TonalityGUIDE - Tonal music basic theory for Undergraduates
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There is a close relationship between any two keys whose tonics are a perfect fifth apart (the page on fifth-related major keys looks at the example of D and A major).

Instead of writing out the notes from these scales on a stave to show this relationship (as on the page on fifth-related major keys) you can write out the note names in a line in order to get an overview of a larger number of fifth-related scales in their wider context.

The easiest way to write all the flattened and sharpened notes of the chromatic scale in a series of perfect fifths is to start with the scale of C major organised as a series of perfect fifths .

Remember to start on F (the subdominant), otherwise you will include the diminished fifth between B and F. You might know a reversible mnemonic for remembering this series: Father Charles Goes Down And Ends Battle

Then, write the same series out twice more. The first series should be flats, the second the natural notes of C major and the third series should all be sharps as below. The result is a long series of perfect fifths starting at Fb and finishing on B#:

Any group of seven notes on this series is a major scale starting on the second note of that group. As you can see below, D and A are very closely related with six notes overlapping. It is therefore easy to modulate from one to another - you have to change just one note.

It is harder to modulate smoothly from A flat major to A major which are a long way apart on this series of perfect fifths, and have no notes in common.

Flatwards and Sharpwards
As you modulate from right to left on this series, you accumulate more sharps, so this is known as a sharpwards modulation. As you move from left to right, on the other hand, you accumulate more flats, so this is known as a flatwards modulation. As with diatonic progression by fifth, modulating by descending fifth (i.e. flatwards) decreases tension while modulating by ascending fifths (sharpwards) increases tension. This is partly because the dominant in a flatwards modulation is diatonic in the old key whereas it is not in a sharpwards modulation.

Hang on a minute! Aren't C# and G# the same note as Db and Ab??
This issue is too much of a minefield for a quick aside, although the box at the bottom of the page gives a brief explanation. The long and the short of it is that our tuning system is a distortion that allows tonal music to keep modulating up or down this series of fifths and eventually arrive back in the same place.

Unlike the series in the second diagram, which can be expanded infinitely using double and triple sharps and flats, the modern tuning system is called equal temperament and results in a closed circle of fifths as below.

The circle of fifths is useful, because it let you see quickly which keys are closely related to each other. Ab and A major, for example, are a long way from each other on the circle of fifths, so they will not have many notes in common, whereas F# and Db have lots of notes in common. One of the most common modulations is by one step around the circle of fifths in either direction - sharpwards or flatwards. You can find some examples in the relevant section of longer progressions.

Be careful not to confuse the modulatory circle of fifths with the diatonic chains of fifths that stay in the same key.

Closed circle of fifths

Infinite series of fifths

If you carry on flatwards around the circle of fifths from Db major you arrive at Gb major (in brackets on the diagram). In modern tuning, Gb is the same as F# major and this is called enharmonic equivalence. The two notes are 'spelt' differently, but they sound the same.

Tuning and Temperament
  • if you tuned a set of strings in ascending perfect fifths starting in C you would never arrive back at C after 7 octaves like you do on the piano - the resulting B# would in fact be considerably sharper
  • in the days when music did not modulate much or cover a particularly wide range this was not a problem
  • as music has ranged more widely over the last half millenium, a standard system has gradually evolved in which fifths have been made slightly smaller or tempered so that 12 fifths fit within the 7 octaves (the same applies to all the other intervals)
  • this system is called equal temperament and it closes the circle so that, in the example above, C# = Db and Gb = F#
  • part of the controversy arises over which composers would have been familiar with which tuning systems or temperaments, because equal temperament is one of many competing systems that have been used over the centuries
  • there are many books and web sites that discuss this issue in detail