A scale is a sequence of pitches. Each of these pitches is a tone (not to be confused with other meanings of the term tone). Each tone can be identified by its degree, its position in the scale, and is often expressed as the interval it makes with the tonic, the first tone of the scale. The interval between two consecutive tones is a step.
Scales are typically organized in ascending or descending order. In most scales, the descending form is identical to the ascending form but in the opposite order. However, some scales have different forms in both directions, such as the melodic minor scale. A reentrant scale features at least one negative step, going backwards relative to the general direction of the scale. Although a reentrant scale is not strictly ascending or descending, its ascending and descending forms are determined by its general direction.
Most scales are periodic, featuring a step pattern that repeats after a given interval, the period, often the octave or a fraction thereof. A scale's step pattern can be condensed in the form of a scale signature. Periodic scales may be rotated into different modes by designating a different tone as the tonic. Aperiodic scales also exist, notably the harmonic series.
Concrete and abstract scales
An absolute concrete scale, defines exact frequencies for all of its tones. For example, the 12edo diatonic scale in the key of C major with a reference pitch of A4 = 440 Hz is an absolute concrete scale. The 12edo diatonic scale in the key of A (natural) minor with a reference pitch of A4 = 440 Hz is equivalent to the previous scale: since C major and A minor are relative keys, only the tonal center moves. Note that the information provided by the key and reference pitch may be described differently in some contexts: for instance, one could assume the brightest mode (by mode height) by default and only provide the root frequency.
A relative concrete scale defines exact intervals between all of its tones, but does not define the frequency of any tone. In other words, it is a set of absolute concrete scales which are equivalent up to transposition. Starting from the previous example, removing either the key (C major) or reference pitch (A4 = 440 Hz), or both, results in a relative concrete scale.
An abstract scale contains at least one variable interval, often constrained within certain boundaries to ensure that important scale properties are preserved. In other words, it is a set of (relative or absolute) concrete scales which have equivalent structures within the limits of the variable's boundaries. Some scales have variable intervals that vary in relation with each other; that is taken into account by the variable boundaries, which are not assumed to be constant. Starting from the relative concrete scale example, removing any information would turn the scale into an abstract scale. For instance, the "diatonic scale" is an abstract scale which refers to all relative concrete diatonic scales in various tunings (and keys): in this particular case, the size of the generator is bounded by the generator range that produces the diatonic scale's characteristic step pattern, and every other interval is defined in relation to the generator. Other scales may have less technical definitions, such as broad interval ranges for each degree, which is typical of traditional scales.
In practice, considering that there are physical limitations to the accuracy of an instrument's tuning, especially in the context of acoustic instruments, a tolerance should be allowed to avoid labelling as abstract what would otherwise be concrete scales. Similarly, playing out of tune with relation to a given concrete scale would correspond to playing outside of that tolerance. The concept of just-noticeable difference (JND) may be helpful to determine a reasonable tolerance.
- Main article: Scale properties simplified
Relation to tuning systems and chords
A tuning system defines the set of discrete pitches used to tune an instrument in a composition. A scale can be built by taking a subset of pitches from a given tuning system. Furthermore, a chord can be built by taking a subset of pitches from a given scale.
Scales are generally treated with a greater focus on melody, while chords are generally treated with a greater focus on harmony. That said, the boundary between the two is fuzzy, and some musicians use the term scale-chord to refer to a set of pitches that is treated both as a scale and as a chord.