Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth

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L. Nizami, "Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth," Paper 10387, (2020 October.). doi:
L. Nizami, "Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth," Paper 10387, (2020 October.). doi:
Abstract: A major question in sensory science is how a sensation of magnitude F (such as loudness) depends upon a sensory stimulus of physical intensity I (such as a sound-pressure-wave of a particular root-mean-square sound-pressure-level). An empirical just-noticeable sensation difference (ΔF)_{j} at F_{j} specifies a just-noticeable intensity difference (ΔI)_{j} at I_{j}. Intensity differences accumulate from a stimulus-detection threshold I_{th} up to a desired intensity I. Likewise, the corresponding sensation differences are classically presumed to accumulate, accumulating up to F(I) from F(I_{th}), a non-zero sensation (as suggested by hearing studies) at Ι_{th}. Consequently, sensation growth F(I) can be obtained through Fechnerian integration. Therein, empirically-based relations for the Weber Fraction, ΔI/I, are individually combined with either Fechner’s Law ΔF = Β or Ekman’s Law (ΔF/F) = ɡ; the number of cumulated steps in I is equated to the number of cumulated steps in F, and an infinite series ensues, whose higher-order terms are ignored. Likewise classically ignored are the integration bounds I_{th} and F(I_{th}). Here, we deny orthodoxy by including those bounds, allowing hypothetical sensation-growth equations for which the differential-relations ΔF(I) = F(I+ ΔI) - F(I) or (ΔF(I)/F(I)) = (F(I+ΔI) - F(I))/F(I) do indeed return either Β or ɡ, for linear growth of sensation F with intensity I. Also, 24 sensation-growth equations F(I), which had already been derived by the author likewise using bounded Fechnerian integration (12 equations for the Weber Fraction (ΔI/I), each combined with either Fechner’s Law or with Ekman’s Law), are scrutinized for whether their differential-relations return either Β or ɡ respectively, particularly in the limits (ΔI/I) << 1 and the even-more-extreme limit (ΔI/I) → 0, both of which seem unexplored in the literature. Finally, some relevant claims made by Luce and Edwards (1958) are examined under bounded Fechnerian integration: namely, that three popular forms of the Weber Fraction, when combined with Fechner’s Law, produce sensation-growth equations that subsequently return the selfsame Fechner’s Law. Luce and Edwards (1958) prove to be wrong.

@article{nizami2020non-classical,,
author={nizami, lance},
journal={journal of the audio engineering society},
title={non-classical, bounded fechnerian integration for loudness: contrary to luce and edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth},
year={2020},
volume={},
number={},
pages={},
doi={},
month={october},}
@article{nizami2020non-classical,,
author={nizami, lance},
journal={journal of the audio engineering society},
title={non-classical, bounded fechnerian integration for loudness: contrary to luce and edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth},
year={2020},
volume={},
number={},
pages={},
doi={},
month={october},
abstract={a major question in sensory science is how a sensation of magnitude f (such as loudness) depends upon a sensory stimulus of physical intensity i (such as a sound-pressure-wave of a particular root-mean-square sound-pressure-level). an empirical just-noticeable sensation difference (δf)_{j} at f_{j} specifies a just-noticeable intensity difference (δi)_{j} at i_{j}. intensity differences accumulate from a stimulus-detection threshold i_{th} up to a desired intensity i. likewise, the corresponding sensation differences are classically presumed to accumulate, accumulating up to f(i) from f(i_{th}), a non-zero sensation (as suggested by hearing studies) at ι_{th}. consequently, sensation growth f(i) can be obtained through fechnerian integration. therein, empirically-based relations for the weber fraction, δi/i, are individually combined with either fechner’s law δf = β or ekman’s law (δf/f) = ɡ; the number of cumulated steps in i is equated to the number of cumulated steps in f, and an infinite series ensues, whose higher-order terms are ignored. likewise classically ignored are the integration bounds i_{th} and f(i_{th}). here, we deny orthodoxy by including those bounds, allowing hypothetical sensation-growth equations for which the differential-relations δf(i) = f(i+ δi) - f(i) or (δf(i)/f(i)) = (f(i+δi) - f(i))/f(i) do indeed return either β or ɡ, for linear growth of sensation f with intensity i. also, 24 sensation-growth equations f(i), which had already been derived by the author likewise using bounded fechnerian integration (12 equations for the weber fraction (δi/i), each combined with either fechner’s law or with ekman’s law), are scrutinized for whether their differential-relations return either β or ɡ respectively, particularly in the limits (δi/i) << 1 and the even-more-extreme limit (δi/i) → 0, both of which seem unexplored in the literature. finally, some relevant claims made by luce and edwards (1958) are examined under bounded fechnerian integration: namely, that three popular forms of the weber fraction, when combined with fechner’s law, produce sensation-growth equations that subsequently return the selfsame fechner’s law. luce and edwards (1958) prove to be wrong.},}

TY - paper
TI - Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth
SP -
EP -
AU - Nizami, Lance
PY - 2020
JO - Journal of the Audio Engineering Society
IS -
VO -
VL -
Y1 - October 2020
TY - paper
TI - Non-classical, bounded Fechnerian integration for loudness: contrary to Luce and Edwards, initial loudness-difference-size stipulations are only recouped for linear loudness growth
SP -
EP -
AU - Nizami, Lance
PY - 2020
JO - Journal of the Audio Engineering Society
IS -
VO -
VL -
Y1 - October 2020
AB - A major question in sensory science is how a sensation of magnitude F (such as loudness) depends upon a sensory stimulus of physical intensity I (such as a sound-pressure-wave of a particular root-mean-square sound-pressure-level). An empirical just-noticeable sensation difference (ΔF)_{j} at F_{j} specifies a just-noticeable intensity difference (ΔI)_{j} at I_{j}. Intensity differences accumulate from a stimulus-detection threshold I_{th} up to a desired intensity I. Likewise, the corresponding sensation differences are classically presumed to accumulate, accumulating up to F(I) from F(I_{th}), a non-zero sensation (as suggested by hearing studies) at Ι_{th}. Consequently, sensation growth F(I) can be obtained through Fechnerian integration. Therein, empirically-based relations for the Weber Fraction, ΔI/I, are individually combined with either Fechner’s Law ΔF = Β or Ekman’s Law (ΔF/F) = ɡ; the number of cumulated steps in I is equated to the number of cumulated steps in F, and an infinite series ensues, whose higher-order terms are ignored. Likewise classically ignored are the integration bounds I_{th} and F(I_{th}). Here, we deny orthodoxy by including those bounds, allowing hypothetical sensation-growth equations for which the differential-relations ΔF(I) = F(I+ ΔI) - F(I) or (ΔF(I)/F(I)) = (F(I+ΔI) - F(I))/F(I) do indeed return either Β or ɡ, for linear growth of sensation F with intensity I. Also, 24 sensation-growth equations F(I), which had already been derived by the author likewise using bounded Fechnerian integration (12 equations for the Weber Fraction (ΔI/I), each combined with either Fechner’s Law or with Ekman’s Law), are scrutinized for whether their differential-relations return either Β or ɡ respectively, particularly in the limits (ΔI/I) << 1 and the even-more-extreme limit (ΔI/I) → 0, both of which seem unexplored in the literature. Finally, some relevant claims made by Luce and Edwards (1958) are examined under bounded Fechnerian integration: namely, that three popular forms of the Weber Fraction, when combined with Fechner’s Law, produce sensation-growth equations that subsequently return the selfsame Fechner’s Law. Luce and Edwards (1958) prove to be wrong.

A major question in sensory science is how a sensation of magnitude F (such as loudness) depends upon a sensory stimulus of physical intensity I (such as a sound-pressure-wave of a particular root-mean-square sound-pressure-level). An empirical just-noticeable sensation difference (ΔF)_{j} at F_{j} specifies a just-noticeable intensity difference (ΔI)_{j} at I_{j}. Intensity differences accumulate from a stimulus-detection threshold I_{th} up to a desired intensity I. Likewise, the corresponding sensation differences are classically presumed to accumulate, accumulating up to F(I) from F(I_{th}), a non-zero sensation (as suggested by hearing studies) at Ι_{th}. Consequently, sensation growth F(I) can be obtained through Fechnerian integration. Therein, empirically-based relations for the Weber Fraction, ΔI/I, are individually combined with either Fechner’s Law ΔF = Β or Ekman’s Law (ΔF/F) = ɡ; the number of cumulated steps in I is equated to the number of cumulated steps in F, and an infinite series ensues, whose higher-order terms are ignored. Likewise classically ignored are the integration bounds I_{th} and F(I_{th}). Here, we deny orthodoxy by including those bounds, allowing hypothetical sensation-growth equations for which the differential-relations ΔF(I) = F(I+ ΔI) - F(I) or (ΔF(I)/F(I)) = (F(I+ΔI) - F(I))/F(I) do indeed return either Β or ɡ, for linear growth of sensation F with intensity I. Also, 24 sensation-growth equations F(I), which had already been derived by the author likewise using bounded Fechnerian integration (12 equations for the Weber Fraction (ΔI/I), each combined with either Fechner’s Law or with Ekman’s Law), are scrutinized for whether their differential-relations return either Β or ɡ respectively, particularly in the limits (ΔI/I) << 1 and the even-more-extreme limit (ΔI/I) → 0, both of which seem unexplored in the literature. Finally, some relevant claims made by Luce and Edwards (1958) are examined under bounded Fechnerian integration: namely, that three popular forms of the Weber Fraction, when combined with Fechner’s Law, produce sensation-growth equations that subsequently return the selfsame Fechner’s Law. Luce and Edwards (1958) prove to be wrong.

Author:
Nizami, Lance
Affiliation:
Independent Research Scholar, Palo Alto, CA, USA
AES Convention:
149 (October 2020)
Paper Number:
10387
Publication Date:
October 22, 2020Import into BibTeX
Subject:
Perception
Permalink:
http://www.aes.org/e-lib/browse.cfm?elib=20924