ABOUT INTONATION - Misha Amory N.B.: The following notes are only intended as a theoretical backup to what your own ear tells you: intuition should always be trusted over theory when it comes to intonation! To define two terms: 1. A "pure" interval is one without "beats", that is, one whose pitches can both be found on the same overtone series. 2. A "comma" is equal to one-ninth of a whole tone; it is also the difference between 12 stacked perfect fifths and seven stacked octaves. The two approaches to intonation which have been widely used in the 20th century are known as "equal- tempered" and "Pythagorean", or "expressive". EQUAL TEMPERAMENT: This system divides the octave up into 12 absolutely equal semitones, and lets the chips fall where they may. What happens with this system is: perfect fifths that are narrower than pure, and perfect fourths that are wider than pure. Major intervals will be slightly wide of pure and minor intervals slightly narrow of pure. PYTHAGOREAN/EXPRESSIVE: In this system there are two kinds of semitone: the chromatic and the diatonic. The chromatic semitone is any semitone where the letter name does not change, for instance C to C#. The diatonic semitone is one where the letter name changes, e.g. C# to D. The chromatic semitone is five commas in width, and the diatonic semitone is only four. In addition to this, the expressive system advocates absolutely pure perfect fourths and fifths; the problem of the 12 stacked fifths is ignored, on the reasonable assumption that so much stacking would be a rare occurence anyway. In this system, major intervals are very wide of pure, and minor intervals (including the minor second, or "diatonic semitone") are very narrow. My point is that since each of these systems is a compromise, exclusive use of either system would be unjustifiable. The equal-tempered system answers best to music with a strong triadic context, especially a melody over a triadic accompaniment (e.g. the opening of the slow movement of Schubert's Arpeggione Sonata), and to freely atonal music. The expressive system answers best to harmonically non-functional notes (e.g. chromatic neighbor notes), melody without accompaniment, unisons/octaves in ensemble work, etc. One way to apply this theory directly to practice is to explore the range of an interval, for instance a major sixth, playing versions of that range from pure to as wide as you can bear it, to better acquaint your ear with these intervallic shadings. Remember: major intervals will only shade wide of pure, minor intervals narrow of pure. This is like upgrading the prescription for a pair of eyeglasses: you are bringing the interval into focus and learning to distinguish much finer shadings. Don't overdo it though, because you'll go out of your mind. USEFUL TECHNIQUES IN PRACTICING INTONATION 1. Horizontal Pitch Memory: It is a good practice to be able to keep in your memory the intonation of the pitches which you have just played, especially in passages where a pitch returns shortly an octave higher. Cf. Brahms f minor, opening of the first movement. 2. Triadic Context: If you are playing arpeggiated melodies, or melodies over a triadic or seventh-chord accompaniment, have the whole chord "glowing" in your inner ear as a relevant context to the melody. This prevents isolated or obsessive attention to the intonation of each pitch, which can be a big obstacle. 3. Impose a tonality in your mind if it is briefly implied within an atonal context, e.g. the opening of Bartók Concerto, the opening of the first movement of Hindemith 25 #1, or the middle of the third movement of the same. This is useful to give pitches a clearer relation to one another in your ear, even if they don't have that relation over the long term in the work. 4. Perfect intervals are your friends, because they are the most narrowly defined of the intervals; when a passage is out of tune but it's not clear why, look to the perfect intervals (real or implied) as the key to improvement. They can be used to form a structure for the remaining intervals in the passage. Also use open strings to check pitches when they are a perfect interval away. Because perfect intervals are so clear, they are also the ones most likely to sound out of tune!