Diaschismatic Guitar

Description
Diaschismatic guitar is designed to play makam music with guitar technique. It's made by luthier Uğur Keçecioğlu from Mothership Guitars, Istanbul, by modifying an Ibanez. It has 5 strings and 22 frets to the octave. It's fretless after the octave fret, fingerboard raised by an ebony sheet for easier fretless playing.

Fret spacings follow a pattern of alternating larger (B) and smaller (F) intervals.

B = 65.78 cents (~0.66 frets in a regular guitar)
F = 41.06 cents (~0.41 frets in a regular guitar)

Technical Details
Turkish classical music tuning does not temper out syntonic comma, which means two whole tones and a major third are not the same.


This feature creates a melodic diversity not found in Western music. There are many different musical modes called “makams”, each having its own feeling and mood.

Syntonic comma (or simply comma) has a frequency ratio of 81:80 (~21.5 cents), which requires a fret spacing of about 1/5 of a semitone in a regular guitar. Diaschismatic temperament creates larger fret spacings by slightly flattening the perfect fifth, without compromising of major third. A guitar with larger fret spacings is easier to play, especially to play chords.

Diaschismatic temperament tempers out the diaschisma, the ratio of 2048:2025, which is the interval between the chromatic semitone (25:24) and Pythagorean semitone (256:243). Chromatic semitone itself is the interval between major third and minor third, and Pythagorean semitone is the semitone achieved through circle of fifths.

Since these two semitones are the same, diaschismatic guitar has a regular pattern of alternating larger and smaller intervals; semitones and commas. Semitones are denoted by B (from "bakiye" in Turkish music theory) and commas are denoted by F (from "fazla", also called "koma"). 


B = 65.78 cents, and F = 41.06 cents. There are 12 B's and 10 F's in an octave, leading to 22 frets in total. 

Calculation

Diaschismatic temperament may have several varieties with different values for perfect fifths and major thirds. All satisfy the following equation (p: perfect fifth, m: major third, v: octave, given as logarithms).

   4p + 2m = 3v

Since B is the interval between major third and minor third, and minor third is the interval between perfect fifth and major third, so;

   
   B = m - (p - m) = 2m - p.

F is the syntonic comma, the interval between two whole tones and a major third. A whole tone is the interval between a perfect fifth and a perfect fourth, or two perfect fifths and an octave. So a whole tone equals to 2p - v. Two whole tones are 4p - 2v. So, 


   F = 4p - 2v - m.

This diaschismatic guitar uses just major thirds (exactly equal to 386.31 cents). The octave is given as 1200 cents by definition, perfect fifth is calculated as 706.84 cents (flattened by 4.89 cents -a quarter of diaschisma- from just value). The calculation gives the values for B and F as 65.78 cents and 41.06 cents respectively. 


Another variety can be 34 equal division of octave (34-EDO). It is achieved by taking B = 2 and F = 1 units. In terms of cents, B = 70.59 cents and F = 35.29 cents. It can be useful for describing intervals and scales in terms of simple integers.

The values of some natural intervals in diaschismatic temperament are as follows:



Interval
Formula
34-EDO
Octave
12B + 10F
34
Perfect Fifth
7B + 6F
20
Perfect Fourth
5B + 4F
14
Major Third
4B + 3F
11
Minor Third
3B + 3F
9
Augmented Second
3B + 2F
8
Whole Tone
2B + 2F
6
Minor Whole Tone
2B + F
5
Major Semitone
B + F
3
Minor Semitone
B
2

Some of most common makams’ scales in terms of 34-EDO:
Rast
 6-5-3-6-6-5-3
Nihavend
 6-2-6-6-2-6-6
Kürdi
 2-6-6-6-2-6-6
Segâh
 3-6-5-6-3-6-5
Hüseyni
 5-3-6-6-5-3-6
Uşşak
 5-3-6-6-2-6-6
Sabâ
 5-3-4-8-2-6-6
Hicaz
 3-8-3-6-5-3-6
Hicaz Hümayun
 3-8-3-6-3-5-6
Hicazkâr
 3-8-3-6-3-8-3
Karcığar
 5-3-6-3-8-3-6
Hüzzam
 3-6-3-8-3-6-5
Acemaşiran
 6-6-2-6-6-6-2
Suzinak
 6-5-3-6-3-8-3
Nikriz
 6-3-8-3-6-3-5