55 Tertial-Quintal Sabat Euler Lattice

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

About this tuning

55-tone 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-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 Rational Intonation scale, based on the primary roots of major and minor triads in the "Harmonium for Ben Johnston" section of Marc Sabat's string quartet Euler Lattice Spirals Scenery‍. The tertial diatonic tones are each divided in the same symmetric pattern of 9 "commasA comma is a small difference in pitch between the same interval in different tuning systems.Visit the link to learn more." To divide the interval C 1/1 - D 9/8 add two notes above C: 81/80 and 25/24, then add two more notes one Syntonic Comma above each of these. Similarly, add two notes below D: 10/9 and 27/25; then, add two more notes a Syntonic Comma below each of these. To divide the diatonic limmas E-F and B-C, add two successive Syntonic Commas upward from the lower note and two commas downward from the higher note, reaching a point of enharmonic near-equivalence (these two "pairs" of proximal enharmonics extend the tuning from 53 to 55 notes per octave).

Learning more

• Marc Sabat - Euler Lattice Spirals Scenery (musical example)
• Marc Sabat - Euler Lattice Spirals Scenery score (pdf)

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 tertial (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) comma. 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 quintal 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 septimal 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 undecimal 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 tridecimal 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.