查看“︁梅尔刻度”︁的源代码

{{cleanup-jargon|time=2013-04-26T10:06:09+00:00}}

[[Image:Mel-Hz_plot.svg|right|thumb|300px|梅尔与赫兹的对应图]]
[[Image:A440.png|thumb|A440 {{audio|A440.mid|Play}}. 440 Hz = 549.64 mels]]

'''梅尔刻度'''（又稱'''Mel尺度'''，{{lang-en|Mel scale}}）是一種基于[[頻率 (物理學)|頻率]]定义的非線性刻度单位，表示人耳对[[音高]](pitch)等距變化的感官，由{{Link-en|Stanley Smith Stevens|Stanley Smith Stevens|Stevens}}、{{Link-en|John Volkman|John Volkman|Volkman}} 和Newman于1937年命名。<ref name=stevens1937>
{{cite journal
 |journal     = Journal of the Acoustical Society of America
 |title       = A scale for the measurement of the psychological magnitude pitch
 |author      = Stevens, Stanley Smith; Volkman; John; & Newman, Edwin B.
 |volume      = 8
 |issue       = 3
 |publisher   = 
 |pages       = 185–190
 |issn        = 
 |year        = 1937
 |url         = http://asadl.org/jasa/resource/1/jasman/v8/i3/p185_s1
 |deadurl     = yes
 |archiveurl  = https://archive.today/20130414065947/http://asadl.org/jasa/resource/1/jasman/v8/i3/p185_s1
 |archivedate = 2013-04-14
}}</ref> 

梅爾刻度與線性的頻率刻度赫茲(Hz)之間可以進行近似的數學換算。一个常用的将<math>f</math>赫兹转换为<math>m</math>梅尔的公式是：<ref>{{cite book
 | title = Speech communication: human and machine
 | author = Douglas O'Shaughnessy
 | publisher = Addison-Wesley
 | year = 1987
 | isbn = 978-0-201-16520-3
 | page = 150
 | url = http://books.google.com/books?ei=rhFfSpa-BJOCNsTT4IUG&id=mHFQAAAAMAAJ&dq=Speech+Communications:+Human+and+Machine&q=2595#search_anchor
 | access-date = 2013-04-26
 | archive-date = 2015-03-19
 | archive-url = https://web.archive.org/web/20150319154042/http://books.google.com/books?ei=rhFfSpa-BJOCNsTT4IUG&id=mHFQAAAAMAAJ&dq=Speech+Communications:+Human+and+Machine&q=2595#search_anchor
 | dead-url = no
 }}</ref>

:<math>m = 2595 \log_{10}\left(1 + \frac{f}{700}\right)</math>

梅尔刻度將1000Hz，且高于人耳[[听阈]]值40[[分贝]]的聲音信號，定為1000mel的参考点。在頻率500Hz以上时，随着频率的增加，人耳每感覺到等量的音高變化，所需要的頻率變化愈來愈大。这导致在赫茲刻度500Hz往上的四个[[八度]](一個八度即為兩倍的頻率)，只对应梅尔刻度上的两个[[八度]]。'''Mel'''的名字来源于单词melody，表示这个刻度是基於音高比较而被創造的。

==历史和其他公式==

历史上，存在过各种各样的转换公式。<ref>
{{cite book
 | title =  Foundations of Modern Auditory Theory
 | editor = Jerry V. Tobias
 | publisher = Academic Press
 | volume = 1
 | author = W. Dixon Ward
 | chapter = Musical Perception
 | year = 1970
 | page = 412
 | quote = no one claims yet to have determined 'the' mel scale.}}</ref>  在O'Shaugnessy的书中的常用公式选用不同的对数底可以有不同的表达式：

:<math>m = 2595 \log_{10}\left(1 + \frac{f}{700}\right) = 1127 \log_e\left(1 + \frac{f}{700}\right) \ </math>

对应的逆变换公式是：

:<math>f = 700(10^{m/2595} - 1) = 700(e^{m/1127} - 1) \ </math>

自从Steinberg于1937年出版的基于[[最小可覺差]]音高的刻度曲线和表格<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Positions of stimulation in the cochlea by pure tones
 | author = John C. Steinberg
 | volume = 8
 | issue = 3
 | publisher = 
 | pages = 176–180
 | issn = 
 | year = 1937
 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000008000003000176000001
 }}</ref> 
后，还有许多其他曲线通过不同的实验方法和分析途径被提出，如Fletcher和Munson在1937年<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Relation Between Loudness and Masking
 | url = https://archive.org/details/sim_journal-of-the-acoustical-society-of-america_1937-07_9_1/page/1
 | author = Harvey Fletcher and W. A. Munson
 | volume = 9
 | pages = 1–10
 | year = 1937
 }}</ref>
，Fletcher在1938年<ref>
{{cite journal
 | journal = Journal of the Acoustical Society of America
 | title = Loudness, Masking and Their Relation to the Hearing Process and the Problem of Noise Measurement
 | author = Harvey Fletcher
 | volume = 9
 | pages = 275–293
 | year = 1938
 | issue = 4
 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000009000004000275000001
 }}</ref>
，Steven于1937年<ref name=stevens1937/> 以及 Stevens 和 Volkmann于1940年<ref>
{{cite journal
 | journal = American Journal of Psychology
 | title = The Relation of Pitch to Frequency: A Revised Scale
 | author = Stevens, S., and Volkmann, J.
 | volume = 53
 | issue = 3
 | pages = 329–353
 | year = 1940
 | url = http://www.jstor.org/pss/1417526
 }}</ref>
分别给出的曲线。

在1949年，Koenig发表了一个基于独立的线性部分和对数部分的近似值，取1000Hz作为两个部分的分界点。<ref>
{{cite journal
 | journal = Bell Telephone Laboratory Record
 | title = A new frequency scale for acoustic measurements
 | author = W. Koenig
 | volume = 27
 | pages = 299–301
 | year = 1949
 }}</ref>

Gunnar Fant于1949年发表了当前流行的线性\对数公式，但是有1000Hz的[[截止频率]](corner frequency)。<ref>
Gunnar Fant (1949) "Analys av de svenska konsonantljuden : talets allmänna svängningsstruktur", 
LM Ericsson protokoll H/P 1064
</ref>
Fant于1968年发表了该公式的另一种与对数的[[底数(指数)|底数]]的选择无关的形式：<ref>Fant, Gunnar. (1968). Analysis and synthesis of speech processes. In B. Malmberg (Ed.), ''Manual of phonetics'' (pp. 173-177). Amsterdam: North-Holland.</ref><ref>{{cite book
 | title = Techniques in speech acoustics
 | author = Jonathan Harrington and Steve Cassidy
 | publisher = Springer
 | year = 1999
 | isbn = 978-0-7923-5731-5
 | page = 18
 | url = http://books.google.com/books?id=E1SyZZN8WQkC&pg=PA18
 | access-date = 2013-04-26
 | archive-date = 2015-03-19
 | archive-url = https://web.archive.org/web/20150319142314/http://books.google.com/books?id=E1SyZZN8WQkC&pg=PA18
 | dead-url = no
 }}</ref>

:<math>m = \frac{1000}{\log(2)} \log \left(1 + \frac{f}{1000}\right) \ </math>

1976年，Makhoul与Cosell发表了现在流行的版本，截止频率取为700Hz。<ref>{{citation
 | work = ICASSP 1976
 | title = LPCW: An LPC vocoder with linear predictive spectral warping
 | author = John Makhoul and Lynn Cosell
 | volume = 1
 | publisher = IEEE
 | pages = 466–469
 | year = 1976
 | url = http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1170013
 | accessdate = 2013-04-26
 | archive-date = 2013-07-31
 | archive-url = https://web.archive.org/web/20130731004330/http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1170013
 | dead-url = no
 }}</ref>
Ganchev等人指出："相比于Fant等人的1000Hz的公式，700Hz的公式能够在1000Hz以下更近似于Mel刻度，代价是超过1000Hz时误差更大。"<ref>{{citation
 | work = Proceedings of the SPECOM-2005
 | title = Comparative evaluation of various MFCC implementations on the speaker verification task,
 | author = T. Ganchev, N. Fakotakis, and G. Kokkinakis
 | pages = 191–194
 | year = 2005
 | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.8303
 | accessdate = 2013-04-26
 | archive-date = 2012-10-15
 | archive-url = https://web.archive.org/web/20121015210511/http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.8303
 | dead-url = no
 }}</ref> 但是当频率超过7kHz时,700Hz的版本表现的更好。

这些公式的数据由Beranek于1949年基于Stevens 和 Volkman的曲线被制作成表格：<ref>Beranek, Leo L. (1949). ''Acoustic measurements''. New York: McGraw-Hill.</ref> 

{{Table
 |type = class = "wikitable"
 |title = Beranek 等制表(1949),数据来源于Stevens 和 Volkman(1940)
 | row1 = '''Hz'''  {{!!}} 20 {{!!}} 160 {{!!}} 394 {{!!}} 670 {{!!}} 1000 {{!!}} 1420 {{!!}} 1900 {{!!}} 2450 {{!!}} 3120 {{!!}} 4000 {{!!}} 5100 {{!!}} 6600 {{!!}} 9000 {{!!}} 14000
 | row2 = '''mel'''  {{!!}} 0 {{!!}} 250 {{!!}} 500 {{!!}} 750 {{!!}} 1000 {{!!}} 1250 {{!!}} 1500 {{!!}} 1750 {{!!}} 2000 {{!!}} 2250 {{!!}} 2500 {{!!}} 2750 {{!!}} 3000 {{!!}} 3250
 }}

具有625Hz截断频率的公式由Lindsay和Norman于1977年在《Human information processing: An introduction to psychology》中提出，<ref>Lindsay, Peter H.; & Norman, Donald A. (1977). ''Human information processing: An introduction to psychology'' (2nd ed.). New York: Academic Press.</ref> 但在该书1972年第一版中该公式没有出现：

:<math>m = 2410 \log_{10}(1.6\times10^{-3} f + 1)</math>

大多数的公式能够保证1000 mel对应1000Hz。截断频率(break frequency)，如700Hz、1000Hz或625Hz，是这些公式中唯一的自由参数。一些非MEL听觉频率尺度(auditory-frequency-scale)公式使用了相同的形式，但截断频率低得多，不一定能保障1000mel对应1000Hz，例如1990年Glasberg与Moore提出的[[Equivalent rectangular bandwidth|ERB-rate]]刻度使用的是228.8Hz<ref>B.C.J. Moore and B.R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns" Journal of the Acoustical Society of America 74: 750-753, 1983.</ref> ，1990年Greenwood的“cochlear frequency–place map”则使用165.3Hz作为截断频率。<ref>Greenwood, D. D. (1990). A cochlear frequency–position function for several species—29 years later. ''The Journal of the Acoustical Society of America'', 87, 2592–2605.</ref>

Umesh等人对其他形式的梅尔刻度进行了研究。根据从这些曲线上计算的数据，他们指出，传统的含有对数区域和线性区域的公式，以及其他形式的公式，都不符合Stevens和Volkman的曲线：<ref>
{{citation
 | journal = Proc. ICASSP 1999
 | title = Fitting the mel scale
 | author = Umesh, S. and Cohen, L. and Nelson, D.
 | publisher = IEEE
 | pages = 217–220
 | isbn = 0-7803-5041-3
 | year = 1999
 | url = 
 }}</ref>

{{Table
 |type = class = "wikitable"
 |title = Umesh 等制表(1999),数据来源于Stevens 和 Volkman(1940)
 | row1 = '''Hz'''  {{!!}} 40 {{!!}} 161 {{!!}} 200 {{!!}} 404 {{!!}} 693 {{!!}} 867 {{!!}} 1000 {{!!}} 2022 {{!!}} 3000 {{!!}} 3393 {{!!}} 4109 {{!!}} 5526 {{!!}} 6500 {{!!}} 7743 {{!!}} 12000
 | row2 = '''mel'''  {{!!}} 43 {{!!}} 257 {{!!}} 300 {{!!}} 514 {{!!}} 771 {{!!}} 928 {{!!}} 1000 {{!!}} 1542 {{!!}} 2000 {{!!}} 2142 {{!!}} 2314 {{!!}} 2600 {{!!}} 2771 {{!!}} 2914 {{!!}} 3228
 }}

==参考文献==

{{reflist}} 

==外部链接==

*[http://users.utu.fi/jyrtuoma/speech/Mel2Hz.html Hz–mel, mel–Hz conversion] {{Wayback|url=http://users.utu.fi/jyrtuoma/speech/Mel2Hz.html |date=20121125042523 }} (uses the O'Shaughnessy equation)
*[http://scitation.aip.org/dbt/dbt.jsp?KEY=JASMAN&Volume=8&Issue=3 J. Acoust. Soc. Am. table of contents] for Stevens et al. paper
*[http://www.sfu.ca/sonic-studio/handbook/Mel.html Handbook for Acoustic Ecology] {{Wayback|url=http://www.sfu.ca/sonic-studio/handbook/Mel.html |date=20130312043725 }}

==参见==
*[[巴克刻度]]
*[[梅尔频率倒谱系数]]
*{{Link-en|Fletcher–Munson curves|Fletcher–Munson curves}}

{{Acoustics}}

[[Category:刻度]]
[[Category:心理声学]]