3.1 Tuning systems

Introduction

The first thing musicians must do before they can play together is "tune". For musicians in the standard Western music tradition, this means agreeing on exactly what pitch (what frequency ) is an "A", what is a "B flat" and so on. Other cultures not only have different note names and different scales, they may even have different notes - different pitches - based on a different tuning system. In fact, the modern Western tuning system, which is called equal temperament , replaced (relatively recently) other tuning systems that were once popular in Europe. All tuning systems are based on the physics of sound . But they all are also affected by the history of their music traditions, as well as by the tuning peculiarities of the instruments used in those traditions. Pythagorean , mean-tone , just intonation , well temperaments , equal temperament , and wide tuning .

To understand all of the discussion below, you must be comfortable with both the musical concept of interval and the physics concept of frequency. If you wish to follow the whole thing but are a little hazy on the relationship between pitch and frequency, the following may be helpful: Pitch ; Acoustics for Music Theory ; Harmonic Series I: Timbre and Octaves ; and Octaves and the Major-Minor Tonal System . If you do not know what intervals are (for example, major thirds and perfect fourths), please see Interval and Harmonic Series II: Harmonics, Intervals and Instruments . If you need to review the mathematical concepts, please see Musical Intervals, Frequency, and Ratio and Powers, Roots, and Equal Temperament . Meanwhile, here is a reasonably nontechnical summary of the information below: Modern Western music uses the equal temperament tuning system. In this system, an octave (say, from C to C) is divided into twelve equally-spaced notes. "Equally-spaced" to a musician basically means that each of these notes is one half step from the next, and that all half steps sound like the same size pitch change. (To a scientist or engineer, "equally-spaced" means that the ratio of the frequencies of the two notes in any half step is always the same.) This tuning system is very convenient for some instruments, such as the piano, and also makes it very easy to change key without retuning instruments. But a careful hearing of the music, or a look at the physics of the sound waves involved, reveals that equal-temperament pitches are not based on the harmonics physically produced by any musical sound. The "equal" ratios of its half steps are the twelfth root of two, rather than reflecting the simpler ratios produced by the sounds themselves, and the important intervals that build harmonies can sound slightly out of tune. This often leads to some "tweaking" of the tuning in real performances, away from equal temperament. It also leads many other music traditions to prefer tunings other than equal temperament, particularly tunings in which some of the important intervals are based on the pure, simple-ratio intervals of physics. In order to feature these favored intervals, a tuning tradition may do one or more of the following: use scales in which the notes are not equally spaced; avoid any notes or intervals which don't work with a particular tuning; change the tuning of some notes when the key or mode changes.