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Tuning Systems for Musical Instruments

Early instruments

Perfect tuning can only be achieved for one musical key. In practice we will always want to play music of more than one key on the instrument, so a compromise becomes necessary. This problem has been known for many years and a number of solutions have been produced.

The term 'intonation' is used to describe the precise relative tuning of individual notes relative to each other and often refers to singing or playing technique as well as tuning of the actual instrument. The term 'temperament' is used to define the overall tuning of the instrument as an acceptable compromise. (Temperament = moderation = compromise).

The almost universal system of tuning is now that known as 'equal temperament' which makes every interval from one note to the next equal. This system of tuning is a reasonable compromise and achives one thing perfectly, that is the instrument can be played equally well in any key. The probem is that some of the intervals are not quite right. Much modern music completely ignores this fact, and as listeners we learn to accept the compromise.

I am convinced that something has happened to pop music since the introduction of the modern electronic keyboard in about 1983. Because harmony has been somewhat unsatisfactory a preference had developed for music with little harmony, but more emphasis on rhythm, individual sounds, and on quality of sound reproduction.

Absolute Pitch

Many people believe that 'concert pitch' defines every note. This belief has come about through the almost universal use of equal temperament. Electronic tuning meters use this. Musicians will often tune to any convenient note. It would seem that there is no scope for variations of tuning of individual notes. In fact concert pitch does not define every note, but is based on standardisation of the note A to 440Hz. Any tuning should result in A having the correct pitch.

The table shows pitches relative to C for ease of understanding. The correct method is to use the relative tuning shown, while ensuring that the note A has the correct pitch, i.e. 440 Hz. This requires C to be tuned to 264 Hz for the pure C Major scale, and to 263.18 Hz for Mean Tone tuning.

Small changes in pitch can be measured in cents, which is a modern concept. In simple terms, a cent is a hundredth of a semitone. This assumes that all semitones are equal and is based on the semitone of equal temperament.

Instrument Types

The principles of temperament apply differently to different instruments. It depends on how rigidly the tuning is built into the instrument at the time of its construction. In some cases tuning will be for consideration of the instrument maker. For example, the tuning of flute-like instruments depends on the size and position of the holes. In others it will be for consideration of the player or tuner of the instrument. Traditional keyboard instruments are tunable, but only by a specialist tuner, not the player.

Traditionally temperament has mainly been considered for keyboards. This is because ideally we would like a keyboard to be able to play in all keys without need for retuning. It applies to both pianos and organs. Traditional piano tuners know about temperament. Almost all electronic keyboards use the system of 'equal temperament'. The reason top performers prefer traditional pianos to electronic ones is probably because they can have them tuned to a high standard in a way which suits their needs. Similarly some performers insist on using older types of electronic organs, such as the Hammond, which might have been tuned differently. However, for the majority of music using electronic keyboards the harmony is a compromise.

Diatonic instruments are those which have only the notes of one key. Examples are melodeons, harmonicas and tin whistles. With these it is possible to achieve perfect tuning for that key without any temperament.

Some instruments, for example saxophones, allow the player to vary the pitch of each note slightly, so precise tuning may not matter. This is partly true also of stringed instruments such as violins, where only the tuning of the open strings need be considered.

Fretted instruments such as guitars are fundamentally designed for equal temperament. However there are several issues which affect tuning. The spacing of the frets as determined by the maker. Adjustments of intonation at the bridge, tuning of the open strings, and playing techniques which allow variations of pitch.

Temperaments

The main temperaments worth considering are:

Just temperament

This is the theoretical perfect tuning for a given key, but is only practical for diatonic instruments. For many instruments, notably keyboards, it is not suitable due to the need to play in different keys. There is also a view that the sound may be too pure and lack interest, but this too is mainly applicable to keyboards.

In Just Temperament the 'sharps' are not the same pitch as the 'flats' which are normally regarded as the same note, especially on keyboards.

Equal temperament

This is an all round compromise giving the same relative tuning in all keys. There are twelve identical semitones in each octave. It is a remarkable coincidence that this method provides reasonable approximation for all the important intervals. In particular seven semitones give an almost perfect interval of a fifth, the most important interval of the scale. The result is that all notes have a reasonable pitch.

Mean tone temperament

Mean Tone Temperament attempts to give a better approximation to the theoretically perfect tuning, while allowing a wide variety of different keys to be played in. It does not attempt to equate sharps and flats but provides a number of sharps and flats to allow playing in a number of keys.

Like 'just temperament' this must me defined for a particular key, but allows more flexibility for use in other related keys and so allows temporary modulations of key, and the use of chords which are not absolutely native to the key being played. In practice most modern music uses such chords, and often modulates to different keys within a song.

This tuning is an excellent compromise for playing in a limited number of related keys.

The theory is to tune the instrument so that the major thirds of the key are all perfect. This is done at the expense of slightly flattened fifths. The resulting tuning also improves the minor thirds too. The result is that major and minor chords sound much better.

The tuning can be used for playing in precisely those keys which use the notes which are available, bearing in mind that sharps and flats are different. However most instruments will not allow you to differentiate between, for example G# and Ab, so this will result in limits to the keys which can be played.

It turns out that in any of the major keys, for which the tuning is valid, all intervals are identical to those of any other valid key. In other words, each valid key is a perfect transposition of the others. In particular the intervals within each valid chord, (i.e. the third and the fifth), are the same for each chord. It also follows that all tones are equal, and all semitones are equal. However, a semitone is not half the size of a tone, but a little more.

 


The following table shows the theoretical ideal tuning of a C Major scale, and the errors which occur with Mean Tone Temperament, and Equal Temperament.

 

Comparison of Tuning Systems for the Key of C

 

C Major

Mean Tone

Equal

Note

Rel. Pitch

Rel. Pitch

Error %

Rel. Pitch

Error %

C

1.0

1.0

0

1.0

0

C#

 

1.0449

 

1.0595

 

D

1.125

1.1180

 

1.1225

 

Eb

1.2

1.1963

-0.31

1.1892

-0.9

E

1.25

1.2500

0

1.2599

0.79

F

1.3333

1.3375

0.31

1.3348

0.11

F#

 

1.3975

 

1.4142

 

G

1.5

1.4953

0.31

1.4983

-0.11

G#

 

1.5625

 

1.5874

 

A

1.6667

1.6719

0.31

1.6818

0.91

Bb

 

1.7889

 

1.7818

 

B

 

1.8692

 

1.8877

 

C

2.0

2.0

0

2.0

0

 

Using the selection of notes in the above table for mean tone tuning it is not possible to play, for example, in the key of Eb, which requires the note Ab. Similarly it cannot be used to play in the key of E which requires the note D#.

The following chords sound excellent:

Major chords: Eb, Bb,  F, C, G, D, A, E.

Minor chords: Cm, Gm, Dm, Am, Em, Bm, F#m, C#m.

It is easy to change the tuning system to allow playing in extra keys, but at the expense of others. For example to allow playing in the key of E, at the expense of Bb, retune the note Eb to D#.

This table below shows the required tuning offsets, in cents from equal temperament, to give mean tone temperament.

 

Mean Tone Tuning

Note

Offset

C

+10

C#

-14

D

+3

Eb

+20

E

-4

F

+13

F#

-11

G

+7

G#

-17

A

0

Bb

+17

B

-7

 


Appendix - Calculation methods

Just temperament

This will be used only for diatonic instruments, i.e. those which have only the notes of a single key, and even then it will be used only rarely. Tuning is done so that the important intervals are perfect, i.e. the minor thirds have ratio 6:5 (1.2), major thirds 5:4 (1.25), and fifths 3:2 (1.5).

Equal temperament

There are twelve identical semitone intervals in an octave, and because each increment is a ratio we have to find a number which can be used to multiply the starting note so that after twelve times the result is exactly twice the starting note to give an octave. This number is in mathematical terms the twelfth root of 2, and in practice is approximately 1.06. (Try multiplying by this twelve times and see). Each time we multiply the starting note by this value we get the next note. Now we see a remarkable coincidence. Multiplying by this number seven times we get almost exactly 1.5. This means that the important note, the fifth, is part of this scale which is what we want. We also find that all the other important notes are approximated too.

Mean Tone Temperament

This has the most difficult calculation method. Calculation of note pitches is based on making perfect all the major thirds of all the keys to be played. This single criterion is sufficient to calculate all the notes.

If we start with the note C, successive major thirds will take us to E, then G# and back to C, but will not lead to the other notes. On the other hand a sequence of fifths will eventually reach every note, thus C, G, D, A, E, B, F#, C# etc. and taking fifths downwards, C, F, Bb, Eb etc. If we can calculate the interval for our fifths we can then calculate every other note.

From C we can get to E in two ways. In major thirds and in fifths. We use the perfect interval of 1.25 to define E. The E two octaves above the starting C will be:

2 x 2 x 1.25 = 5

The fifth we use must give this value of E in the series:

C, G, D, A, E

The fifth is therefore required to be the fourth root of 5, i.e. 1.4953 approximately. We see immediately that this is a good approximation to the ideal of 1.5. Using this as the basis we can then calculate all the other notes.


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