Regular diatonic tuning

12-tone  Play , 72-tone (Maneri-Sims notation)  Play , and also (both written the same as 12-tone in Easley Blackwood notation) 17-tone  Play  and 19-tone  Play  regular diatonic scales
T and S in various equal temperaments (*5-tone and 7-tone are the limits of and not regular diatonic tunings)  Play 53  and  Play 31 

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) 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 the ordinary diatonic scales the T's here are tones and the S's are semitones which are half, or approximately half the size of the tone. But in the more general regular diatonic tunings, the two steps can be of any relation within the range between T=171.43 (S=T) and T=240 (S=0) cents (fifth between 685.71 and 720). Note that regular diatonic tunings are not limited to the notes of the diatonic scale which defines them.

One may determine the corresponding cents of S, T, and the fifth, given one of the values:

When the S's reduce to zero (T=240 cents) the result is TTTTT or a five tone equal temperament, As the semitones get larger, eventually the steps are all the same size, and the result is in seven tone equal temperament (S=T=171.43). These two end points are not included as regular diatonic tunings, because to be regular the pattern of large and small steps has to be preserved, but everything in between is included, however small the semitones are, or however similar they are to the whole tones.

"Regular" here is understood in the sense of a mapping from Pythagorean diatonic such that all the interval relationships are preserved.[1] For instance, in all regular diatonic tunings, just as for the pythagorean diatonic:

and so on.

If one continues to increase the size of the S further, so that it is larger than the T, one gets scales with two large steps and five small steps, and eventually, when all the T's vanish the result is SS, so a tritone division of the octave. These scales however are not included as regular diatonic tunings.

All regular diatonic tunings are also Linear temperaments, i.e. Regular temperaments with two generators: the octave and the tempered fifth. One can use the tempered fourth as an alternative generator (e.g. as B E A D G C F, ascending fourths, reduced to the octave), but the tempered fifth is the more usual choice.

All regular diatonic tunings are also Generated collections (also called Moments of Symmetry) and the chain of fifths can be continued in either direction to obtain a twelve tone system F C G D A E B F# C# G# D# A# where the interval F#-G is the same as B - C etc, another moment of symmetry with two interval sizes. A chain of seven fifths generates a chromatic semitone, for instance from F to F# and the pattern of chromatic and diatonic semitones is CDCDDCDCDCDD or a permutation of it where the C is the chromatic semitone, and D is the diatonic semitone e.g. from E to F between notes five steps apart in the cycle. Here, the seven equal system is the limit as the chromatic semitone tends to zero, and the five tone system in the limit as the diatonic semitone tends to zero.

Range of recognizability

The regular diatonic tunings include all linear temperaments within Easely Blackwood's "Range of Recognizabilty" in his The Structure of Recognizable Diatonic Tunings[2] for diatonic tunings with

However, his "range of recognizability" is more restrictive than "regular diatonic tuning". For instance, he requires the diatonic semitone to be at least 25 cents in size. See [3] for a summary.

Significant regions within the range

When the fifths are a little flatter than the 700 cents of the diatonic subset of twelve equal, then we are in the region of the historical meantone tunings, which distribute or temper out the syntonic comma. This includes

When the fifths are exactly 3/2, or around 702 cents, the result is the Pythagorean diatonic tuning.

For fifths slightly narrower than 3/2, the result is a Schismatic temperament, where the temperament is measured in terms of a fraction of a schisma - the amount by which a chain of eight fifths reduced to an octave is sharper than the just minor sixth 8/5. So for instance, a 1/8 schizma temperament will achieve a pure 8/5 in an ascending chain of eight fifths. 53 equal temperament achieves a good approximation to Schismatic temperament.

At around 703.4-705.0 cents, with fifths mildly tempered in the wide direction, the result is major thirds with ratios near 14/11 (417.508 cents) and minor thirds around 13/11 (289.210 cents).

At 705.882 cents, or tempered in the wide direction by 3.929 cents, the result is the diatonic scale in 17 tone equal. Beyond this point, the regular major and minor thirds approximate simple ratios of numbers with prime factors 2-3-7, such as the 9/7 or septimal major third (435.084 cents) and 7/6 or septimal minor third (266.871 cents). At the same time, the regular tones more and more closely approximate a large 8/7 tone (231.174 cents), and regular minor sevenths the "harmonic seventh" at the simple ratio of 7/4 (968.826 cents). This septimal range extends out to around 711.111 cents or 27-ed2, or a bit further.

That leaves the two extremes, what we could call:

Diatonic scales constructed in equal temperaments can have fifths either wider or narrower than a just 3/2. Here are a few examples:

Syntonic temperament and timbre

Regular diatonic tunings have also been referred to as syntonic temperaments.[5] This terminology is used for an instrument able to adjust timbre and use inharmonic partials in order to approximate the smoothness of 5/4 in harmonic timbres using inharmonic instruments with tempered fifths anywhere in the range of the regular diatonic tunings. It is named after the syntonic comma because of the resemblance to quarter comma meantone where the syntonic comma is tempered out. In the syntonic temperament on this instrument, then the syntonic comma is not tempered out, but the partials are adjusted to give a similar sound to the pure major thirds of quarter comma meantone while also preserving its characteristic that the major third is made up of two equally sized tones.

Figure 1: The syntonic temperament's tuning continuum, from (Milne et al. 2007).

Since it's a linear temperament, it can be played on an isomorphic keyboard, on which the fingering-pattern of any given musical structure is the same in any syntonic tuning. The combination of an isomorphic keyboard and continuously variable tuning enables Dynamic tonality, a novel expansion of the framework of tonality.[6]

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 can move towards (approximate) some Just Intonation tunings.[7]

Figure 2: Change in widths of intervals of the syntonic temperament across its tuning continuum (tonic is D).

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.

Research projects regarding the syntonic temperament

Notes

  1. ↑ Denckla, Benjamin Frederick (1995). "Dynamic Intonation for Synthesizer Performance".
  2. ↑ Blackwood, Easley. The Structure of Recognizable Diatonic Tunings. Princeton University Press. ISBN 9780691610887.
  3. ↑ Serafini, Carlo. "THE STRUCTURE OF RECOGNIZABLE DIATONIC TUNINGS by EASLEY BLACKWOOD - a review".
  4. ↑ "1-6 Syntonic Comma Meantone". xenharmonic wiki.
  5. ↑ Milne, A., Sethares, W.A. and Plamondon, J., "Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
  6. ↑ Plamondon, J., Milne, A., and Sethares, W.A., "Dynamic Tonality: Extending the Framework of Tonality into the 21st Century", in Proceedings of the Annual Meeting of the South Central Chapter of the College Music Society.
  7. ↑ Milne, A., Sethares, W.A., Tiedje, S., Prechtl, A., and Plamondon, J., "Spectral Tools for Dynamic Tonality and Audio Morphing", Computer Music Journal, in press.
  8. ↑ "Musica Facta". Archived from the original on 2014-05-17. Retrieved 2015-09-19.
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