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 scalesT 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 such a tuning, then the notes are connected together in a chain of seven fifths, all the same size (TTTS or a permutation of that) which makes it a Linear temperament with the tempered fifth as a generator.
For the ordinary diatonic scales described here, the T-s 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 ¢ (for s = T at the high extreme) and T = 240 ¢ (for s = 0 at the low extreme) in musical cents (fifth, p5, between 685.71 ¢ and 720 ¢). Note that regular diatonic tunings are not limited to the notes of any particular diatonic scale used to describe them.
One may determine the corresponding cents of s, T, and the fifth (p5), given one of the values:
semitone
full tone
perfect fifth
When the (diatonic) semitones, s, are reduced to zero (T = 240 ¢) the octave is T T T T T, 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 extremes are not included as "regular" diatonic tunings, because to be "regular" the pattern of five large and two small steps has to be preserved; everything in between is regular, however small the semitones are without vanishing completely, or however large they become while still being strictly smaller than a whole tone.
"Regular" here is understood in the sense of a mapping from Pythagorean diatone such that all the interval relationships are preserved.[1] For instance, in all regular diatonic tunings, just as for the Pythagorean diatonic:
The notes are connected together through a chain of six fifths reduced to the octave, or equivalently, through ascending fifths and descending fourths (e.g. F C G D A E B in C major).
A chain of three tones spaced in equal-sized fifths (reduced to the octave) generates a whole tone (e.g. C G D).
A sequence of six tones spaced in fourths generates a semitone in the same way (e.g. E A D G C F).
A sequence of five fifths spaced in fifths (e.g. C G D A E) generates a major third, consisting of two whole tones.
A chain of four tones spaced in fourths generates a minor third (A D G C)
and so on; in all those examples the result is "reduced to the octave" (lowered by an octave whenever a note in the sequence exceeds an octave above the starting tone).
If one breaks the rule for "regular" that s must be smaller than T and continues to increase the size of s further, so that it becomes larger than the T, one gets irregular scales with two large steps and five small steps, and eventually, when all the T-s vanish the result is s s, so a division of the octave into tritones. However, these strange scales are only mentioned here to dismiss them; they not 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, and in any case, because fifths and fourths are octave complements, rising by perfect fourths produces the same result as rising by fifths.
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♭B, etc., another moment of symmetry with two interval sizes.
Instead of there being one semitone, S, there are actually two: the chromatic semitone, c, and the diatonic semitone, D; D is another name for S. Three notes spaced by a chromatic and diatonic semitone make a whole tone between the first and the last: c d = d c =T. The small difference in pitch between the two is called a comma, usually prefixed by the name of the tuning system that generates it, such as a syntonic comma (21.5 ¢), or Pythagorean comma (23.5 ¢), or a 53TET comma (22.6 ¢).
A chain of eight notes spaced in fifths generates a chromatic semitone, c, as the space between the first and the last; it is the change of pitch needed to raise a minor tone to a major tone; for instance from E♭ to E. For any tuning, the chromatic semitone is the space between a flat note and its natural, or a natural note and its sharp; between a white key and either the black key above it (if tuned as a sharp) or the black key below it (if tuned as a flat); in most tunings, the two intervals are different. The diatonic semitone, D, called S above, is the change in pitch of a sequence of six notes spaced by fifths, e.g. from E to F or B to C. For any tuning, the diatonic semitone is the relative pitch difference on a standard keyboard between two white keys that have no black key between them. The pattern of chromatic and diatonic semitones is c d c d d c d c d c d d or some mixed-around version of it. 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 Easley Blackwood's "Range of Recognizability" in his The Structure of Recognizable Diatonic Tunings[2] for diatonic tunings with
the fifth tempered to between 4/7 and 3/5 of an octave;
the major and minor seconds both positive;
the major second larger than the minor second.
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 12 tone equal temperament, then we are in the region of the historical meantone tunings, which distribute or temper out the syntonic comma. They include
1/3 comma meantone - achieves pure minor thirds 6/5; fifth is 694.786 cents; closely approximated by the diatonic scale in 19 tone equal temperament
1/6 comma meantone[4] - achieves a rational diatonic tritone 45/32; fifth is 698.371 cents; closely approximated in 55 tone equal temperament
1/11 comma meantone - fifth is 699.99988 cents; almost indistinguishable from 12 tone equal temperament
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 schisma temperament will achieve a pure 8/5 in an ascending chain of eight fifths. 53 tone 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, with fifths tempered in the wide direction by 3.929 cents, the result is the diatonic scale in 17 tone equal temperament. 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 tone equal temperament, or a bit further.
That leaves the two extremes, what we could call:
the "inframeantone" range with fifths between the lower bound for the regular diatonic of 7 tone equal temperament (685.7143 cents) and the range of historical meantones beginning around 1/3-comma or 19 tone equal temperament (694.786 cents), and with the diatonic "semitones" approaching the size of the diatonic whole tone
the "ultraseptimal" range from around 712 cents all the way to the upper bound of the regular diatonic at 720 cents or 5 tone equal temperament, and with very small diatonic semitones
Diatonic scales constructed in equal temperaments can have fifths either wider or narrower than a just 3/2. Here are a few examples:
12 (and its multiples), 19, 31, 53, have fifths narrower than a just 3/2
Syntonic temperament and timbre
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.[5]
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,[6] and tonal effects such as polyphonic tuning bends and dynamic tuning progressions.[7]
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.
Figure 1: The syntonic temperament's tuning continuum, from (Milne et al. 2007)
The notes of the syntonic temperament are best played using the Wicki-Hayden note layout.[8] 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.[7]
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.[9][10]
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 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.
Research projects regarding the syntonic temperament
The research program Musica Facta[11] investigates the musical theory of the syntonic temperament.
The music theory of the Guido 2.0 research project is based on the syntonic temperament. Guido 2.0 seeks to achieve a 10x increase in the efficiency of music education by exposing the invariant properties of music's syntonic temperament (octave invariance, transpositional invariance, tuning invariance, and fingering invariance) with geometric invariance. Guido 2.0 is the Music Education aspect of Musica Facta (above).
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.
Meantone temperaments are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are temperaments in that the fifths are not pure.
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.
The chromatic scale is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125). Two notes that differ by this interval would sound different from each other even to untrained ears, but would be close enough that they would be more likely interpreted as out-of-tune versions of the same note than as different notes. The comma is also referred to as a Didymean comma because it is the amount by which Didymus corrected the Pythagorean major third to a just major third.
In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale(or half of a whole step), visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.
A schismatic temperament is a musical tuning system that results from tempering the schisma of 32805:32768 to a unison. It is also called the schismic temperament, Helmholtz temperament, or quasi-Pythagorean temperament.
In music, a ditone is the interval of a major third. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also called a comma-redundant major third; the smallest is the interval with a ratio of 100:81, also called a comma-deficient major third.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81 : 80), with respect to its just intonation used in Pythagorean tuning ; the result is 3/2 × 1⁄4 = 4√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 21⁄53, or 22.6415 cents, an interval sometimes called the Holdrian comma.
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of 31√2, or 38.71 cents.
In music, 19 equal temperament, called 19 TET, 19 EDO, 19-ED2 or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a frequency ratio of 19√2, or 63.16 cents.
In music, 22 equal temperament, called 22-TET, 22-EDO, or 22-ET, is the tempered scale derived by dividing the octave into 22 equal steps. Each step represents a frequency ratio of 22√2, or 54.55 cents.
In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.
Dynamic tonality is a paradigm for tuning and timbre which generalizes the special relationship between just intonation and the harmonic series to apply to a wider set of pseudo-just tunings and related pseudo-harmonic timbres.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.
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