Syntonic temperament

The syntonic temperament [Milne, A., Sethares, W.A. and Plamondon, J., [http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15 Invariant Fingerings Across a Tuning Continuum] , "Computer Music Journal", Winter 2007, Vol. 31, No. 4, Pages 15-32.] is a system of musical tuning in which the frequency ratio of each musical interval is a product of powers of an octave and a tempered perfect fifth, with the width of the tempered major third being equal to four tempered perfect fifths minus two octaves and the width of the tempered major second being equal to two tempered perfect fifths minus one octave ("i.e.", half the width of the major third).

The syntonic temperament is named after the syntonic comma, as that is the first comma tempered to unison in its comma sequence.

Because it has two generators – the octave and tempered perfect fifth (P5) – it is a rank-2 regular temperament. Because one of its generators is the octave, the terminology of Erv Wilson would describe the syntonic temperament as a linear temperament.

As shown in the fgure at right, the tonally-valid tuning range of the syntonic temperament includes a number of historically-important tunings, such as the currently-popular 12-tone equal division of the octave (12-edo tuning, also known as 12-tone “equal temperament”), the meantone tunings, and Pythagorean tuning. Tunings in the syntonic temperament can be equal (12-edo, 31-edo), non-equal (Pythagorean, meantone), circulating, and closely-related Just Intonation tunings. [Milne, A., Sethares, W.A., Tiedje, S., Prechtl, A., and Plamondon, J., [http://eceserv0.ece.wisc.edu/~sethares/paperspdf/SpectralTools.pdf Spectral Tools for Dynamic Tonality and Audio Morphing] , "Computer Music Journal", submitted.]

From the center of the syntonic temperament’s tonally-valid tuning range, as the width of the P5 widens, the minor second narrows, eventually disappearing in 5-edo; as the width of the P5 narrows, the minor second widens, eventually equaling the major second in 7-edo.

The legend of Figure 2 (on the right side of the figure) shows a stack of P5's centered on D. Each resulting note represents an interval in the syntonic temperament with D as the tonic. The body of the figure shows how the widths (from D) of these intervals change as the width of the P5 is changed across the syntonic temperament's tuning continuum.
* At P5 = 686 cents, the intervals converge on just 7 widths (assuming octave equivalence of 0 and 1200 cents), producing 7-edo.
* At P5 = 695 (19-edo), the gaps between these 19 intervals are all equal, producing 19-edo tuning.
* At P5 = 696.6 (31-edo), a stack of 31 such intervals would show equal gaps between each such interval, producing 31-edo tuning.
* At P5 = 700 (12-edo), the sharp notes and flat notes are equal, producing 12-edo tuning.
* etc....
* at P5 = 720 cents, the pitches converge on just 5 widths, producing 5-edo.

Notes