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Window 81

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Calculator

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Calculator

Calculator

Use this for calculations - with special options useful for scales work

Value(s)

Enter a value as ratios, decimals, scale notation... depending on selection...

Enter any number of values separated by commas. The result of the calculation will immediately appear in the results field.

Select what type of value it is, enter the values, and choose how to show the results.

You can also formulae and this applies in nearly all the Tune Smithy fields where you can enter numbers.

You can use phi, or g for the golden ratio, pi for pi, and a wide range of functions:

Most commonly needed are : ^ for powers so 2^3 is two to the power of three, i.e. 8. Brackets to any number of levels, so things like (3/2) / (6/5) are valid.

You can use sqrt for square root so sqrt(2) is the same as 2^0.5 or 2^(1/2) - which is the interval of the tritone in twelve equal temperament - as cents it will be shown as 600.0 cents.

Other functions recognised include: sin, cos, tan sinh, cosh, tanh, sec, cosec, cot, acos, asin,

Trig functions are in radians so sin(pi/2) is the sin of 90 degrees, i.e. 1.

If you want a trig value of an angle in degees, do it as

sin(angle*pi/180) e.g. 90 degrees as sin (90*pi/180), and similarly for the other trig functions that expect angles.

Then there's log, and exp.

You can also use Jn(m) for the nth bessel function (e.g. J2(pi) )

Ratios use wide 64 bit integers for the top and bottom numbers. The largest number treated as an exact whole number is 9,223,372,036,854,775,807 (nineteen digits, value 2^63 - 1)

Nearly all the places in Tune smithy where you can enter scale values accept ratios at this level of precision.

So ratios like 2^62/3^33 can still be evaluated as ratios of exact integers in most areas of Tune Smithy - which is useful for some calculations needed by microtonalists, e.g. some of the xenharmonic bridges etc.

Values are

Format for the values - when set to scale notation, you can use any recognised scale notation...

When set to scale notation you aren't restricted to the current selection for the scale notation.

Whatever the notation you are currently using to display values, a value like 3/2 is always read as a ratio, a value like 702.0 as cents (so long as you have decimal point = cents selected), and formula can be used like 3^3/2^4.

The other notations such as the lattice notations and alternative cents notations are also recognised - just enter the values in the same way that they are displayed in Tune Smithy. It is done in such a way that each one is unique so Tune Smithy will always be able to recognise what you enter no matter how you currently have scale values dispayed.

Evaluate each entry separately

For comma separated lists of values, evaluate each one separately...

Try a scale for instance:

1/1, 9/8, 5/4, 4/3, 3/2 ,5/3 ,16/9, 2/1

Probably the most useful for most :-)

Be sure to remember the commas.

Multiply

Multiply all the comma separated values together to give the result

e.g. for 2,3,5 finds 2*3*5 = 30

Divide

Multiply all except last, then divide by the last one

e.g. for 2,3,5 finds 2*3/5 = 6/5

To power of last

Multiply all except the last one, take the result to the power of the last

E.g. for 2,3,5 it will find (2*3)^5 = 7776

Add

Adds the comma separated values

for 2,3, 5, finds 10

Subtract

Add all except last, then subtract the last one

e.g. for 2,3,5 finds 2 + 3 - 5 = 0

Multiply by last

Adds all except last, multiplies result by last

2,3, 5 finds (2+3) * 5 i.e. 25

Result(s)

Result of calc - use the Show it as drop list to choose how to show these

Show it as

How to show the result of your calculation...

Most of these will be familiar to most who have interest in musical intervals, harmonic series, microtonal music etc..

A brief summary for those new to them - the ratios notation is usually used for ratios of frequencies, so e.g. 660 Hz is a 3/2 above 440 Hz. The cents notation shows the size of a musical interval in hundredths of a (twelve equal) semitone. So there are 1200 equally spaced cents to an octave.

So for instance 5/4 the ratio eg. between 440 Hz and 550 Hz turns out to be about 386 cents, or slightly under a third of an octave (the twelve equal major third at 400 cents is exactly a third of an octave, but that is because it is fractionally sharp compared with the harmonic series based pure ratio 5/4).

If you develop an interest in microtonal music it usually doesn't take long before one encounters these cents and ratios.

However, one of the notations here is a bit more specialised

.................The prime Exponents notation.....................

The Prime Exponents (Monzo) notation here will be new to many. It shows the ratio in a lattice notation using [ and >. It may take some getting used to but is quite simple really at least as used here.

It lists the powers of each prime in increasing order, 2, 3, 5 etc.

So for instance 5/4 is 5/2^4. Since 1/2^4 is the same as 2^-4, then that can also be written as 5^1 * 2^-4

Writing that in increasing prime order, and noting that 3 isn't present so should be raised to the 0th power, it is

2^-2 * 3^0 * 5^1

Now remove the prime numbers and put the brackets and commas in and you've got the Monzo:

[-2, 0, 1> = 5/4.

It takes a bit of getting used to but many find it a useful way to see at a glance what the prime content of a ratio is.

As for why they use [ and >, that leads into higher algebra.

It's not actually my own subject area of mathematics particularly, as I'm more of a geometer / logician myself. But I gather there are close connections with the Bra-ket notation used in quantum mechanics. There are left angled type constructions as well <,,] which algebraically inclined scales theoriests use as operators on the ratios, just as they do in Q.M. But as to what those are, that gets you into rather intricate complexities to do with algebraic methods for lattice constructions. I'm sure you will get interested help over at the yahoogroups tuning-math forum if you want to find out more about it.

Scale Notation Number Opts...

Set the notation such as cents, ratios etc used for display of all scale values...

Also has various options to configure how other numbers (volumes etc) are displayed in Tune Smithy.

Ratio Opts ...

Options for display of ratios - factorise, lattice notation, etc.

Ratio Opts ...

Number of decimal places for the Results for the Calculator

Help = F1

Click for help for this window. Or F1. Other opts: Shift , Alt, Ctrl + click...

F1 or click shows the help for the current window in your web browser.

Some windows may have no help yet in which case the help icon is shown crossed out with a red line.

Shift + F1 or Shift + Click brings up the tool tips extra help window (this window) to show any extra help for a tool tip.

You can tell if a tool tip has extra help if it ends ... like this one.

Ctrl + F1 or Ctrl + click takes you to the list of keyboard shortcuts for Tune Smithy.

Alt + F1 or Alt + click (alternatively Caps lock physically held down + F1 or Click) takes you to the on-line page at the robertinventor.com web site about the current main window task - which gives a short introduction to it for newbies to the program. If there is no on-line page specific to a task, takes you to the main tune smithy page on the web site.

Since the help for Tune Smithy is currently a bit out of date and needs to be redone completely for the new 3.0 release, then you may find the on-line page for some of the newer tasks particularly useful.

Organise Windows = F2

Reset, or Save settings for this window. RIGHT CLICK for all windows menu...

Shows the Organise windows window - which you can use to reset all the parameters for the current window.

You can also use it to save the settings for just this window, or open previously saved parameters for just this window.

Also has a drop list of all the windows and their shortcuts, and related options - some to do with the menu listing, and some to do with window resizing and minimising.

Other Dialog Star

Tip of the day - For All category - right click for neighbouring windows...

Left click for a tip of the day in this category.

Right click to see a menu of neighbouring windows.

The neighbours are the ones you most often move to after this one or within a minute of this one, arranged by popularity.

So as you continue to use FTS, it will learn your habits, and the neighbouring windows listed here, should be the ones you most often visit after this one.

Less <<

Shows this window with either less space, less options, or alternative layout

Neighbours, and Previous - Up - Next

Scale Notation and Number Options (Ctrl + 18)

Scales Drop List (Ctrl + 110)

Help For Tool Tip - Shift + F1 (Ctrl + 141)

Organise windows (Ctrl + 51)

Scale and Arpeggio as text (Ctrl + 104 or Ctrl + T)

N.B. This list of neighbours may change when these pages are updated.

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