Comma Calculations: Practical Math for the 53-Tone System

The previous articles explain what a Holdrian comma is and why 53 divisions work. This one is about doing the arithmetic — converting commas to cents, cents to frequency ratios, and using those conversions to build and verify real makam scales.

The core formula

Established One Holdrian comma is 1/53 of an octave. Since one octave equals 1200 cents, the conversion is:

1 comma = 1200 / 53 ≈ 22.6415 cents

To convert any number of commas to cents, multiply:

cents = commas × (1200 / 53)

To go the other direction — cents to commas — divide:

commas = cents × (53 / 1200)

Established The exact value of 1200/53 is the repeating decimal 22.641509433962..., which means every comma-to-cents conversion involves an irrational truncation. This is harmless in practice — the rounding errors are far below audible thresholds — but it matters when you are checking your arithmetic against published tables and need to know why your fifth decimal place disagrees.

From commas to frequency ratios

Established Cents measure intervals logarithmically. To get the actual frequency ratio — the number you multiply a frequency by to reach the higher pitch — you convert cents back to a ratio:

ratio = 2^(cents / 1200)

Since cents = commas × (1200 / 53), this simplifies to:

ratio = 2^(commas / 53)

This is the formula implemented in practice tools like Meshk. For a tanini (whole tone, 9 commas):

2^(9/53) ≈ 1.12486

Compare this to the Pythagorean whole tone (9:8 = 1.12500). The difference — about 0.014% — is inaudible.

Going the other direction, if you know a frequency ratio and want to find its comma value:

commas = 53 × log₂(ratio)

For the pure fifth (ratio 3:2):

53 × log₂(1.5) = 53 × 0.58496 ≈ 31.003

The pure fifth falls almost exactly on 31 commas. This is the fundamental reason the 53-tone system works: the most important interval in Pythagorean tuning maps to an almost-integer number of commas.

The five AEU intervals

Established Before building a scale, you need the five interval sizes that the Arel-Ezgi-Uzdilek system uses as building blocks. Here they are with their cent and ratio equivalents:

Practice The letters B, S, K, T, and A are the shorthand used in Turkish conservatory pedagogy. When you see a cesni described as "T K S T" you are reading a sequence of 9 + 8 + 5 + 9 = 31 commas. Recognizing these building blocks makes scale construction mechanical.

Worked example: building Rast from commas

Established Rast is the foundational Turkish makam. Its ascending scale is built from two cesni: a Rast besli (pentachord, T + K + S + T = 9 + 8 + 5 + 9 = 31 commas) on the karar, and a Rast dortlu (tetrachord, T + K + S = 9 + 8 + 5 = 22 commas) on the guclu. Together they span the full octave: 31 + 22 = 53 commas.

The cumulative comma values, and their conversions:

Established Check: does the fifth (Neva, 31 commas) land where it should? The 53-TET value is 701.89 cents; a pure Pythagorean fifth is 701.96 cents. The error is 0.07 cents — roughly 1/30 of a cent, far below the 2-5 cent threshold of human pitch discrimination.

Established Does the fourth (Cargah, 22 commas) hold up? 498.11 cents vs. a pure fourth of 498.04 cents. Error: 0.07 cents. Again negligible.

Practice Note that Segah at 384.91 cents sits much closer to a just major third (5:4 = 386.31 cents, error 1.40 cents) than to a Pythagorean major third (81:64 = 407.82 cents, error 22.91 cents). This is one place where the 53-tone system quietly approximates just intonation rather than strict Pythagorean tuning — a feature, not a bug, since the Segah perde has always been described as a "soft" or "neutral" third.

Verifying against Pythagorean ratios

Established If the 53-tone system genuinely approximates Pythagorean tuning, then comma-derived intervals should closely match ratios built from stacked pure fifths. Here is the comparison for the intervals that matter most:

Established The fifths, fourths, and whole tones match to within fractions of a cent. But notice the thirds: 17 commas (384.91 cents) is nowhere near the Pythagorean major third (407.82 cents). That is because 17 commas maps instead to the just major third neighborhood (386.31 cents). The 53-tone system is versatile enough to approximate both Pythagorean and just intervals — different comma counts access different ratio families. This is precisely why it became the chosen framework: it can represent the variety of intervals that makam actually uses, not just the Pythagorean subset.

Comparing intervals across makams

Practice Once you can convert commas to cents, comparing intervals across makams becomes straightforward. Consider the third degree in three different makams:

The difference between the Rast third and the Nihavend third is 4 commas (90.57 cents) — almost exactly one bakiye. The difference between Rast and Cargah is 1 comma (22.64 cents). These are the fine distinctions that give each makam its identity, and comma arithmetic makes them concrete rather than vague.

Practical calculation tips

Practice For quick mental arithmetic, the approximation 1 comma ≈ 23 cents is close enough for most purposes (true value: 22.64). This lets you estimate in your head: a kucuk mücenneb (5 commas) is about 115 cents, a tanini (9 commas) is about 207 cents. The errors from this rounding (about 2 cents per 5 commas) are at the edge of audibility.

For spreadsheet or calculator work, use the exact formulas:

• Commas to cents: = commas * 1200 / 53
• Commas to ratio: = POWER(2, commas / 53)
• Cents to commas: = cents * 53 / 1200
• Ratio to commas: = 53 * LOG(ratio, 2)
• Cents to Hz: = refHz * POWER(2, cents / 1200) (where refHz is your reference frequency, typically 440 Hz for A4)

Common pitfalls

Practice Three errors come up repeatedly when working with comma calculations:

Rounding accumulation. If you round 1200/53 to 22.64 and multiply by 53, you get 1199.92 instead of 1200. Always use 1200/53 as a fraction in your formulas, not a rounded decimal. Spreadsheets and programming languages handle this naturally if you write commas * 1200 / 53 rather than commas * 22.64.

Confusing 53-TET with pure Pythagorean. The 53-tone system approximates Pythagorean tuning — it is not identical to it. The difference is Mercator's comma (about 3.615 cents spread across all 53 steps). For scale construction and interval comparison, this gap is negligible. For acoustics research or precise synthesis, it is not. Know which context you are in.

Treating commas as absolute pitches. A comma count describes an interval — a distance between two pitches — not a fixed frequency. "Dugah is at 9 commas" means 9 commas above Rast, wherever Rast happens to be. Different performers, instruments, and traditions place Rast at different absolute pitches. The comma system is relative, like the Western concept of a "key" — it tells you the interval structure, not the concert pitch.