The conversion of loudness -- big headache

In order to draw the dissonant curve of a spectrum, loudness is a considered factor in the equation. The measurement of loudness in the equation is in sone. This loudness scale is based on a large number of laboratory tests in which people were asked to adjust the intensity of one pure tone until they judged it to be 'half the loudness' of another tone of the same pitch and fixed intensity. This scale is subjective based. However, the human ears respond to loudness irregularly. We are tuned to listen to the vocal range and mid-high frequency. The equal-loudness contour displays the pattern of our hearing. With the same sound pressure level (SPL), with different frequencies, the loudness we perceive is different. On the other hand, we cannot distinguish the loudness difference in high volume (high SPL); but we can sense more loudness change in lower SPL. The measurement of SPL is in Pascal and is often presented in logarithmic scale (dB), while sone is a linear scale.

Sethares (2010) offers an approximated solution of translating the SPL in dB (dBSPL) to sone as (2^(SPL/10))/16. This is anchored at 1kHz in 40dB, also known as the anchor of dB(A) weighting for soft volume. It is a good middle ground of daily preceded sound, but this is disregarded other much higher or lower frequencies, and SPL, as well as other properties, affect how we perceive loudness.


Apart from frequency, the duration of a tone (envelope) also affects how we perceive loudness. We will feel the noise louder with longer exposure and vice versa. With the same tone, duration and loudness, if the tone is played in crescendo, it would sound louder than decrescendo, since the hearing has a certain delay of perceiving sound.

With the intangible parameters above, it is hard to come up with a scale that can handle psycho-acoustic and objectivity at the same time. The loudness scale sone is only a subjective measurement vary from an individual's age, cultural background. It is hard to translate dB SPL and sone with each other since they are in a different realm. 

Right now I have a recorded sample of a tubular bell played in A3. Here is the spectrum:
281.3Hz   -26.7 dB
538.5Hz      -25.6 dB
880Hz         -13.7 dB
1290Hz        -9 dB
1761Hz        -11.6 dB
2284Hz       -21 dB 

The spectrum lists out all the sine tone that stands out. It shows the difference of the amplitude in dB. If the loudest sine tone 1290Hz sounds in 60dBSPL, it can subtract other partials' amplitude.
281.3Hz42.3dB
538.5Hz43.4dB
880Hz55.3dB
1290Hz60dB
1761Hz57.4dB
2284Hz48dB

The measurement of Sethares' formula asks for sone for loudness, which considers our actual hearing experience, the equal-loudness contour. The BS ISO 226:2003 offers formulas and lookup tables to translate dBSPL to phon to sone, but again it is a rough method to translate in not a linear or logarithmic method. If the frequency is lower than 100Hz and amplitude lower than 40dBSPL an error may occur. The calculated result from ISO 226:

HzdBSone
281.342.30.3255653034
538.543.40.9406046763
88055.32.644761556
1290603.205657358
176157.42.460933922
2284481.912985628


Campbell, M. & Greated, C. (1998) The musician’s guide to acoustics. Oxford: Oxford University Press.


  Anon (2003) BS ISO 226:2003: Acoustics. Normal equal-loudness-level contours. British Standards Institute.


Sethares, W. A. (2010) Tuning, timbre, spectrum, scale. 2nd ed. London: Springer.

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