Subject: Re: Frequency to Mel Formula From: "Richard M. Warren" <rmwarren@xxxxxxxx> Date: Tue, 28 Jul 2009 15:45:48 -0500 List-Archive:<http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>--Apple-Mail-31--76189881 Content-Type: text/plain; charset=WINDOWS-1252; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable On 23 July, Dick Lyon began his email to the list with the statement =93 = =20 I would like to understand the history of the Mel scale, formulas for =20= it, and its relation to other scales;=94. To the best of my knowledge there have been a total of three Mel scale =20= studies and only two of these were based on half-pitch judgments, =20 unlike the Sone scale that has involved many studies (almost all using =20= half-loudness judgments). Stevens, Volkmann, and Newman (1937) reported the first Mel scale =20 study using half-pitch judgments for standards from 125 to 12,000 Hz. =20= In the next study, Stevens and Volkmann (1940) bisected intervals =20 between higher and lower frequencies over three ranges from 40 to =20 1,000, 200 to 6,500, and 300 to 12,000 Hz. The third study by Siegel =20= (1965) obtained half-pitch judgments for standard frequencies from 92 =20= to 9,200 Hz. The two half-pitch studies are in quite good agreement, while the =20 bisection procedure differed from the other two. This discussion will =20= concentrate on the data obtained in the half-pitch studies. In Wever=92s book, Theory of Hearing (1949, pages 338-339), he stated =20= that the data of Stevens et al. indicated that for low tones, which =20 Wever considered to be frequencies up to about 5 kHz, =93two intervals =20= of pitch that seem equal will embrace the same numbers of cycles.=94 In = =20 other words, within this range according to Wever, half pitch was =20 equal to half frequency. While this is only approximate, Siegel=92s =20 data are closer to this octave relation. Since his study is not =20 generally known, the data he reported are given as follows, with the =20 standard frequencies employed followed by the percent judged half in =20 parentheses: 92 Hz (60%), 165 (54%), 510 (50%), 920 (46%), 1650 =20 (48%), 5100 (47%), 9200 (43%). [If anyone wants a copy of a table =20 showing the half-pitch frequencies along with a figure published by =20 Siegel that compares his and Stevens, Volkmann, and Newman=92s data, I =20= will be pleased to send them a PDF file]. It can be seen that when both standard and comparison frequencies lie =20= within the familiar range of orchestral instruments, half pitch =20 appears to be approximately half frequency. A basis for this relation =20= may be found in Helmholtz=92s statement that =93The musical scale is at = it =20 were the divided rod, by which we measure progressions in pitch, as =20 rhythm measures progression in time.=94 (1954, page 252). He pointed =20= out that the octave is the basic unit of musical scales and that this =20= interval is employed in the music of virtually all cultures. He also =20= stated that the octave is the only interval that is completely =20 consonant: The harmonics of a higher octave are also harmonics of a =20 lower one, so that they blend perfectly with no discordant beats. A =20 discussion concerning the use of the octave as a possible basis for =20 half-pitch judgments is discussed more fully in a chapter by Warren =20 (2008). Incidentally, on 23 July, Don Greenwood=92s email (using the Re: =20 Auditory Digest thread) stated that Mel studies by Stevens and his =20 colleagues appeared to be biased (as he indicated, he had worked on =20 the Mel scale in Stevens=92s lab). He went on to say that =93If anyone =20= wants a Mel scale they should do it over, controlling carefully for =20 order bias and using plenty of subjects - more than in the past - and =20= using both musicians and non-musicians=85=94. If anyone does intend to do this, I would suggest the single-judgment =20= procedure which avoids experimental biases (and certainly uses plenty =20= of subjects) as used for half-loudness judgments and described by =20 Warren (2008) in the chapter dealing with both the Sone and Mel scales. As to the use of musicians as participants, it might be relevant that =20= in their half-pitch experiment, Stevens et al. reported that one of =20 their five participants was a trained musician: he =93reported an =20 inability to disregard octaves and other musical intervals when =20 setting the second tone at half the pitch of the first.=94 There is =20 more direct evidence involving other sensory scales indicating that, =20 participants having special familiarity with a physical scale =20 associated with the stimulus, use that scale in judging half-sensory =20 magnitude. Thus, it has been reported from three different labs that =20= listeners having experience using sound-level meters could not avoid =20 using half decibels as half loudness, and they could not be used as =20 participants (Ham & Parkinson, 1932; Laird, Taylor & Wille, 1932; and =20= Rowley & Studebaker, 1969). For example, 30 dB would seem to be half =20= as loud as 60 dB even when trying consciously to ignore the physical =20 scale. A similar observation was reported by Warren and Warren (1958) =20= in their study of half- brightness judgments of visual fields. =20 Undergraduate participants chose one-quarter luminance as half =20 brightness. However, an individual working in visual physiology, who =20= had considerable experience using an illuminometer, chose one-half =20 luminance even though he tried to judge subjective rather than =20 physical magnitude. References: Ham, L.B., and Parkinson, J.S. 1932. Loudness and intensity =20 relations,=94 Journal of the Acoustical Society of America, 3, 511-534. Helmholtz, H.L.F. 1954. On the Sensations of Tone as a Physiological =20= Basis for the Theory of Music. New York: Dover, 1954. (Reprint of =20 2nd English edition of 1885, A.J. Ellis, translator, based upon the =20 3rd German edition (1870) and rendered conformal with the 4th German =20 edition (1877)). Laird, D.A., Taylor, E., Wille, H.H., Jr. 1932. =93The apparent =20 reduction of loudness,=94 Journal of the Acoustical Society of America, =20= 3, 393-401. Rowley, R.R., and Studebaker, G.A. 1969. =93Monaural loudness-=20 intensity relationships for a 1,000-Hz tone,=94 Journal of the =20 Acoustical Society of America, 45, 1186-1192. Siegel, R.J. 1965. =93A Replication of the Mel Scale of Pitch,=94 =20 American Journal of Psychology, 78 (4), 615-620. Stevens, S.S., and Volkmann, J. 1940. =93The Relation of Pitch to =20 Frequency: A Revised Scale,=94 American Journal of Psychology, 53 (3), =20= 329-353. Stevens, S.S., Volkmann, J., and Newman, E.B. 1937. =93A scale for the = =20 measurement of the psychological magnitude of pitch,=94 Journal of the =20= Acoustical Society of America, 8, 185-190. Warren, R.M. 2008. Auditory Perception: An Analysis and Synthesis. =20= Cambridge, UK: Cambridge University Press (see Chapter 4 =93Judging =20 Auditory Magnitude: The Sone Scale of Loudness and the Mel Scale of =20 Pitch,=94 pp. 107-125). Warren, R.M., and Warren, R.P. 1958. =93Basis for judgments of =20 relative brightness,=94 Journal of the Optical Society of America, 48, =20= 445-450. Wever, E.G. 1949. Theory of Hearing. New York: Wiley. Dick Warren Richard M. Warren Research Professor and Distinguished Professor Emeritus Department of Psychology University of Wisconsin-Milwaukee PO Box 413 Milwaukee, WI 53201= --Apple-Mail-31--76189881 Content-Type: text/html; charset=WINDOWS-1252 Content-Transfer-Encoding: quoted-printable <html><body style=3D"word-wrap: break-word; -webkit-nbsp-mode: space; = -webkit-line-break: after-white-space; "><div><div><div = style=3D"margin-top: 0px; margin-right: 0px; margin-bottom: 0px; = margin-left: 0px; min-height: 14px; "><span class=3D"Apple-style-span" = style=3D"-webkit-text-stroke-width: -1; ">On 23 July, Dick Lyon began = his email to the list with the statement =93 I would like to understand = the history of the Mel scale, formulas for it, and its relation to other = scales;=94.<span style=3D"mso-spacerun: = yes"> </span></span></div></div><div style=3D"word-wrap: = break-word; -webkit-nbsp-mode: space; -webkit-line-break: = after-white-space; "><div><div style=3D"word-wrap: break-word; = -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; = "><br><div apple-content-edited=3D"true"> <span class=3D"Apple-style-span"= style=3D"border-collapse: separate; color: rgb(0, 0, 0); font-family: = Helvetica; font-size: 16px; font-style: normal; font-variant: normal; = font-weight: normal; letter-spacing: normal; line-height: normal; = orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; = white-space: normal; widows: 2; word-spacing: 0px; = -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: = 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: = auto; -webkit-text-stroke-width: 0; "><span class=3D"Apple-style-span" = style=3D"border-collapse: separate; color: rgb(0, 0, 0); font-family: = Helvetica; font-size: 13px; font-style: normal; font-variant: normal; = font-weight: normal; letter-spacing: normal; line-height: normal; = orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; = widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; = -webkit-border-vertical-spacing: 0px; = -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: = auto; -webkit-text-stroke-width: 0px; "><div style=3D"word-wrap: = break-word; -webkit-nbsp-mode: space; -webkit-line-break: = after-white-space; "><div><div><span class=3D"Apple-style-span" = style=3D"font-size: 14px; ">To the best of my knowledge there have been = a total of three Mel scale studies and only two of these were based on = half-pitch judgments, unlike the Sone scale that has involved many = studies (almost all using half-loudness = judgments).</span></div></div></div></span></span></div><br><div><div = style=3D"word-wrap: break-word; -webkit-nbsp-mode: space; = -webkit-line-break: after-white-space; "><div class=3D"MsoNormal">Stevens,= Volkmann, and Newman (1937) reported the first Mel scale study using = half-pitch judgments for standards from 125 to 12,000 Hz.<span = style=3D"mso-spacerun: yes"> </span>In the next study, Stevens and = Volkmann (1940) bisected intervals between higher and lower frequencies = over three ranges from 40 to 1,000, 200 to 6,500, and 300 to 12,000 = Hz.<span style=3D"mso-spacerun: yes"> </span>The third study by = Siegel (1965) obtained half-pitch judgments for standard frequencies = from 92 to 9,200 Hz.<span style=3D"mso-spacerun: = yes"> </span></div><div class=3D"MsoNormal"><o:p></o:p></div> <div = class=3D"MsoNormal"> <o:p></o:p></div> <div class=3D"MsoNormal">The = two half-pitch studies are in quite good agreement, while the bisection = procedure differed from the other two.<span style=3D"mso-spacerun: = yes"> </span>This discussion will concentrate on the data obtained = in the half-pitch studies.<o:p></o:p></div> <div = class=3D"MsoNormal"> <o:p></o:p></div> <div class=3D"MsoNormal">In = Wever=92s book, <i>Theory of Hearing</i><span style=3D"font-style:normal">= (1949, pages 338-339), he stated that the data of Stevens et al. = indicated that for low tones, which Wever considered to be frequencies = up to about 5 kHz, =93two intervals of pitch that seem equal will = embrace the same numbers of cycles.=94<span style=3D"mso-spacerun: = yes"> </span>In other words, within this range according to Wever, = half pitch was equal to half frequency.<span style=3D"mso-spacerun: = yes"> </span>While this is only approximate, Siegel=92s data are = closer to this octave relation.<span style=3D"mso-spacerun: yes"> = </span>Since his study is not generally known, the data he reported are = given as follows, with the standard frequencies employed followed by the = percent judged half in parentheses:<span style=3D"mso-spacerun: = yes"> </span></span><span style=3D"font-size:11.0pt">92 Hz (60%), = 165 (54%), 510 (50%), 920 (46%), 1650 (48%), 5100 (47%), 9200 = (43%).<span style=3D"mso-spacerun: yes"> </span></span>[If anyone = wants a copy of a table showing the half-pitch frequencies along with a = figure published by Siegel that compares his and Stevens, Volkmann, and = Newman=92s data, I will be pleased to send them a PDF = file].<o:p></o:p></div><p class=3D"MsoNormal" = style=3D"margin-right:.5in;tab-stops:2.0in 2.5in 3.0in 3.5in 4.0in 4.5in = 5.0in">It can be seen that when both standard and comparison frequencies = lie within the familiar range of orchestral instruments, half pitch = appears to be approximately half frequency.<span style=3D"mso-spacerun: = yes"> </span>A basis for this relation may be found in Helmholtz=92s= statement that =93The musical scale is at it were the divided rod, by = which we measure progressions in pitch, as rhythm measures progression = in time.=94 (1954, page 252).<span style=3D"mso-spacerun: yes"> = </span>He pointed out that the octave is the basic unit of musical = scales and that this interval is employed in the music of virtually all = cultures.<span style=3D"mso-spacerun: yes"> </span>He also stated = that the octave is the only interval that is completely consonant:<span = style=3D"mso-spacerun: yes"> </span>The harmonics of a higher = octave are also harmonics of a lower one, so that they blend perfectly = with no discordant beats.<span style=3D"mso-spacerun: yes"> = </span>A discussion concerning the use of the octave as a possible basis = for half-pitch judgments is discussed more fully in a chapter by Warren = (2008).<span style=3D"mso-spacerun: yes"> </span></p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal"><o:p></o:p></p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal">Incidentally, on 23 July, = Don Greenwood=92s email (using the Re: Auditory Digest thread) stated = that Mel studies by Stevens and his colleagues appeared to be biased (as = he indicated, he had worked on the Mel scale in Stevens=92s lab).<span = style=3D"mso-spacerun: yes"> </span>He went on to say that =93If = anyone wants a Mel scale they should do it over, controlling carefully = for order bias and using plenty of subjects - more than in the past - = and using both musicians and non-musicians=85=94.</p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal"><o:p></o:p></p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal">If anyone does intend to do = this, I would suggest the single-judgment procedure which avoids = experimental biases (and certainly uses plenty of subjects) as used for = half-loudness judgments and described by Warren (2008) in the chapter = dealing with both the Sone and Mel scales.<span style=3D"mso-spacerun: = yes"> </span></p><p class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal"><o:p></o:p></p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal">As to the use of musicians = as participants, it might be relevant that in their half-pitch = experiment, Stevens et al. reported that one of their five participants = was a trained musician: he =93reported an inability to disregard octaves = and other musical intervals when setting the second tone at half the = pitch of the first.=94<span style=3D"mso-spacerun: yes"> = </span>There is more direct evidence involving other sensory scales = indicating that, participants having special familiarity with a physical = scale associated with the stimulus, use that scale in judging = half-sensory magnitude.<span style=3D"mso-spacerun: yes"> = </span>Thus, it has been reported from three different labs that = listeners having experience using sound-level meters could not avoid = using half decibels as half loudness, and they could not be used as = participants (Ham & Parkinson, 1932; Laird, Taylor & Wille, = 1932; and Rowley & Studebaker, 1969).<span style=3D"mso-spacerun: = yes"> </span>For example, 30 dB would seem to be half as loud as = 60 dB even when trying consciously to ignore the physical scale.<span = style=3D"mso-spacerun: yes"> </span>A similar observation was = reported by Warren and Warren (1958) in their study of half- brightness = judgments of visual fields.<span style=3D"mso-spacerun: yes"> = </span>Undergraduate participants chose one-quarter luminance as half = brightness.<span style=3D"mso-spacerun: yes"> </span>However, an = individual working in visual physiology, who had considerable experience = using an illuminometer, chose one-half luminance even though he tried to = judge subjective rather than physical magnitude.</p><p = class=3D"MsoBodyTextIndent" = style=3D"text-indent:0in;line-height:normal"><o:p></o:p></p> <div = class=3D"MsoNormal">References:</div><div = class=3D"MsoNormal"><o:p></o:p></div> <div = class=3D"MsoNormal"> <o:p></o:p></div> <div class=3D"MsoNormal">Ham, = L.B., and Parkinson, J.S.<span style=3D"mso-spacerun: yes"> = </span>1932.<span style=3D"mso-spacerun: yes"> </span>Loudness and = intensity relations,=94 <i>Journal of the Acoustical Society of = America</i><span style=3D"font-style:normal">, 3, 511-534. = </span></div><div class=3D"MsoNormal"><br></div><div = class=3D"MsoNormal"><span style=3D"font-style:normal">Helmholtz, = H.L.F.<span style=3D"mso-spacerun: yes"> </span>1954.<span = style=3D"mso-spacerun: yes"> </span><i>On the Sensations of Tone = as a Physiological Basis for the Theory of Music</i><span = style=3D"font-style:normal">.<span style=3D"mso-spacerun: yes"> = </span>New York:<span style=3D"mso-spacerun: yes"> </span>Dover, = 1954.<span style=3D"mso-spacerun: yes"> </span>(Reprint of = 2<sup>nd</sup> English edition of 1885, A.J. Ellis, translator, based = upon the 3<sup>rd</sup> German edition (1870) and rendered conformal = with the 4<sup>th</sup> German edition (1877)). = </span></span></div><div class=3D"MsoNormal"><br></div><div = class=3D"MsoNormal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal">Laird, D.A., Taylor, E., Wille, H.H., = Jr.<span style=3D"mso-spacerun: yes"> </span>1932.<span = style=3D"mso-spacerun: yes"> </span>=93The apparent reduction of = loudness,=94 <i>Journal of the Acoustical Society of America</i><span = style=3D"font-style:normal">, 3, 393-401. = </span></span></span></div><div class=3D"MsoNormal"><br></div><div = class=3D"MsoNormal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal">Rowley, = R.R., and Studebaker, G.A.<span style=3D"mso-spacerun: yes"> = </span>1969.<span style=3D"mso-spacerun: yes"> </span>=93Monaural = loudness-intensity relationships for a 1,000-Hz tone,=94 <i>Journal of = the Acoustical Society of America</i><span style=3D"font-style:normal">, = 45, 1186-1192. <span style=3D"color: windowtext; = "></span></span></span></span></span></div><div = class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; ">Siegel, R.J.<span style=3D"mso-spacerun: = yes"> </span>1965.<span style=3D"mso-spacerun: yes"> = </span>=93A Replication of the Mel Scale of Pitch,=94 <i>American = Journal of Psychology</i></span><span style=3D"color: windowtext; ">, 78 = (4), 615-620. </span></span></span></span></span></div><div = class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; ">Stevens, S.S., and Volkmann, J. 1940.<span = style=3D"mso-spacerun: yes"> </span>=93The Relation of Pitch to = Frequency: A Revised Scale,=94 <i>American Journal of = Psychology</i><span style=3D"font-style: normal">, 53 (3), 329-353. = </span></span></span></span></span></span></div><div = class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; "><span style=3D"font-style: = normal">Stevens, S.S., Volkmann, J., and Newman, E.B.<span = style=3D"mso-spacerun: yes"> </span>1937.<span = style=3D"mso-spacerun: yes"> </span>=93A scale for the measurement = of the psychological magnitude of pitch,=94 <i>Journal of the Acoustical = Society of America</i><span style=3D"font-style:normal">, 8, 185-190. = </span></span></span></span></span></span></span></div><div = class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; "><span style=3D"font-style: normal"><span = style=3D"font-style:normal">Warren, R.M.<span style=3D"mso-spacerun: = yes"> </span>2008.<span style=3D"mso-spacerun: yes"> = </span><i>Auditory Perception:<span style=3D"mso-spacerun: yes"> = </span>An Analysis and Synthesis</i><span = style=3D"font-style:normal">.<span style=3D"mso-spacerun: yes"> = </span>Cambridge, UK: Cambridge University Press (see Chapter 4 =93Judging= Auditory Magnitude:<span style=3D"mso-spacerun: yes"> </span>The = Sone Scale of Loudness and the Mel Scale of Pitch,=94 pp. 107-125). = </span></span></span></span></span></span></span></span></div><div = class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; "><span style=3D"font-style: normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal">Warren, = R.M., and Warren, R.P.<span style=3D"mso-spacerun: yes"> = </span>1958.<span style=3D"mso-spacerun: yes"> </span>=93Basis for = judgments of relative brightness,=94 <i>Journal of the Optical Society = of America</i><span style=3D"font-style:normal">, 48, 445-450. = </span></span></span></span></span></span></span></span></span></div= ><div class=3D"MsoNormal"><br></div><div class=3D"MsoNormal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"color: windowtext; "><span style=3D"font-style: normal"><span = style=3D"font-style:normal"><span style=3D"font-style:normal"><span = style=3D"font-style:normal">Wever, E.G.<span style=3D"mso-spacerun: = yes"> </span>1949. <i>Theory of Hearing</i><span = style=3D"font-style:normal">.<span style=3D"mso-spacerun: yes"> = </span>New York:<span style=3D"mso-spacerun: yes"> = </span>Wiley.</span></span></span></span></span></span></span></span></spa= n></span></div><div class=3D"MsoNormal"><br></div><p class=3D"MsoNormal" = style=3D"tab-stops:80.0pt">Dick Warren<span class=3D"Apple-style-span" = style=3D"font-size: 13px; "><font class=3D"Apple-style-span" = size=3D"4"><span class=3D"Apple-style-span" style=3D"font-size: 14px; = "><b></b></span></font></span></p><p class=3D"MsoNormal" = style=3D"tab-stops:80.0pt"><span class=3D"Apple-style-span" = style=3D"font-size: 13px; "><div><div><font class=3D"Apple-style-span" = size=3D"4"><span class=3D"Apple-style-span" style=3D"font-size: 14px; = "><b>Richard M. Warren</b></span></font> </div><div>Research = Professor </div><div> and Distinguished Professor = Emeritus</div><div>Department of Psychology</div><div>University of = Wisconsin-Milwaukee</div><div>PO Box 413</div><div>Milwaukee, WI = 53201</div></div></span></p></div></div></div></div></div></div></bo= dy></html>= --Apple-Mail-31--76189881--