This excerpt is from my book The Theory of Rebetiko, where I delve into the modal system of Greek music as it relates to the rebetiko genre. To access the complete text—including detailed discussions, tables, illustrations, and musical transcriptions of the examples—visit this page and explore the rich world of Greek modal music.
Microtones and Interval Measurement
Most Greek music is melodically defined by the use of musical intervals that differ from the tone and semitone of equal temperament. These intervals require dividing the octave into smaller units and are commonly referred to as microtones. To describe and quantify microtones, an appropriate unit of measurement is necessary. The most commonly used unit is the cent, which is equal to 1:100 of an equal-tempered semitone. The cent value of a musical interval is calculated using the formula: 1200log2x, where x represents the ratio expressing the interval. It is important to emphasize that microtones are just one—and not the only—defining characteristic of Greek modes. While they are not strictly indispensable, the entire Greek modal system can be effectively adapted to equal temperament. However, such adaptation inevitably results in a noticeable loss of pitch nuance (See From Modes to Dromi).
Determining the Pitches
To understand the Greek modal system, it is essential to first identify the pitches that constitute it. These pitches are determined by the intervallic ratios between the degrees of two historically significant scales: the Pythagorean scale and the natural scale. Both scales serve as the starting point for the development of the two primary diatonic scales in Greek music: the hard diatonic scale and the soft diatonic scale.
The Pythagorean Scale
Pythagoras (570 B.C. – ca. 495 B.C.) identified the octave interval with the ratio 2:1 (1200 cents) and the fifth interval with the ratio 3:2 (ca. 702 cents). By subtracting the octave from the fifth, Pythagoras derived the fourth (2 ÷ 3:2 = 4:3, ca. 498 cents). He then calculated the remaining degrees of a diatonic scale by combining sums and differences of these ratios. The result was a scale consisting of two types of intervals:
the Pythagorean tone, derived from the difference between the Pythagorean fifth and fourth (3:2 ÷ 4:3 = 9:8, ca. 204 cents);
the Pythagorean diatonic semitone or, more precisely, limma, derived from the difference between the Pythagorean fourth and two Pythagorean tones (4:3 ÷ 9:8 ÷ 9:8 = 256:243, ca. 90 cents).
Associating the I degree with C and arranging the other degrees in a sequence that mirrors a modern major scale, the result is the Pythagorean C scale.
Pythagorean C scale. The intervallic structure is represented in ratios relative to the I degree and between each degree (left), and in cents (right).
The differences between the intervals of the Pythagorean scale and the intervals of the modern major scale can be considered negligible, as they do not exceed the just noticeable difference (JND), i.e., the smallest interval the human ear can perceive between two pitches. For this reason, in Greek music, we can represent the pitches of the Pythagorean C scale on the staff—at least for now—without requiring additional accidentals. However, it is important to observe that the pitches of the notes written on the staff are not identical to the pitches in equal temperament, but only approximate. For completeness, the following intervals should also be included:
the Pythagorean chromatic semitone or, more precisely, apotome, derived from the difference between the Pythagorean tone and the limma (9:8 ÷ 256:243 = 2187:2048, ca. 114 cents);
the ditonic comma or Pythagorean comma, derived from the difference between the apotome and the limma (2187:2048 ÷ 256:243 = 531441:524288, ca. 23.46 cents).
The Hard Diatonic Scale
The hard diatonic scale (briefly: hard scale) consists of the same intervals as the Pythagorean scale, namely the Pythagorean tone (9:8, ca. 204 cents) and the limma (256:243, ca. 90 cents). While the staff notation for the hard scale could theoretically match that of the Pythagorean scale, in the Greek modal system it is more practical to construct the model of the hard scale starting from D and arranging the intervals differently.
Hard C scale modeled after the Pythagorean C scale (first line) and hard D scale (second line). The intervallic structure is presented both as ratios relative to the first degree and between consecutive degrees (left), and in cents (right).
The Natural Scale
Dissonances between the I degree and certain other degrees of the Pythagorean scale led first Archytas (428 B.C. – 360 B.C.) and then Didymus (ca. 63 B.C. – 10 A.D.) to propose new ratios for the major third interval (5:4 instead of 81:64) and minor third interval (6:5 instead of 32:27). This resulted in the modification of the Pythagorean scale by adjusting the ratios of the III degree (5:4 instead of 81:64), the VI degree (5:3 instead of 27:16), and the VII degree (15:8 instead of 243:128) relative to the I degree. This revised scale was later adopted by Claudius Ptolemy (ca. 100 – ca. 168) and, many centuries later, by Gioseffo Zarlino (1517-1590), and is nowadays referred to as the Zarlinian scale or the natural scale.
Natural C scale. The intervallic structure is presented in ratios relative to the I degree and between each degree (left), and in cents (right).
The differences between the intervals of the natural scale and the intervals of the modern major scale, and especially the intervals of the Pythagorean scale, are far from negligible: for the III, VI, and VII degrees, the just noticeable difference (set at 10 cents) is significantly exceeded. For this reason, in order to represent certain pitches more accurately on the music staff, it will be necessary to use special accidentals that differ from those typically used in our music system. In summary, the intervals that form the natural scale are:
the Pythagorean tone (9:8, ca. 204 cents);
the Didymus’ tone, derived from the difference between the Didymus’ third and the Pythagorean tone (5:4 ÷ 9:8 = 10:9, ca. 182 cents);
the Didymus’ diatonic semitone, derived from the difference between the Pythagorean fourth and the Didymus’ third (4:3 ÷ 5:4 = 16:15, ca. 112 cents).
For completeness, the following intervals should also be included:
the Didymus’ chromatic semitone, derived from the difference between the Didymus’ tone and the Didymus’ diatonic semitone (10:9 ÷ 16:15 = 25:24, ca. 70 cents; the value is ca. 70.67 cents, but it is rounded to 70 cents, not 71, to ensure the sum with the Didymus’ diatonic semitone, 112 cents, is equal to the exact value of the Didymus’ tone: 70 + 112 = 182 cents);
the syntonic comma or Didymus’ comma, derived from the difference between the Pythagorean tone and the Didymus’ tone (9:8 ÷ 10:9 = 81:80, ca. 21.5 cents);
the schisma, derived from the difference between the Pythagorean comma and Didymus’ comma (531441:524288 ÷ 81:80 = 32805:32768, ca. 1.95 cents).
The Soft Diatonic Scale
The soft diatonic scale (briefly: soft scale) consists of certain intervals from the natural scale, namely the Pythagorean tone (9:8, ca. 204 cents), the Didymus’ tone (10:9, ca. 182 cents), and the Didymus’ diatonic semitone (16:15, ca. 112 cents). The soft scale differs from the natural scale solely in the pitch of the VI degree, whose ratio relative to the I degree mirrors that of the Pythagorean scale and, by extension, the hard scale (27:16). As with the hard scale, it is preferable in the Greek system to construct the model of the soft scale starting from D.
Soft C scale modeled after the Natural C scale (first line) and soft D scale (second line). The intervallic structure is presented both as ratios relative to the I degree and between consecutive degrees (left), and in cents (right).
The Commas
The cent provides a unit of measurement with nearly absolute precision, but it is also somewhat unintuitive due to the need to work with large numbers. […] In the context of Greek music and similar traditions, such as Ottoman classical music, the most practical and precise solution is to divide the octave into 53 equal microtones, contrasting with the 12 semitones of equal temperament. This approach is not new: as early as the 17th century, the mathematician Nicolaus Mercator (1620-1687) theorized dividing the octave into 53 equal parts, referring to these divisions as the artificial comma to distinguish them from the Pythagorean and Didymus’ commas. Mercator’s artificial comma—which we will now refer to simply as the comma—has a value of approximately 22.64 cents. To calculate the comma value of a musical interval, the formula 53log2x is used, where x is the ratio expressing the interval.
The great advantage of using the comma as a unit of measurement in the Greek modal system is its capacity to simplify the representation of musical intervals by using smaller, more intuitive numbers instead of complex cent values. This makes intervallic structures more accessible and easier to comprehend. Additionally, the precision provided by the comma is more than adequate for practical musical applications, with deviations smaller than 1 cent—well within the 10 cents threshold of the just noticeable difference. The practical convenience of using the comma becomes evident when comparing intervallic structures in the hard scale and soft scale in both commas and cents.
Hard and soft D scale: the intervallic structure is presented in commas (first line) and in cents (second line).
The Accidentals
The accidentals of equal temperament, such as the sharp (#) and flat (b), are insufficient to accurately represent the pitches of certain notes in the Greek modal system. This limitation stems from the fact that the pitches in this system do not align perfectly with the standardized divisions of equal temperament. As a result, it becomes necessary to introduce new accidentals and reinterpret existing ones.
Accidentals in the Greek modal system and their theoretical pitch change.
The accidentals commonly used in the Greek modal system are the same as those found in Ottoman classical music theory. Each accidental is named according to the number of commas it modifies. […] A graphic representation of a Pythagorean tone interval and its division into 9 commas illustrates that it is impossible to divide this interval exactly into two equal semitones, as it is in equal temperament.
While the use of specific accidentals may suggest the ability to notate the pitch of every note with great precision, in reality, significant discrepancies often exist between theoretical notation and performance practice. The actual modification of pitches indicated by these accidentals should be understood as variable and subject to considerable fluctuation. Although a detailed description of each accidental and its contexts of use is beyond the scope of this discussion, several general observations can be made:
the flat theoretically indicates a pitch change of -5 commas on the staff, but can vary between -5 and -4 commas;
the tetra-flat theoretically indicates a pitch change of -4 commas on the staff, but can vary between -5 and -3 commas;
the mono-flat is the most variable of all the accidentals, theoretically indicating a pitch change of -1 comma on the staff, but ranging between 0 and -4 commas;
the sharp theoretically indicates a pitch change of +4 commas on the staff, but can vary between +4 and +5 commas.
Regarding the mono-sharp and penta-sharp, these accidentals are relatively rare in Greek music, and their notation on the staff is often considered optional. In fact, they primarily appear in contexts involving unusual transpositions of elements or modes, or when describing specific microtonal nuances in a theoretical context. The actual realization of these nuances is either optional or implied in performance practice.
Musical, Textual, and Modal Notation
The Greek modal system spans across two octaves, ranging from G3 to G5 in scientific pitch notation. Sometimes, it can be useful to represent notes using textual notation, eliminating the need for a staff. In this method, the note name is written followed by any applicable accidental, with an underline for notes below middle C (e.g., A̲) and an overline for those at or above the upper octave (e.g., C̅). This approach offers a simple and intuitive way to reference notes without requiring their placement on a staff.
Example of textual notation in the Greek modal system.
A more traditional approach, common to many musical cultures (e.g., Persian, Ottoman, Arabic), assigns a specific name to each note within the system. However, these names do not denote absolute pitches (e.g., C, D, E, etc.), but rather a position within the system, as defined by specific intervals. The reference point is typically the fretboard of a stringed instrument—such as the tanbûr for the Turks, or the tabouras or laouto politiko for the Greeks—where each fret corresponds to a named position. These names are independent of the instrument’s tuning and do not necessarily correspond to the actual pitch produced.
For Arabs, Greeks, and Turks, the key note of the system is rast. Arabs and Greeks identify rast with middle C (C4), while the Turks associate it with G (G4). However, although the Turks also use the G-clef, in practice, rast corresponds to concert D in their Bolahenk transposition, which is the most commonly used in Ottoman classical music. The identification of the rast position—and by extension, all other positions—with a specific note does not preclude transposing the entire system or an individual piece to begin from any other note. What remains essential is the preservation of the intervallic relationships between all pitches, starting from the position established as rast.
Modal notation in the Greek system (first line) and the Ottoman system with Bolahenk transposition (second line; concert pitches in parentheses).
