Intonation Demystified

Minor variations of pitch are, in fact, part of what gives a string section its “warm” sound.

Tuning up to the pitch from below is easier to hear and also results in a greater pitch stability, due to a more equal string tension on either side of the nut.

Controlling the speed of an interval's beats is how keyboard temperaments are set by ear.

For example, the major sixth E-C# will result in an “A” an octave and a fifth below the “E”.

See Fritz Gearhart , “The Use of ‘Tartini Tones’ in Teaching”, Strings, ???, 2007.

Greek philosopher (c. 6th century BC) who theorized that the octave divides most perfectly into 12 semitones.

This is called a Pythagorean Third, and the amount of the deviation from pure is called the “Pythagorean comma”. It results between the cello's C string and the violin's E string when a string quartet tunes with pure fifths. Many quartets address this problem by slightly narrowing the fifths of their open strings, either by ear or by using a tuner.

Keyboard instruments in Baroque period-instrument ensembles are typically tuned in an unequal or “well-tempered” system, in which two or more sizes of narrowed fifths produce thirds ranging in quality from nearly pure in the primary keys to more “tense” in the farther-afield ones, reflecting a more vertical approach to intonation.

Or with some that have moveable, gut frets, for example lutes and gambas.

This is why the cellist should give the tuning pitch.

Occasionally the lowest root note will have to adjust to the presence of open strings in other parts.

Galamian Ivan , and Neumann Frederick , “Contemporary Violin Technique, Volume 1”, Galaxy Music Corp. Boston.

The circles each represent a position, starting at the outside with first, illustrating the cyclic nature of the finger.