) = 1.7918...
log(5) = 1.6094...
log(6)-log(5) = 0.1823...
log(2) = 0.6931
(log(6)-log(5))/log(2) = 0.2630...

and you need 0.2630... volts of input to obtain the pitch ratio 6/5. If you used base 10 logarithms the numbers would start out different but end up with the same answer.

Bonus calculation: multiply voltage (which is number of octaves) by 1200 to get cents, or divide cents by 1200 to get voltage.
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Post by Phil999 »

interesting, mskala, especially that the base of the logarithm doesn't matter as long as one uses the same base for the entire equation.

Thank you (although I probably won't be able to apply this in the real world since I don't have such a sequencer).
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Post by mskala »

Phil999 wrote:interesting, mskala, especially that the base of the logarithm doesn't matter as long as one uses the same base for the entire equation.
That follows from the core properties of logarithms. log_{base x}(y) = ln(y)/ln(x) where ln() is the natural log. Then if you substitute a different base x for all the logs in the equation I posted, then you're multiplying or dividing the numerator and denominator by the same number, and it cancels out. It may help to think of the different "bases" of logarithms as different units of measure, like feet and metres; measure the same object with different units and you get different numbers, but if you're interested in ratios and proportions, you end up getting the same practical answers from either as long as you're consistent about it.
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Post by slow_riot »

Another way to do it is:

Put your frequency ratio expressed as a floating point number instead of a fraction into this calculator.

http://www.sengpielaudio.com/calculator-centsratio.htm

Obtain the cents value, divide by 1200 (on a sequencer an octave made up of 1200 cents is represented by 1 volt).

e.g. for 6/5

6/5 = 1.2

is an interval of 315.641287 cents.

divided by 1200 this gives 0.26303440583333333333333333333333 V
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Eichburger
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Post by Eichburger »

Thanks mskala and slow_riot. That makes complete sense. Better dust off the old scientific calculator...

I'm fed up with trying to use tuners to set sequencer voltages, and wondered if it would be easier to use my multimeter to measure the voltage directly. will give it a try and see how I get on.
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Post by Phil999 »

mskala wrote:It may help to think of the different "bases" of logarithms as different units of measure, like feet and metres; measure the same object with different units and you get different numbers, but if you're interested in ratios and proportions, you end up getting the same practical answers from either as long as you're consistent about it.
thank you very much for this lesson or reminder.
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Post by maf »

Thanks for all these useful tips!
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Post by flo »

I made this chart for (C-based) 12 tone just intonation some time ago; if you have access to a sequencer that let's you dial in precise voltage offsets, it's easy to get JI going...

Image

Cheers :guinness:
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Post by adam »

do the values change for each octave though? i thought each one was different
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Post by flo »

Not for 12 tone just intonation.

However, they change if you use another key (other than C). The beauty of CV is that you can just "transpose" (modulate) to the desired key with an offset and still retain those same numbers (in other words, key modulation and the actual scale can be kept separate and thus avoid the modulation problems of JI with "traditional" instruments).
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Post by adam »

thats good - the roland grooveboxes had alternate tuning modes iirc, 505, d2 etc anyway, seems like this kind of thing should be much easier now with mcu's running the show, higher resolution dacs etc
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Post by ndkent »

flo wrote:Not for 12 tone just intonation.

However, they change if you use another key (other than C). The beauty of CV is that you can just "transpose" (modulate) to the desired key with an offset and still retain those same numbers (in other words, key modulation and the actual scale can be kept separate and thus avoid the modulation problems of JI with "traditional" instruments).
People learning certainly wonder, if Just sounds better then why aren't we using it instead of ET all the time?

The reality is when you use a tuning with perfect ratios (or alternately closer to perfect ratios), it only works best with the key you calculate with. So looking above, if you calculate for C as your root, your results for something played in C will superb. So as I understand it, if you look on a "Circle of Fifths" chart, the keys closest to what you calculated for will sound ok though slightly less perfect, then subsequently the further you get, the worse it goes. So take C, basically F and G (fifth above and fifth below) will sound pretty good. You really hear in in chords you make playing in the key. But Gb or F# is the furthest and will sound terribly out of tune if you play a chord in those keys.

So as mentioned, yes, you certainly can take these ratios and voltage offset them so you are now pitched in Ab or whatever. Though one does have to figure out when and exactly how physically to implement that offset.
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Post by mskala »

Yeah, there are issues for instrument design - if your instrument can't change its tunings on the fly and you tune it for just intonation in one key, then it'll only be just in that one key and no others. However, there are other reasons to prefer a non-just tuning - like being able to do tritone substitution and other jazz harmony stuff. You may also want to use dissonance on purpose, and a tuning intended to make everything as consonant as possible isn't a natural fit to that.
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Post by flo »

I concur with you in all those points, ndkent and mskala :tu: :guinness:
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Post by jkjelec »

Maybe linear (Hz/V) and exponential (V/Oct) VCOs are being discussed at various times in this thread? Most of the responses are relevant to Hz/V systems, but flo's chart and slow riot's ratios are relevant to V/Oct systems. :hmm:
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Post by mskala »

The thread title says V/oct, and I don't think any responses here are relevant to linear (Hz/V) systems. In a Hz/V system you would multiply voltages to achieve an interval, not add them.
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Eichburger
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Post by Eichburger »

mskala wrote:Yeah, there are issues for instrument design - if your instrument can't change its tunings on the fly and you tune it for just intonation in one key, then it'll only be just in that one key and no others. However, there are other reasons to prefer a non-just tuning - like being able to do tritone substitution and other jazz harmony stuff. You may also want to use dissonance on purpose, and a tuning intended to make everything as consonant as possible isn't a natural fit to that.
I know I mentioned just intonation in the thread title but for me its about getting away from ET tuning and its is nothing to do with 'accurate' tuning.

The fact that there are 'errors' in ET tuning are less of an issue for me than the fact that those errors are always the same. I tend to tune 'fast and loose' and am perfectly happy with errors as long as they sound ok for the particular piece.

I also am keen to use the 'notes between the notes' such as harmonic sevenths and septimal thirds.

More than all of this I want to break out of the whole mindset of scales/chord progressions/modulation so not being able to move key is really not an issue. All intervals that can be created are there to be played with if you escape the straight jacket of western musical theory and it seems pointless to me to try and shoe horn alternative tunings back into that system.
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