IntervalCalc^TM

IntervalCalc^TM is a calculator which is arranged so that it is very useful for working with musical tuning systems. It converts cents to ratios and back — yes. And so can any other calculator or slide rule that can handle logarithms. It can also describe those ratios using terms in english. That’s pretty neat too, sure.

But what makes IntervalCalc^TM extra special is the way it is set up. Every field you can type in can handle not just regular numbers, but mathematical expressions as well. By mathematical expressions, I mean things like 7*13 and (7:5)^(1/13) (seven times thirteen and seven-fifths raised to the one-thirteenth power.) And if that’s not cool enough, the individual numbers within your expression can be incremented and decremented very quickly by using the up and down arrows. A subtle change from the usual method of entering numbers, but what a world of difference it makes.

I wouldn’t have gone to the trouble of writing a custom application to do this simple task if it didn’t generate a massive improvement in my ability to search for interesting tuning systems — which it does. And it can do so for you too if you are a composer or theoretician who is interested in composing interesting, mind-blowing, haunting, striking, mesmerizing, intense, or just plain beautiful music.

For example, let’s say you have become enamored with the lovely minor septimal tritone interval (7:5) and you want to use it as the base for an entire tuning system. If you want to see what sort of just ratios you approximate by dividing the ratio of 7:5 into — say — 13 equal parts, you can do so very easily. You start in ratio-to-cents mode and type in 7 and 5. Right away, you see that that interval is equal to 582.51 cents. Next you move into cents-to-ratio mode — you’ll see that the 582.51 cents has been automatically entered in the edit field. Now you can edit the entry to read 582.51*1/13. Hit the calculate button and you see that that interval is 4.1 cents flat of the just ratio 13:12. Put the cursor next to the number ‘1’ and press the up arrow (or use the mouse wheel if your mouse has one). The 1 increments to 2. The calculation again shows a nearby just interval and the difference. Press the up arrow again — the 2 increments to 3. When the interval has a name in english (such as the fifth, 3:2; or the subminor 3rd, 7:6), that name is also displayed. Very quickly you can zoom through all of the intervallic resources you will have by basing a tuning on the 13th root of 7/5 — or any other base. (In the thirteenth root of 7:5, you’d quickly discover that you have more than a bucketful of awesome harmonic and melodic possibilities available.)

You just have to try it to see how much more efficient this method is than using a regular calculator or spreadsheet. It’s a genuinely useful musical tool.

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Below we see IntervalCalc^TM working in Ratio to Cents mode. This mode takes a ratio — like 2:1, 3:2, 4:3, 5:4, 6:5, 7:6, or even 7:5, 7:4, 12:7, or 13:11 — and converts that value to cents, the unit which is used in music as a way of precisely measuring intervals. There are 1200 cents in the octave which is a ratio of 2:1.

Here we see that the interval of 7 to 5 is equivalent to 582.51 cents. Also, we see 7 cycles of a sine wave and 5 cycles of a sine wave, then what it looks like when those two waveforms are added together.

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Below we see IntervalCalc^TM working in Cents to Ratio mode. This mode not only shows which ratio is nearest the value in cents which is typed in, but it also describes that interval in English if it can.

• Example 1: If you give it the interval of 700 cents, it will tell you that 700 cents is two cents flat of a pure fifth — which is the ratio of 3 to 2 (3:2).
• Example 2: Type in 1200 cents, and it tells you that there are 1200 cents in an octave and that the octave is just another name for the ratio of 2 to 1 (2:1).
• Example 3 (shown): Here we see IntervalCalc^TM identifying the interval which is 6/13 of the septimal tritone — turns out that that interval is a mere two cents sharp from the subminor third — that extra flat third used in jazz which is a ratio of 7 to 6 (7:6).

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Other Features

• IntervalCalc^TM also reduces expressions (shows what they are equal to) when you press "CMD-=". Expressions with ratios and integers get reduced to lowest terms. In other words, if you type in 11:3-5:12 and reduce, you will get 13:4.
• Holding down the shift button while using the arrow keys to increment or decrement a number in an expression causes the number to change in steps of 10 instead of 1.
• Graphical display can be drawn using sine, square, or sawtooth waveforms.

More about Expressions

IntervalCalc is a general purpose calculator, including logarithmic and trigonometric functions.
Examples of expressions it can handle:

3 log 100

log (3:2) / log (2^(1/1200))

sin (2 pi / 360 * 51.72)

2 + 7 choose 3 * 14

1 + factorial(11)

e^4

"71077345" base 9

"11111110" base 2

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Obtaining IntervalCalc 1.7:

• IntervalCalc is no longer available. This page is retained as a reference of features for those still using it.