Source: https://jmrplens.github.io/phonometry/buildings/rooms/stage-and-uncertainty/ # Stage support and uncertainty (ISO 3382-1) Every other page about ISO 3382-1 measures what a listener hears. This one covers the two things the standard says about the measurement itself: what a musician on the platform hears of their own instrument, and how much of a measured reverberation time belongs to the room rather than to the noise the room was excited with. *The fourth cell is this page: on the platform the microphone sits a metre from the source, and everything else reflecting has to be further than two.* ## 1. Support: the hall answering the player A musician a metre from their own instrument hears the direct sound and, some milliseconds later, whatever the platform and the enclosure send back. Annex C makes that a ratio, measured with the source and the microphone 1,0 m apart at the same height, with nothing reflecting within 2 m: $$ ST_\mathrm{Early} = 10 \lg \left[ \frac{\int_{0,020}^{0,100} p^2(t)\ \mathrm{d}t} {\int_{0}^{0,010} p^2(t)\ \mathrm{d}t} \right] \text{dB}, \qquad ST_\mathrm{Late} = 10 \lg \left[ \frac{\int_{0,100}^{1,000} p^2(t)\ \mathrm{d}t} {\int_{0}^{0,010} p^2(t)\ \mathrm{d}t} \right] \text{dB}. $$ Early support relates to ensemble, the ease of hearing the other players; late support to how much of the hall comes back. Both divide by the same direct sound, so their difference is a property of the hall and does not move with how loudly the instrument was played. *The three windows, the two supports, and what the excitation costs.* ```python import numpy as np from phonometry import room fs = 48000 # A unit direct arrival, one reflection in the early window and one in the # late one: each support is then 20 lg of that reflection's amplitude. response = np.zeros(round(2.0 * fs)) response[100] = 1.0 response[100 + round(0.050 * fs)] = 0.2 response[100 + round(0.400 * fs)] = 0.1 result = room.stage_support(response, fs, limits=None) print(result.early.round(4)) # [-13.9794] = 20 lg 0.2 print(result.late.round(4)) # [-20.] = 20 lg 0.1 result.plot() # both supports against the Table C.1 ranges ``` Table C.1 gives the typical ranges as −24 dB to −8 dB for the early support and −24 dB to −10 dB for the late one, and prints "Not known" for both just-noticeable differences, so this module has none. C.2.4 asks for the 250 Hz to 2 kHz octave bands, which is what `limits` defaults to, and puts the standard deviation of one band in one position at 1 dB and of the frequency- and position-averaged single number at 0,3 dB. That pair closes: 1 dB over the root of four bands in three positions is 0,2887 dB, which rounds to the printed figure, and it holds for exactly those twelve readings. ## 2. Two integration limits that are in the equations and not in the prose The left-hand panel above shows two gaps, and neither is described by the sentence beside the equation that makes it. C.2.1 calls the early support "the reflected energy within the first 0,1 s", but Equation (C.1) starts at 20 ms, not at the 10 ms where the direct window ends. The interval between them counts for nothing, and taking the prose at its word instead adds 17 % more energy in a 2 s hall, worth 0,68 dB against the 1 dB standard deviation C.2.4 estimates for a single reading. C.2.2 calls the late support "the reflected energy after the first 0,1 s" with no upper bound at all, but Equation (C.2) stops at one second. That one costs 0,01 dB at $T = 2$ s, where the decay is already 30 dB down by then, and grows to 1,0 dB at $T = 8$ s: it separates cathedrals, not concert halls. The equations govern, and [the errata register](https://jmrplens.github.io/phonometry/reference/errata/) records both. ```python import numpy as np from phonometry import room fs = 48000 response = np.zeros(round(2.0 * fs)) response[100] = 1.0 response[100 + round(0.050 * fs)] = 0.2 response[100 + round(0.400 * fs)] = 0.1 # An arrival in the 10 ms to 20 ms gap, and one past the one-second ceiling. neither = response.copy() neither[100 + round(0.015 * fs)] = 5.0 neither[100 + round(1.500 * fs)] = 5.0 plain = room.stage_support(response, fs, limits=None) loud = room.stage_support(neither, fs, limits=None) print(np.allclose(plain.early, loud.early)) # True print(np.allclose(plain.late, loud.late)) # True ``` ## 3. What the excitation costs A reverberation time measured with interrupted noise is a measurement of a random process, and Clause 7.1 says how much of the answer that randomness owns: $$ \sigma(T_{20}) = 0{,}88\, T_{20} \sqrt{\frac{1 + 1{,}90/n}{N B T_{20}}}, \qquad \sigma(T_{30}) = 0{,}55\, T_{30} \sqrt{\frac{1 + 1{,}52/n}{N B T_{30}}}, $$ with $B$ the filter bandwidth, $n$ the decays measured in each position and $N$ the independent positions, meaning source and receiver combinations rather than receivers alone. ```python from phonometry import room # The 1 kHz octave: the clause puts B at 0,71 times the mid-band frequency. print(room.filter_bandwidth(1000.0)) # 710.0 Hz print(room.filter_bandwidth(1000.0, 3)) # 230.0 Hz sigma = room.reverberation_time_standard_deviation(2.0, 710.0, positions=12) print(round(float(sigma), 6)) # 0.009044 s print(round(100.0 * float(sigma) / 2.0, 3)) # 0.452 % ``` Two things fall out of the printed form that are worth knowing before planning a survey. **It grows as the square root of the decay time.** The $T$ in the prefactor and the $T$ under the radical leave half a power between them, so a long reverberation time carries a larger absolute uncertainty and a smaller relative one. Four times the decay time is twice the standard deviation, not four times it. **The integrated impulse response is not free of it.** Clause 7.2 says the method behaves like an average of ten interrupted-noise decays per position, not like the infinity the theory gives, and that is what `decays` defaults to. The difference is 7 % on $\sigma(T_{30})$ and 9 % on $\sigma(T_{20})$, which is not a rounding. ```python from phonometry import room ten = room.reverberation_time_standard_deviation(2.0, 710.0, decays=10) many = room.reverberation_time_standard_deviation(2.0, 710.0, decays=10**9) print(round(float(ten) / float(many), 4)) # 1.0733 ``` The two coefficient pairs are not arbitrary. ISO 3382-2:2008 prints the same result in general form, $\sigma(T)/T = G \sqrt{(1 + H/n)/(N B T)}$, with a table of $G$ and $H$ against the evaluation range and against $\gamma = T/T_\mathrm{det}$. The 88 % with $H = 1{,}90$ and the 55 % with $H = 1{,}52$ are its 20 dB and 30 dB rows in the $\gamma = 5$ column, an assumption ISO 3382-1 does not print. ## 4. Where a short decay stops being measurable Clause 7.3 puts two lower limits on a decay time measured by traditional forward analysis, and both are normative: $$ B\,T > 16, \qquad T > 2\, T_\mathrm{det}. $$ The first is the filter's own: a band of width $B$ cannot resolve a decay faster than its impulse response. The second is the averaging detector's, and it drops out when there is no detector, which is the case for the backward integration of 5.3.3. ```python from phonometry import room print(round(float(room.minimum_reliable_reverberation_time(88.75)), 4)) # 0.1803 s print(round(float(room.minimum_reliable_reverberation_time(2840.0)), 4)) # 0.0056 s print(room.minimum_reliable_reverberation_time(710.0, 0.05)) # 0.1 s ``` The 125 Hz octave binds hardest, because it is the narrowest: 16 over 88,75 Hz is 180 ms, and that is a bound the decay has to clear rather than one it may equal, so a room at or below 180 ms cannot have its 125 Hz reverberation time measured by forward analysis at all. ISO 3382-2:2008, 7.3 NOTE relaxes the product to `B T > 4` when the filtering is time-reversed, which is what [`room_parameters`](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/) does with `zero_phase=True`. ## See also - [Room Acoustic Parameters](https://jmrplens.github.io/phonometry/buildings/rooms/room-acoustics/): the decay times this page puts an uncertainty on. - [Sound strength G (ISO 3382-1)](https://jmrplens.github.io/phonometry/buildings/rooms/sound-strength/): how loud the hall is, against the same source at 10 m in a free field. - [Spatial impression (ISO 3382-1)](https://jmrplens.github.io/phonometry/buildings/rooms/spatial-impression/): where the sound comes from, measured with a second microphone. - [Measuring the Room Impulse Response](https://jmrplens.github.io/phonometry/buildings/rooms/room-impulse-response/): the ISO 18233 acquisition of the responses this page measures. - API reference: [`room.auditorium`](https://jmrplens.github.io/phonometry/reference/api/rooms/auditorium/) and [`room.acoustics`](https://jmrplens.github.io/phonometry/reference/api/rooms/acoustics/). ## Standards ISO 3382-1:2009, Annex C (informative): the early and late stage support of Equations (C.1) and (C.2), their Table C.1 ranges and the two standard deviations of C.2.4. And its normative Clause 7: the standard deviation of a measured reverberation time from Equations (4) and (5) with the printed filter bandwidths, and the lower limits of Equations (6) and (7). The two coefficient pairs are the 20 dB and 30 dB rows of the general form ISO 3382-2:2008 prints as its Equation (A.4). Validated against closed-form window geometries and the printed coefficients in the [conformance report](https://jmrplens.github.io/phonometry/reference/conformance/); the two integration limits the prose of Annex C leaves out are in the [errata register](https://jmrplens.github.io/phonometry/reference/errata/).