Both the Sabine and Eyring equations predict the same quantity — RT60, the time for sound to decay 60 dB — yet they can produce answers that differ by 30% or more in the same room. Understanding when and why they diverge is essential for any practitioner doing RT60 calculations. This article uses a single recording studio to demonstrate the difference numerically, including the Millington-Sette equation as a third comparator.
The Room
A medium-sized recording control room:
- Length: 6.5 m
- Width: 5.0 m
- Height: 2.8 m
- Volume (V): 6.5 × 5.0 × 2.8 = 91.0 m³
- Total surface area (S): 2 × (6.5×5.0 + 6.5×2.8 + 5.0×2.8) = 2 × (32.5 + 18.2 + 14.0) = 129.4 m²
Surface Inventory and Materials
| Surface | Dimensions | Area (m²) | Material |
|---|---|---|---|
| Floor | 6.5 × 5.0 | 32.5 | Hardwood over concrete |
| Ceiling | 6.5 × 5.0 | 32.5 | Fabric-wrapped 100 mm mineral wool |
| Front wall | 5.0 × 2.8 | 14.0 | Angled 50 mm studio foam panels (80% coverage) + 2 mm hardwood (20%) |
| Rear wall | 5.0 × 2.8 | 14.0 | 150 mm broadband absorber panels (full coverage) |
| Side wall A | 6.5 × 2.8 | 18.2 | 100 mm mineral wool panels (60%) + 2 mm hardwood (40%) |
| Side wall B | 6.5 × 2.8 | 18.2 | 100 mm mineral wool panels (60%) + hardwood (40%) |
Front wall breakdown: foam = 0.80 × 14.0 = 11.2 m²; hardwood = 0.20 × 14.0 = 2.8 m² Side walls each: mineral wool = 0.60 × 18.2 = 10.9 m²; hardwood = 0.40 × 18.2 = 7.3 m²
Absorption Coefficients
| Material | 125 Hz | 250 Hz | 500 Hz | 1000 Hz | 2000 Hz | 4000 Hz |
|---|---|---|---|---|---|---|
| Hardwood floor (25 mm on battens) | 0.15 | 0.11 | 0.10 | 0.07 | 0.06 | 0.07 |
| Fabric-wrapped 100 mm mineral wool (ceiling) | 0.55 | 0.80 | 0.95 | 0.98 | 0.98 | 0.95 |
| 50 mm studio foam panels | 0.15 | 0.35 | 0.70 | 0.92 | 0.97 | 0.96 |
| 150 mm broadband absorber (rear wall) | 0.75 | 0.95 | 0.98 | 0.99 | 0.99 | 0.98 |
| 100 mm mineral wool panels | 0.45 | 0.75 | 0.95 | 0.98 | 0.97 | 0.95 |
| 2 mm hardwood sheet (reflective) | 0.10 | 0.08 | 0.06 | 0.05 | 0.04 | 0.03 |
Step 1 — Calculate Total Absorption Area A
For each surface, compute Si × αi at each octave band.
At 500 Hz (Key Band)
| Surface | Area (m²) | Material | α (500 Hz) | Si × αi |
|---|---|---|---|---|
| Floor | 32.5 | Hardwood | 0.10 | 3.25 |
| Ceiling | 32.5 | 100 mm mineral wool | 0.95 | 30.88 |
| Front wall — foam | 11.2 | 50 mm studio foam | 0.70 | 7.84 |
| Front wall — hardwood | 2.8 | 2 mm hardwood | 0.06 | 0.17 |
| Rear wall | 14.0 | 150 mm broadband | 0.98 | 13.72 |
| Side A — mineral wool | 10.9 | 100 mm mineral wool | 0.95 | 10.36 |
| Side A — hardwood | 7.3 | 2 mm hardwood | 0.06 | 0.44 |
| Side B — mineral wool | 10.9 | 100 mm mineral wool | 0.95 | 10.36 |
| Side B — hardwood | 7.3 | 2 mm hardwood | 0.06 | 0.44 |
| Total A (500 Hz) | 129.4 | 77.46 m² |
Average absorption coefficient: ᾱ = A/S = 77.46 / 129.4 = 0.599
This is a highly absorptive room.
Full Octave-Band A Values
Applying the same calculation at all bands:
| Band (Hz) | A (m²) | ᾱ = A/S |
|---|---|---|
| 125 | 38.97 | 0.301 |
| 250 | 60.90 | 0.471 |
| 500 | 77.46 | 0.599 |
| 1000 | 83.97 | 0.649 |
| 2000 | 84.25 | 0.651 |
| 4000 | 82.26 | 0.636 |
Step 2 — Sabine Calculation
Per ISO 3382-2:2008 §A.1: RT60 = 0.161 × V / A
V = 91.0 m³
| Band (Hz) | A (m²) | RT60 Sabine (s) |
|---|---|---|
| 125 | 38.97 | 0.161 × 91.0 / 38.97 = 0.376 s |
| 250 | 60.90 | 0.161 × 91.0 / 60.90 = 0.241 s |
| 500 | 77.46 | 0.161 × 91.0 / 77.46 = 0.189 s |
| 1000 | 83.97 | 0.161 × 91.0 / 83.97 = 0.174 s |
| 2000 | 84.25 | 0.161 × 91.0 / 84.25 = 0.174 s |
| 4000 | 82.26 | 0.161 × 91.0 / 82.26 = 0.178 s |
Mid-frequency Sabine RT60 = (0.189 + 0.174) / 2 = 0.182 s
Step 3 — Eyring Calculation
Per ISO 3382-2:2008 §A.2: RT60 = 0.161 × V / (−S × ln(1 − ᾱ) + 4mV)
For rooms below 500 m³, air absorption (4mV) is negligible and set to zero.
The only change from Sabine is replacing A (= S × ᾱ) with −S × ln(1 − ᾱ).
Let's work through 500 Hz:
- ᾱ = 0.599
- 1 − ᾱ = 0.401
- ln(0.401) = −0.913
- −S × ln(1 − ᾱ) = −129.4 × (−0.913) = 118.1 m²
- RT60 = 0.161 × 91.0 / 118.1 = 14.65 / 118.1 = 0.124 s
Full Octave-Band Eyring Results
| Band (Hz) | ᾱ | 1 − ᾱ | ln(1 − ᾱ) | −S × ln(1 − ᾱ) | RT60 Eyring (s) | RT60 Sabine (s) | Difference |
|---|---|---|---|---|---|---|---|
| 125 | 0.301 | 0.699 | −0.358 | 46.32 | 0.316 s | 0.376 s | −16% |
| 250 | 0.471 | 0.529 | −0.636 | 82.30 | 0.178 s | 0.241 s | −26% |
| 500 | 0.599 | 0.401 | −0.913 | 118.1 | 0.124 s | 0.189 s | −34% |
| 1000 | 0.649 | 0.351 | −1.047 | 135.5 | 0.108 s | 0.174 s | −38% |
| 2000 | 0.651 | 0.349 | −1.052 | 136.1 | 0.108 s | 0.174 s | −38% |
| 4000 | 0.636 | 0.364 | −1.011 | 130.8 | 0.112 s | 0.178 s | −37% |
Mid-frequency Eyring RT60 = (0.124 + 0.108) / 2 = 0.116 s
The Sabine prediction of 0.182 s is 57% higher than the Eyring prediction of 0.116 s.
Step 4 — Millington-Sette Calculation
The Millington-Sette equation (also called the Millington equation) uses:
RT60 = 0.161 × V / (−Σ(Si × ln(1 − αi)))
Instead of using a single average ᾱ, it applies the logarithm to each surface's individual absorption coefficient before summing. This matters when surfaces have very different absorption levels.
At 500 Hz:
| Surface | Area (m²) | α | 1 − α | ln(1 − α) | −Si × ln(1 − α) |
|---|---|---|---|---|---|
| Floor (hardwood) | 32.5 | 0.10 | 0.90 | −0.105 | 3.42 |
| Ceiling (100 mm MW) | 32.5 | 0.95 | 0.05 | −2.996 | 97.37 |
| Front — foam | 11.2 | 0.70 | 0.30 | −1.204 | 13.48 |
| Front — hardwood | 2.8 | 0.06 | 0.94 | −0.062 | 0.17 |
| Rear wall | 14.0 | 0.98 | 0.02 | −3.912 | 54.77 |
| Side A — MW | 10.9 | 0.95 | 0.05 | −2.996 | 32.66 |
| Side A — hardwood | 7.3 | 0.06 | 0.94 | −0.062 | 0.45 |
| Side B — MW | 10.9 | 0.95 | 0.05 | −2.996 | 32.66 |
| Side B — hardwood | 7.3 | 0.06 | 0.94 | −0.062 | 0.45 |
| Sum | 235.43 m² |
RT60 Millington-Sette (500 Hz) = 0.161 × 91.0 / 235.43 = 14.65 / 235.43 = 0.062 s
The three predictions at 500 Hz:
- Sabine: 0.189 s
- Eyring: 0.124 s
- Millington-Sette: 0.062 s
Full Octave-Band Millington-Sette Results
| Band (Hz) | −Σ(Si × ln(1−αi)) | RT60 M-S (s) | RT60 Eyring (s) | RT60 Sabine (s) |
|---|---|---|---|---|
| 125 | 54.16 | 0.271 s | 0.316 s | 0.376 s |
| 250 | 133.8 | 0.110 s | 0.178 s | 0.241 s |
| 500 | 235.4 | 0.062 s | 0.124 s | 0.189 s |
| 1000 | 283.6 | 0.052 s | 0.108 s | 0.174 s |
| 2000 | 285.1 | 0.052 s | 0.108 s | 0.174 s |
| 4000 | 274.9 | 0.053 s | 0.112 s | 0.178 s |
Why the Results Diverge So Dramatically
The mathematical divergence stems from Jensen's inequality: for a concave function f(x) = −ln(1−x), the function value at the mean is less than the mean of the function values.
In plain terms: mixing near-perfect absorbers (α = 0.95–0.98) with reflective surfaces (α = 0.06–0.10) in the same room produces a much lower effective absorption than you get by averaging the coefficients first.
In this studio:
- The ceiling alone has α = 0.95 at 500 Hz. Millington-Sette treats this as −ln(0.05) = 3.00 per unit area.
- Eyring uses ᾱ = 0.599, giving −ln(0.401) = 0.913 per unit area of total surface.
- That 3.3× difference in the log term for the most absorptive surface drives the large gap.
Which Equation to Use?
| Room Type | Average ᾱ | Recommended Equation | Error if Wrong |
|---|---|---|---|
| Bare concrete room | < 0.05 | Sabine | < 5% |
| Office with standard ceiling tiles | 0.10–0.20 | Sabine (acceptable) | 5–15% |
| Open-plan with carpet + ceiling | 0.20–0.35 | Eyring preferred | 15–25% |
| Well-treated meeting room | 0.35–0.50 | Eyring | 20–30% |
| Recording control room | 0.50–0.70 | Millington-Sette | 50–100% |
| Anechoic chamber | > 0.90 | Millington-Sette or measurement | Sabine fails |
Practical Implications for This Studio
If the designer had used Sabine and specified treatment to achieve a target mid-frequency RT60 of 0.15 s, they would have under-specified the treatment. The room, built to Sabine's prediction, would actually measure closer to 0.11–0.12 s (Eyring) — drier than intended.
Conversely, if a studio designer targets a specific "live end, dead end" RT60 profile and measures 0.12 s at 500 Hz while Sabine predicts 0.19 s, the discrepancy is expected and correct — it is not a measurement error.
The lesson is straightforward: Sabine is a conservative predictor in highly absorptive rooms. It always overestimates RT60 when ᾱ > 0.2. For recording studios, vocal booths, and any space where controlling the lower limit of RT60 is the design goal, use Eyring at minimum and Millington-Sette where surfaces have widely varying absorption coefficients.