# Elements of Music

10 11 MEEt thE IntErvals and the circle of fifths A musical interval is the distance between two tones, and although tuned slightly differently from culture to culture, the same intervals are broadly found all over the world. Intervals can be thought of in two ways The first is as an evercontracting series of simple frequency ratios, so that the first interval the octave is a 21 relationship, the second the fifth is 32, and the third the fourth is 43. Then follow the major third of 54, the minor third of 65, the second of 98 or 109, and yet smaller intervals, with names like the quarter tone, shruti, li, comma, apotome, and microtone, depending upon the era and culture. The second way of looking at this series is to compare intervals to the fundamental to which they all relate. This approach results in an octave 21, a fifth 31, another octave 41, a major third 51, another fifth 61, a seventh 71, another octave 81, a second 91, a third 101, and a tritone 111, etc. If the first view is relative, with each partial compared to its nearest neighbor, then the second view is absolute, as intervals take an absolute value compared to the fundamental. Both views are useful when taking an analytical approach to the construction of musical scales and the melodies that ultimately derive from them. Notice the octave, fifth, and major third appearing in both systems. All scales around the world broadly contain these intervals in some form, with their precise tuning revealing slight variations and nuances in instrument construction and cultural tastes. Those using the first approach tune their instruments and derive their scales by the relationship of each overtone to one another, while others using the second approach relate intervals to their fundamental. Above left The circle of fifths is a diagram of common tones. Chromatic notes sharps and flats are added in the order of perfect fifths to preserve the whole and half step relationships in the major scale built upon any note. Any consecutive seven fifths on the circle yield all the notes in any one of the twelve major scales. Accidentals occur whenever an alteration to a scale must be made to preserve the syntax of the alphabet. Opposing notes on the wheel form a tritone, the symmetrical interval that cuts the octave in half. Above right A table of the intervals present in the key of C major, the white notes on a piano. Intervals in C major Right A symbolic system of glyphs used in this book for describing musical intervals and their relative sizes. Note the shapes expanding and contracting with the size of the interval. Minor and diminished intervals are smaller by one semitone than their respective major and Perfect counterparts. Augmented intervals are one semitone larger than perfect. Major and minor are sometimes referred to as hard and soft dur and moll. The lower part of the diagram shows the symmetrical nature of the consonances of the intervals. Unison, the fourth, fifth and octave are perfect. Thirds and sixths are imperfect consonances, while seconds and sevenths are dissonances. The C ircle of Fifths Consonances