Resources / Theory / Pitch systems

Pitch systems

The single most important page here. Get this wrong and everything downstream is wrong.

Three incompatible systems

These are not three views of the same thing. A Turkish neutral third and an Arabic neutral third are different pitches, and both differ from anything on a piano.

The Turkish system (Arel–Ezgi–Uzdilek)

Established The octave divides into 53 commas. One comma (koma, Holdrian comma) = 1200/53 ≈ 22.6415 cents.

This isn't arbitrary. 53-tone equal temperament approximates just intonation remarkably well:

The fifth is accurate to a fourteenth of a cent. That's why the system survives.

Named intervals

Established

Note there is no single "semitone." Bakiye (4) and küçük mücennep (5) are both "half steps" and choosing between them is a real musical decision.

Accidentals

Established AEU uses five sharps and five flats, altering by 1, 4, 5, 8, and 12 commas respectively. This is why Turkish scores look alarming to Western readers — there are ten accidental glyphs, not two.

The Arabic system

Established Arabic notation uses the half-flat and half-sharp, dividing the octave into 24 equal quarter-tones of 50 cents each.

Practice This is a notational convention, not a description of performance. The actual pitches vary substantially by region and by performer. The "quarter-tone" in Egyptian Rast sits nowhere near an exact 50-cent division. Musicians treat the notation as an approximate pointer, and the real pitch is transmitted by ear.

This is a genuine and well-known gap between Arabic notation and Arabic practice. An app that renders Arabic maqam at exact 24-TET is rendering the notation, not the music. Worth stating explicitly in the UI.

Where theory and practice diverge — the Uşşak problem

Practice The clearest documented case. AEU theory places the second degree of Uşşak (the segâh perde relative to dügâh) at 8 commas ≈ 181 cents above the tonic.

Measured performance places it consistently lower — commonly cited in the region of 140–160 cents, sometimes lower still, and varying with melodic direction and with the individual performer.

Implications for the app:

1. Don't render one authoritative dot. Render a band.
2. The AEU value is worth showing as theory, labelled as such.
3. Where practice values exist from measurement studies, show them as the default.
4. The gap itself is pedagogically valuable — it teaches that this music is transmitted by ear and that notation trails behind.

The same tension applies to Rast's third and seventh, and to Segâh and Hüzzam.

Conversion reference

cents  = 1200 · log2(f / f_ref)
commas = cents / 22.6415
freq   = f_ref · 2^(cents/1200)

Fretless neck position from the nut, as a fraction of scale length:

x = 1 − 2^(−cents/1200)

At the octave (1200 ¢) this gives exactly 0.5 — halfway along the string, as expected.

Practical consequence for the instrument

Established On a 600 mm scale length, a comma displaces the finger by:

For comparison, a full semitone at the nut is 33.7 mm. So a comma is roughly a quarter of a semitone's distance — larger than you'd guess, and well within conscious finger control.

That's the encouraging part. The hard part is that the target moves. The correct placement of a given perde shifts with the makam, with melodic direction, and with regional style. There is no fixed millimetre mark to memorise, which is why fretless players learn this by ear and why the app's audio accuracy matters more than its visual accuracy. The delay-line interpolation bug in the prototype is therefore not a cosmetic issue.

Sources

• Karl Signell, Makam: Modal Practice in Turkish Art Music (Seattle: Asian Music Publications, 1977; repr. Da Capo Press) — the standard English-language account of the Arel–Ezgi–Uzdilek system, its 53-comma structure, and named intervals
• Ozan Yarman, "79-Tone Tuning & Theory for Turkish Maqam Music" (PhD dissertation, Istanbul Technical University Turkish Music State Conservatory, 2008) — a detailed critique of AEU comma values measured against performance practice
• Barış Bozkurt, Ozan Yarman, M. Kemal Karaosmanoğlu, and Can Akkoç, "Weighing Diverse Theoretical Models on Turkish Maqam Music Against Pitch Measurements: A Comparison of Peaks Automatically Derived from Frequency Histograms with Proposed Scale Tones," Journal of New Music Research 38, no. 1 (2009): 45–70 — the pitch-histogram measurement data behind the theory/practice gap described for Uşşak, Rast, Segâh and Hüzzam
• William Holder, A Treatise of the Natural Grounds and Principles of Harmony (London, 1694) — origin of the "Holdrian comma" as the name for the 1/53-octave unit later formalised in Turkish theory
• Scott L. Marcus, Music in Egypt: Experiencing Music, Expressing Culture (Oxford University Press, 2006) — on the gap between Arabic 24-tone notation and actual quarter-tone performance practice