page_title: Regular diatonic tuning
section_title: Syntonic temperament and timbre
hierarchical_section_title: Regular diatonic tuning / Syntonic temperament and timbre
caption_attribution_description: Syntonic temperament's tuning continuum
context_page_description: A regular diatonic tuning is any musical scale consisting of "tones" and "semitones" arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
context_section_description: The term syntonic temperament describes the combination of 
  the continuum of tunings in which the tempered perfect fifth (P5) is the generator and the octave is the period;
  Comma sequences that start with the syntonic comma (i.e., in which the syntonic comma is tempered to zero, making the generated major third as wide as two generated major seconds); and
  the "tuning range" of P5 temperings in which the generated minor second is neither larger than the generated major second, nor smaller than the unison.
  This combination is necessary and sufficient to define a set of relationships among tonal intervals that is invariant across the syntonic temperament's tuning range. Hence, it also defines an invariant mapping -- all across the tuning continuum -- between (a) the notes at these (pseudo-Just) generated tonal intervals, and (b) the corresponding partials of a similarly-generated pseudo-Harmonic timbre. Hence, the relationship between the syntonic temperament and its note-aligned timbres can be seen as a generalization of the special relationship between Just Intonation and the Harmonic Series.
  Maintaining an invariant mapping between notes and partials, across the entire tuning range, enables Dynamic tonality, a novel expansion of the framework of tonality, which includes timbre effects such as primeness, conicality, and richness, and tonal effects such as polyphonic tuning bends and dynamic tuning progressions.
  If one considers the syntonic temperament's tuning continuum as a string, and individual tunings as beads on that string, then one can view much of the traditional microtonal literature as being focused on the differences among the beads, whereas the syntonic temperament can be viewed as being focused on the commonality along the string.
  The notes of the syntonic temperament are best played using the Wicki-Hayden note layout. Because the syntonic temperament and the Wicki-Hayden note-layout are generated using the same generator and period, they are isomorphic with each other; hence, the Wicki-Hayden note-layout is an isomorphic keyboard for the syntonic temperament. The fingering-pattern of any given musical structure is the same in any tuning on the syntonic temperament's tuning continuum. The combination of an isomorphic keyboard and continuously variable tuning supports Dynamic tonality as described above.
  As shown in the figure 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 Just.
  The legend of Figure 2 (on the right side of the figure) shows a stack of P5s 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 ≈ 685.7 cents Play , the intervals converge on just 7 widths (assuming octave equivalence of 0 and 1200 cents), producing 7-edo. S/T = 0.
  At P5 ≈ 694.7 Play  (19-edo), the gaps between these 19 intervals are all equal, producing 19-edo tuning. S/T = 2/3.
  At P5 ≈ 696.8 Play  (31-edo), a stack of 31 such intervals would show equal gaps between each such interval, producing 31-edo tuning. S/T = 3/5.
  At P5 = 700.0 Play  (12-edo), the sharp notes and flat notes are equal, producing 12-edo tuning. S/T = 1/2.
  At P5 ≈ 701.9 Play  (53-edo), a stack of 53 such intervals - each just 3/44 of a cent short of a pure fifth - makes 31 octaves, producing 53-edo tuning. S/T = 4/9.
  etc....
  at P5 = 720.0 cents Play , the pitches converge on just 5 widths, producing 5-edo. S/T = 1.
caption: Figure 1: The syntonic temperament's tuning continuum, from (Milne et al. 2007)
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