Total loudness is the integral of specific loudness over critical-band rate

Evidence

As usual, we call the value we require “specific loudness” with the symbol N’. Loudness, N, is then the integral of specific loudness over critical-band rate (see Fig. 8.16 lower right) or in mathematical expression: 1 24 Bark N= N’dz, o where the integral is taken over all critical-band rates. — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1250

Because we have seen that the two tones influence each other in the creation of the total loudness, even though they are spectrally separated, it may be useful to treat total loudness as an integral of a value that we have to find, but which can be drawn as a function of critical-band rate. If such an integral leads to the loudness that is given in sone, the value we require has to have the unit of sone/Bark. In this case, total loudness, given as the integral of this value over critical-band rate, will lead to the unit of sone. — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1249

Because critical bandwidth plays an important role in loudness, the loudness on the slopes of the frequency selectivity characteristic of our hearing system may also play an important role. The excitation level versus criticalband rate pattern represents the pattern that describes not only the influence of critical bandwidth, but also that of the slopes. Therefore, it seems reasonable to use the excitation level versus critical-band rate pattern as a basis from which the loudness of the complex may be constructed. Because we have seen that the two tones influence each other in the creation of the total loudness, even though they are spectrally separated, it may be useful to treat total loudness as an integral of a value that we have to find, but which can be drawn as a function of critical-band rate. If such an integral leads to the loudness that is given in sone, the value we require has to have the unit of sone/Bark. — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1248

As usual, we call the value we require “specific loudness” with the symbol N’. Loudness, N, is then the integral of specific loudness over critical-band rate (see Fig. — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1250

Categorization

• DEFINE: Names the central construct ‘specific loudness’ (N’) and defines total loudness as its integral over the Bark scale.
• corroborated: 5 independent passages support this claim — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1250; Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1249; Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1248
• XREF: connects to: Loudness rises once noise bandwidth exceeds the critical band at constant SPL.; Loudness stays constant below critical bandwidth then grows beyond it — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 1034; Loudness is the summed specific loudness across critical-band rates, not the raw spectral level (+9 more)

Broader Connections

• Loudness is the summed specific loudness across critical-band rates, not the raw spectral level #corroborates (0.95)
• Total loudness requires linear scales before integration is valid #qualifies (0.75) — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1283
• Loudness grows when sound energy spills beyond a single critical band #corroborates (0.72) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 667
• Loudness grows with bandwidth via area tradeoff except at extreme levels #qualifies (0.70) — Professor Dr.-Ing. Hugo Fastl, Professor Dr.-Ing. Eberhard, Psychoacoustics- Facts and Models, 1288
• Wider bandwidth increases perceived loudness at equal sound pressure #corroborates (0.70) — Jennifer Lentz, Psychoacoustics- Perception of Normal…, 868