Wolf Intervals
A whole number of just perfect fifths will never add up to a whole number of octaves, because they are incommensurable (see Fundamental theorem of arithmetic). If a stacked-up whole number of perfect fifths is to close with the octave, then one of the fifths must have a different width than all of the others. For example, to make the 12-note chromatic scale in Pythagorean tuning close at the octave, one fifth must be out of tune by the Pythagorean comma; this altered fifth is called a wolf fifth.
Wolf intervals are an artifact of keyboard design. This can be shown most easily using an isomorphic keyboard, such as that shown in Figure 2.
On an isomorphic keyboard, any given musical interval has the same shape wherever it appears, except at the edges. Here's an example. On the keyboard shown in Figure 2, from any given note, the note that's a perfect fifth higher is always up-and-rightwardly adjacent to the given note. There are no wolf intervals within the note-span of this keyboard. The problem is at the edge, on the note E♯. The note that's a perfect fifth higher than E♯ is B♯, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A♯, hence maintaining the keyboard's consistent note-pattern). Because there is no B♯ button, when playing an E♯ power chord, one must choose some other note, such as C, to play instead of the missing B♯.
Even edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has enharmonically-distinct notes (Milne, 2007). For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge-condition, from E♯ to C, is not a wolf interval in 12-ET, 17-ET, or 19-ET; however, it is a wolf interval 26-ET, 31-ET, and 50-ET. In these latter tunings, using electronic transposition could keep the current key's notes on the isomorphic keyboard's white buttons, such that these wolf intervals would very rarely be encountered in tonal music, despite modulation to exotic keys.
Isomorphic keyboards expose the invariant properties of the meantone tunings of the syntonic temperament isomorphically (that is, for example, by exposing a given interval with a single consistent inter-button shape in every octave, key, and tuning) because both the isomorphic keyboard and temperament are two-dimensional (i.e., rank-2) entities (Milne, 2007). One-dimensional N-key keyboards can expose accurately the invariant properties of only a single one-dimensional N-ET tuning; hence, the one-dimensional piano-style keyboard, with 12 keys per octave, can expose the invariant properties of only one tuning: 12-ET.
When the perfect fifth is exactly 700 cents wide (that is, tempered by approximately 1/11 of a syntonic comma, or exactly 1/12 of a Pythagorean comma) then the tuning is identical to the familiar 12-tone equal temperament. This appears in the table above when R = 2/1.
Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments and eventually equal temperament became more popular.
Using standard interval names, twelve fifths equal six octaves plus one augmented seventh; seven octaves are equal to eleven fifths plus one diminished sixth. Given this, three "minor thirds" are actually augmented seconds (for example, B♭ to C♯), and four "major thirds" are actually diminished fourths (for example, B to E♭). Several triads (like B–E♭–F♯ and B♭–C♯–F) contain both these intervals and have normal fifths.
Read more about this topic: Meantone Temperament
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