12 Quintal (center C)

For a general introduction to Just (Rational) Intonation and these presets, please visit Introduction to Just or Rational Intonation (JI).

About this tuning

Commonly known as Ptolemaic tuning, after Claudius Ptolemy, who identified the sequence of proportions 8:9, 9:10, 15:16. The two tones of slightly different size combine to produce a major third in the proportion 4:5, which combines with the semitone to produce a perfect fourthPitches a fourth apart have a 3:4 frequency ratio (in just intonation) or a difference of 500 cents (in 12 tone equal temperament).Visit the link to learn more 3:4. The diatonic major scale with central pitch C combines two conjunct "tetrachords," one rising from G to C, and another rising from C to F, followed by a 8:9 wholetone between F and G to reach an octave 1:2. This tuning produces concordant major triads in the proportion 4:5:6 on C, F, and G. Similarly, the melodic minor scale (Aeolian mode) centered on C is produced by tuning three concordant minor triads in the proportion 10:12:15 on C, F, and G, deriving the notes E♭, A♭, B♭. The two remaining notes of the 12-tone chromatic scale are derived as the Neapolitan D♭ (diatonic minor 2nd, a perfect fifth below A♭) and the secondary Dominant F♯ (augmented fourth, a major third above D). Note that the interval from D to A is NOT a perfect fifthPitches a fifth apart have a 2:3 frequency ratio (in just intonation) or a difference of 700 cents (in 12 tone equal temperament).Visit the link to learn more. To obtain a concordant major triad on D, A must be raised by a Syntonic commaA comma is a small difference in pitch between the same interval in different tuning systems.Visit the link to learn more in the proportion 80:81 (approximately 21.5 cents).

Learning more

• 5-limit at the Xenharmonic Wiki‍
• Marc Sabat - Claudius Ptolemy (musical example)
• Wolfgang von Schweinitz - KLANG auf Schön Berg La Monte Young (musical example)

Legend of ASCII Notations

The following ASCII are used to write note names in rational intonation:

• ♭ ♯ for flat and sharp These standard symbols are used in different ways according to context. In Rational Intonation and near-harmonic EDOs, these refer to notes ordered by perfect fifthsPitches a fifth apart have a 2:3 frequency ratio (in just intonation) or a difference of 700 cents (in 12 tone equal temperament).Visit the link to learn more in a series or spiral, extending indefinitely. Double flats precede flats, followed by naturals and continue on to sharps, etc., always moving by 2:3 ratios. In this case, flats are generally lower than enharmonically related sharps, differing by a tertialIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more (PythagoreanPythagorean tuning is an approach to tuning based on stacking pure fifths (i.e. fifths with an exact frequency ratio of 2:3).Visit the link to learn more) commaA comma is a small difference in pitch between the same interval in different tuning systems.Visit the link to learn more. In historical Meantone tunings, on the other hand, perfect fifths are generally tempered, i.e., intentionally made smaller that 2:3 ratios, so that major and minor thirds more closely represent the quintalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more ratios 4:5 and 5:6. As a result, sharps and flats represent smaller degrees of alteration; sharps are often lower than enharmonically related flats. This difference may be observed by comparing the spelling and order of note names in 41edo (a nearly tertial tuning) and 43edo (nearly 1/5 comma Meantone).

• v ^ for quintal (syntonic) comma alteration In Rational Intonation, these represent alteration upward or downward by the quintal comma ratio 80:81 (ca. 21.5 cents, or about 1/9 tone).

• < > for septimal comma In Rational Intonation, these represent alteration upward or downward by the septimalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more comma ratio 63:64 (ca. 27.3 cents, or about 1/7 tone).

• d t for undecimal comma In Rational Intonation, these represent alteration upward or downward by the undecimalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more comma ratio 32:33 (ca. 53.3 cents, or about 1/4 tone).

• d| t| for tridecimal comma In Rational Intonation, these represent alteration upward or downward by the tridecimalIn Just/Rational Intonation tunings, ratios are often categorized by the prime numbers that constitute them. The Latin-derived terms "tertial" and "quintal," initiated by composer Catherine Lamb, are used to semantically describe the color and function qualities of the specific primes referenced (3 and 5), as well as the chordal and scale structures these primes generate. Higher primes have traditionally been named in similar manner, i.e. "septimal" (7), "undecimal" (11), "tridecimal" (13), etc.Visit the link to learn more comma ratio 26:27 (ca. 65.3 cents, or about 1/3 tone).

• `` ´´ for 17 In Rational Intonation, these represent alteration upward or downward by the ratio 2176:2187 (ca. 8.7 cents).

• ` ´ for 19 In Rational Intonation, these represent alteration upward or downward by the ratio 512:513 (ca. 3.4 cents).

• ° after a number refers to an overtonal harmonic relationship (23° partial refers to the frequency ratio 1:23 from a fundamental).

• u preceding a number refers to an undertonal subharmonic relationship (u23 refers to a frequency obtained by multiplying a reference frequency by 1/23).

For EDO systems, the standard notation uses a degree and division, i.e. 1\31 for the first step of 31edo. If another interval is being divided, it may be placed in brackets or angle brackets afterwards, i.e. 1\11 (3/2) for Wendy Carlos beta.