19 equal temperament

History and use

Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory. The ratio of four minor thirds to an octave ( or 62.565 cents – the "greater" diesis) was almost exactly a nineteenth of an octave. Interest in such a tuning system goes back to the 16th century, when composer Guillaume Costeley used it in his chanson Seigneur Dieu ta pitié of 1558. Costeley understood and desired the circulating aspect of this tuning.

In 1577, music theorist Francisco de Salinas discussed meantone, in which the tempered perfect fifth is 694.786 cents. Salinas proposed tuning nineteen tones to the octave to this fifth, which falls within one cent of closing. The fifth of 19 EDO is 694.737 cents, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error, so Salinas' suggestion is, for purposes relating to human hearing, functionally identical to 19 EDO.

In the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50 EDO.

The composer Joel Mandelbaum wrote on the properties of the 19 EDO tuning and advocated for its use in his Ph.D. thesis: Mandelbaum argued that it is the only viable system with a number of divisions between 12 and 22, and furthermore, that the next smallest number of divisions resulting in a significant improvement in approximating just intervals is . Mandelbaum and Joseph Yasser have written music with 19 EDO. : cited by

Easley Blackwood stated that 19 EDO makes possible "a substantial enrichment of the tonal repertoire".

Notation

for 19 equal temperament: Intervals are notated similarly to the intervals that approximate them. Aside from double sharps or double flats, only the note pairs E♯ & F♭ and B♯ & C♭ are enharmonic equivalents (modern sense). ]]

Notes in 19-EDO can be represented with the letter notation or staff notation of 12-EDO; however, enharmonic notes in 12-EDO are different pitches in 19-EDO (e.g. C♯ and D♭ are different pitches in 19-EDO though they sound the same in 12-EDO). Additionally in 19-EDO, B♯ is the note between B and C, enharmonic with C♭; E♯ is the note between E and F, enharmonic with F♭. Asides from those two cases, there are no other enharmonic equivalent notes in 19-EDO without the use of double sharps or double flats.

This article uses the notation described above.

Interval size

Here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series; the difference column measures in cents the distance from an exact fit to these ratios.

For reference, the difference from the perfect fifth in the widely used 12 TET is 1.955 cents flat, the difference from the major third is 13.686 cents sharp, the minor third is 15.643 cents flat, and the (lost) harmonic minor seventh is 31.174 cents sharp.

:{| class="wikitable" style="text-align:center;" | | |- C♭ F♭ |- |}

:{| class="wikitable sortable" style="text-align:center;" ! Interval name ! Size (steps) ! Size (cents) ! Midi ! Just ratio ! Just (cents) ! Midi ! Error (cents) |- | Octave | 19 | 1200 | | 2:1 | 1200 |

| | 27:14 | 1137.04 |

| | 48:25 | 1129.33 |

| | 15:8 | 1088.27 |

| | 9:5 | 1017.60 |

| | 12:7 | 933.13 |

| | 5:3 | 884.36 |

| | 8:5 | 813.69 |

| | 25:16 | 772.63 |

| | 14:9 | 764.92 |

| | 3:2 | 701.96 |

| | 13:9 | 636.62 |

| | 10:7 | 617.49 |

| | 7:5 | 582.51 |

| | 18:13 | 563.38 |

| | 4:3 | 498.04 |

| | 125:96 | 456.99 |

| | 13:10 | 454.12 |

| | 9:7 | 435.08 |

| | 5:4 | 386.31 |

| | 6:5 | 315.64 |

| | 7:6 | 266.87 |

| | 15:13 | 247.74 |

| | 9:8 | 203.91 |

| | 10:9 | 182.40 |

| | 13:12 | 138.57 |

| | 14:13 | 128.30 |

| | 15:14 | 119.44 |

| | 16:15 | 111.73 |

| | 21:20 | 84.46 |

| | 25:24 | 70.67 |

| | 28:27 | 62.96 | | +0.20 |}

A possible variant of 19-ED2 is 93-ED30, i.e. the division of 30:1 in 93 equal steps, corresponding to a stretching of the octave by 27.58¢, which improves the approximation of most natural ratios.

Scale diagram

Because 19 is a prime number, repeating any fixed interval in this tuning system cycles through all possible notes; just as one may cycle through 12-EDO on the circle of fifths, since a fifth is 7 semitones, and number 7 is coprime to 12.

[[Mode (music)|Modes]]

[[Ionian mode]] ([[major scale]])

[[Dorian mode]]

[[Phrygian mode]]

[[Lydian mode]]

[[Mixolydian mode]]

[[Aeolian mode]] ([[natural minor scale]])

[[Locrian mode]]

References

References

1. (Winter 2007). "Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum". [[Computer Music Journal]].
2. [[Joseph Yasser]]. "A Theory of Evolving Tonality".
3. Heino, Arto Juhani. "Artone 19 Guitar Design".