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Harmonics and just temperament

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Play a low note on a stringed instrument. Then lightly touch the middle of the string with one hand while continuing to bow (or pluck) with the other, and you will get the note an octave above. Touch a third of the way across, and you get the note an octave and a fifth above. A quarter of the way across, and you get to two octaves.

In the next octave you get the major third, fifth again, and then the seventh harmonic is a note which isn't used in our system, so sounds out of tune to us (ratio 7/4). It is used in jazz. Then you get another octave again, this time three octaves above.

The notes get gradually closer together as you get higher.

[help_files/harmonic_series_first_five_octaves.ts harmonic_series_first_five_octaves.ts]

[tune_smithy_links.htm Tune smithy links options]

[[Image:harmonic_series_conventional_score.gif|571px|Score of the harmonic series from C upwards]]

To show the notes like this in FTS, use Bs | Tune , and select Score from the drop list, then play the tune for the .ts file.

Here is another way to show them.

Harmonic series.gif

This time on a special score which lets you position notes accurately according to pitch (the dashed lines are all Ds). To show the notes in this way you select Bs | Tune | Options | Nudge to pitch .

For a conventional score with sharps and flats, see Steven Langford's The Overtone Series Demonstrated in the Key of C.

In fact, all these notes are present in the original low note, and if you have keen hearing, you might be able to hear some of them, as its "overtones". All you are doing by touching the string lightly is to select out some of the overtones by damping other ones so that they stop sounding. (Some instruments, notably the piano, and the church bell, have extra non harmonic partials, which are out of tune with the fundamental and contribute to the characteristic piano, and bell sounds).

If you are a string player yourself, try sounding an overtone, then play the original note, and see if you can hear the overtone still sounding.

Since all the overtones belong to the same note, they sound good together. [harmonic_series_notes_sound_well_together.htm More details] .

It can sometimes be confusing that ordinals such as "fifth" are used in two ways in this subject.

Fifth, used on its own, refers to the fifth note of the major scale. In this case, the fifth note you reach if you sing a major scale upwards from the note that starts the harmonic series.

When one wishes to talk about harmonics, one says third harmonic, fifth harmonic, ....

The other thing one has to get used to is that notes that are an octave apart sound as if they are the "same note", only higher. So the third, sixth and twelfth harmonics are all the "same note"; they are all fifths. They are the octave and a fifth, two octaves and a fifth, and three octaves and a fifth above the first harmonic, and those notes are all the fifth note of the major scale in their respective octaves.

The fifth note is the same in both minor and major scales, but the third differs, so one also talks about the major or minor third. If one says "third" on its own, this is understood to be the major third unless the context is clearly about minor thirds.

To add to the potential for confusion, by coincidence, the fifth harmonic is the third note of the major scale, and the third harmonic is the fifth note of the major scale. Hopefully that is understandable now.

A pure major chord, as it is most usually described, is

1/1 5/4 3/2 2/1

Here 3/2 is the third harmonic, originally in the second octave, and divided by two to bring it down into range. The 5/4 is the fifth harmonic, in the third octave. It's divided by four this time, because you have to drop it down by two octaves to get the note back into the same octave as the 1/1, 3/2 and 2/1.

If you multiply a ratio by two, or divide it by two, you get the "same note".

Starting from C, the notes of the harmonic series are:

 C  C  G  C  E  G  -  C  D  E  -  G  -  -  B  C
 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

The ones shown with dashes are out of tune in the ordinary twelve-tone scales.

[help_files/in_tune_harmonic_series.ts in_tune_harmonic_series.ts] (played on 'cello)

In tune harmonic series.gif

We've already got all the white notes except for F and A.

Here it is again, this time showing only the new notes each time, all transposed down into the same octave (and ending with the octave as an extra note). The original harmonic series notes are shown in brackets.

[help_files/in_tune_harmonic_series_new_notes.ts in_tune_harmonic_series_new_notes.ts] (harmonic series: 1 3 5 9 15 2).

In tune harmonic series new notes.gif (base clef)

In sol fah, the notes are doh, soh, mi, re, ti (then doh' to end it)

Here it is with all the notes. Tune drops back to the fundamental after each of the "out of tune" notes. They aren't really out of tune; notes of the harmonic series always sound good together, but they are certainly more adventurous.

[help_files/all_harmonic_series_new_notes.ts all_harmonic_series_new_notes.ts] (harmonic series: 1 3 5 7 1 9 11 1 13 1 15 2).

All harmonic series new notes.gif

(sharps in blue)

The adventurous notes are 7/4, 11/8, and 13/8.

We can now express all the new notes as ratios. For instance, the B is 15 times the fundamental. You need to drop it down a few octaves to get it into the same octave as the fundamental, which you do by dividing by two as many times as needed. The result is 15/8.

The other notes are D = 9/8, E = 10/8 (= 5/4), and G = 3/2.

Here are the notes of the scale so far in ascending order:

1/1 9/8 5/4 3/2 15/8 2/1

As a tune smithy file

[help_files/diatonic_notes_from_harmonic_series.ts diatonic_notes_from_harmonic_series.ts] (played on violin this time).

We can find the missing F because it's a fifth below C, and the A because it's a third above the F (alternatively, as a fifth below E).

Using 3/2 for the fifth, and 5/4 for the major third, the F is 2/3, or moving it up by an octave, 4/3. The A is then a third above that, so it's ratio is 4/3 times 5/4, which is 5/3.

As a tune smithy file

[help_files/diatonic_notes_from_harmonic_series_with_F_and_A.ts diatonic_notes_from_harmonic_series_with_F_and_A.ts]

Diatonic notes from harmonic series with F and A.gif

(treble clef)

So the complete just temperament diatonic (i.e. major) scale, after adding in 4/3 and 5/3, is

1 9/8 5/4 4/3 3/2 5/3 15/8 2

[help_files/diatonic_notes.ts diatonic_notes.ts]

We can now fill in the black notes, which is usually done using major thirds. We have to multiply by 5/4 to go up by a major third, and divide by it, or equivalently, multiply by 4/5, to go down a major third.

Here is one way to do it:

Going down a major third from the octave C gives 2 times (4/5), so 8/5, which is our A flat / G sharp. Doing the same from the G at 3/2 gives 3/2 times 4/5, which is 6/5, our E flat / D sharp. We can go down from F at 4/3 to get D flat / C sharp as 16/15. Going down a major third from the D at 9/8 gives the B flat / A sharp as 9/10, or after moving it up an octave into range, 9/5.

Notice, that if you went up a major third from E instead of down a major third from C, you would get to G sharp as 5/4 times 5/4, or 25/16 instead of 8/5, which is how a musician sensitive to just intonation major thirds might play it in the scale of E major from this E. So that is another method.

If playing in pure just intonation, you could be forced to play a minute shift in pitch when changing from a G sharp to an A flat.

Enharmonic shift conventional score.gif

These are the notes on a conventional score with flats in yellow, sharps in blue

[[Image:enharmonic_shift_nudge_to_pitch.gif|519px|score for the tune, notes positioned exactly according to pitch]]

The same notes, but this time positioned exactly, by pitch, so that you can see that the G sharp (highest of the two blue notes) is a little flatter than the A flat (lowest of the two yellow notes).

This is what one sounds like: [help_files/enharmonic_shift_for_G_sharp.ts enharmonic_shift_for_G_sharp.ts] (plays major scale in C, then goes down to the A flat in A flat major, then goes up to the G sharp in E major from the E of the just intonation C major scale, then shifts back to the first tuning for the A flat ready to go back up to the original C in A flat major).

We now have all the notes except the F sharp (or G flat). The F sharp is a major third above D at 9/8, so 9/8 times 5/4, which gives 45/32, the most complicated ratio in the just temperament twelve tone scale. Alternatively as a G flat, it would be a major third below B flat (since B flat is the major third in the scale of G flat).

Try [help_files/just_temperament_twelve_tone_constructed_from_diatonic_scale.ts just_temperament_twelve_tone_constructed_from_diatonic_scale.ts]

[help_files/just_temperament_twelve_tone_constructed_from_diatonic_scale.ts [[Image:just_temperament_twelve_tone_constructed_from_diatonic_scale.gif|550px|score for the tune]]]

The notes played are

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2 8/5 3/2 6/5 4/3 16/15 9/8 9/10 9/8 45/32 1/1
C    D   E   F   G   A   B   C  Ab  G  Eb  F     Db   D   Bb   D    F#   C

(sharps in blue, flats in yellow).

So the complete twelve tone scale is

 C     Bb    D    Eb    E    F     F#     G    Ab   A    Bb    B     C
 1   16/15  9/8  6/5  5/4   4/3  45/32   3/2  8/5  5/3  9/5  15/8   2/1

This scale has I, IV and V pure, but II is impure - the D to F minor third and the D to A fifth are both out of tune.

There are many other twelve tone scales one might use:

 C     C#    D    Eb   E    F    F#     G    Ab   A    Bb     B     C
1/1 135/128 9/8  6/5  5/4 27/20 45/32  3/2  8/5 27/16  9/5  15/8   2/1

has I, II and V pure. I made this one just by rotating the scale to start at another degree for the 1/1. Here, looking at the ratios, one can see that the A is a fifth above the D, and the C# is a major third above the A, so the II is pure. The F is a minor third above the D (9/8*6/5). The I fixes the chord C E G, the V fixes G B D, and the II fixes D F A, so all the white notes are fixed once one decides that one wants I, II and V are pure.

However, basically one can only tune a keyboard to pure just intonation if the chords used are very restricted indeed, at least for Western music. When it is possible, then the music sounds great!

It is perfectly possible to tune a keyboard to just intonation for playing Indian classical music, which is usually played in just intonation with a drone. The modern Indian gamut is

 C     Bb    D    Eb    E    F     F#     G    Ab   A    Bb    B     C
1/1  16/15  9/8  6/5   5/4  4/3  45/32   3/2  8/5  27/16 9/5  15/8   2/1

Here the A is a fifth above the D.

One can sing and play in just intonation more readily, as singers can learn to automatically adjust pitch depending on context - known as adaptive just intonation. Also, as a composer, one can choose a fixed just intonation scale, suitable for tuning a keyboard, and write the music in such a way that it only uses just intonation chords - that can be an interesting restriction to impose on oneself.

It is also possible to write a computer program to analyse a complete midi clip and retune all the notes to just intonation, or as near as possible. John de Laubenfels is the outstanding modern pioneer in this field. One can also retune in real time, while playing but this is usually less successful as one may sometimes need to look forward many bars before one can see where a cadence is leading exactly.

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