The Musical Calculator at the bottom of the page will be used to illustrate the features of various musical tunings and temperaments. The calculator may be controlled directly using the Control Panel on the left, but clicking on 'special' text will set it up in a particular way. For instance, click here to show Equal Temperament, relative to itself. The calculator should now be displaying what is called Equal Temperament, showing a set of major scales, with each note displaying its note name. Whenever you come across an underlined section in the text, click on it to illustrate the current point.

If you clicked on the special text above, the calculator will be showing a series of scales centered on the home key of C Major. The scales are laid out like a flattened circle of fifths with the 'near' keys next to the home key and 'far' keys further away. Clicking on any of the key names in the left-hand column will make that the new home key. Try clicking on F#, the top-most key currently displayed. You should see this become the home key, with the keys then ranging from C Major through to B# Major. As in a standard circle of fifths, the diagram will not go further than B#, so clicking C# Major will display its enharmonic key, Db Major. (Practice moving around between the Major keys.)

The relative minor key is shown in the right-hand column. Clicking one of these keys will make it the home key, and display the minor scale, rather than the major. Shifting between major and minor is also possible via the radio buttons in the Control Panel, also offers a third option of displaying the full chromatic scale.

A tuner of an organ or keyboard has to start somewhere, and this 'musical calculator' assumes the key of C as the starting point. Even if the home key is then changed, the tuning will still be relative to the tuning key of C. This simply mirrors the reality that a keyboard cannot be re-tuned on-the-fly and therefore regardless of the key you are playing in, the keyboard will remain tuned to C.

Up to now, the musical calculator has simply shown scales and the relationship between them. Clicking on Cents/Tonic in the Control Panel will show the pitch in cents for each note. (If you are not familiar with the concept of 'cents' read the chapter Ratios and Cents before continuing with this chapter) Since the musical calculator is currently showing Equal Temperament, each note is evenly spaced, 100 cents apart, and the diagram shows that this is true for every key (the big advantage of Equal Temperament).

Every other temperament or tuning leads to a situation where the pitch of intervals is different in different keys. Natural Tuning shows this very well. The note F, the 4th in our tuning key of C, has a pitch of 498 cents, relative to the tonic, but the Bb in the key of F, its equivalent 4th, has a pitch of 520 cents relative to its tonic, a difference which is very audible! This is an unfortunate consequence of Natural Tuning which is explored in more detail in its own chapter, but it is clear to see that many intervals are different in the different keys. What takes a bit of working out is how different each interval is. To make this easier to see, there is a useful option Cents/Tuning Key which shows the interval in cents relative to the same interval in the tuning key of C. In our previous example, it is now easy to see that the F-Bb in the key of F is 21 cents sharp relative to the C-F.

To enable you to hear the difference, the musical calculator has a 'primitive' sound facility. Clicking down on the left mouse button will play the current interval in the currently specified tuning. Releasing the mouse button will play the same interval in the relative tuning. The pitches of the notes being sounded are shown in the status bar at the bottom of the screen, relative to a C at 264Hz (A440). To hear an example, select Pythagorean Tuning relative to Natural Tuning and click the E in the tuning key of C to hear the Pythagorean 3rd as the mouse button is pressed (0-408 cents) and the natural 3rd as the button is released (386 cents). This sharp 3rd is the 'syntonic comma' which was a key feature of medieval music. For a different effect, try the D# in the key of B Major. This Major 3rd is only 2 cents flatter than the 'natural' ratio, so it would be extremely difficult to hear the difference if the notes were played on their own. However, because the tonic is being sounded at the same time, you should be hearing very clear 'beat' when the mouse button is depressed but no beat when the mouse button is released. This is because the difference is only 2 cents between the Pythagorean internal and the natural interval, too close to hear in a line of melody (horizontally) but clearly audible in a chord (vertically)

This difference between intervals in the different keys is going to be crucial to much of the discussion of temperaments and tunings, because it will define how useful a particular tuning or temperament will be. Large differences will render some notes (sometimes whole keys) unusable, which may or may not be important depending on the style of music. To make this point as clear as possible, each of the notes in the diagram is colour coded, according to how far out it is relative to the same interval in the tuning key. Sharp notes go from a dark red, for relatively benign differences, up to 'white hot' for unplayable notes and flat notes go from a relatively okay light-blue through to black, representing notes which should be banished to the outer darkness! If the note is green, there is no difference in the interval and it will sound exactly the same as the equivalent interval in the tuning key

One further useful facility, hovering over a note will display a popup showing all of the detail for that note.

Lastly, a note about enharmonics (if you'll excuse the pun). Looking at note names, the 4th in the key of Db is a Gb. On a keyboard it is obviously the same key as the F#, an enharmonic. However, the way in which pitches are derived for the different tunings and temperaments will lead to a situation where the pitch of these two notes will normally be different. Since the diagram simulates the keys on a keyboard, only one of the enharmonics can be used, and the Gb is actually a F#. The reason for this is that the order in which the keys are tuned goes up and down in 5ths from C, (C,G,D,A,E,B,F#,C#) and (C,F,Bb,Eb,Ab), producing the chromatic scale:

     C - C# - D - Eb - E - F - F# - G - Ab - A - Bb - B - C

All pitches are those of the notes in this chromatic scale, even if the theoretical note name would be an associated enharmonic.