Mel scale
Encyclopedia
The mel scale, named by Stevens
, Volkman
and Newman
in 1937
is a perceptual scale of pitch
es judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency
measurement is defined by assigning a perceptual pitch of 1000 mels to a 1000 Hz
tone, 40 dB
above the listener's threshold. Above about 500 Hz, larger and larger interval
s are judged by listeners to produce equal pitch increments. As a result, four octave
s on the hertz scale above 500 Hz are judged to comprise about two octaves on the mel scale. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons.
A popular formula to convert
hertz into 
mel is:
The corresponding inverse expressions are:
There were published curves and tables on psychophysical pitch scales since Steinberg's 1937
curves based on just-noticeable differences of pitch. More curves soon followed in Fletcher and Munson's 1937
and Fletcher's 1938
and Steven's 1937 and Stevens and Volkmann's 1940
papers using a variety of experimental methods and analysis approaches.
In 1949 Koenig published an approximation based on separate linear and logarithmic segments, with a break at 1000 Hz.
Gunnar Fant proposed the current popular linear/log formula in 1949, but with the 1000 Hz corner frequency.
An alternate expression of the formula, not depending on choice of log base, is noted in Fant (1968):
In 1976, Makhoul and Cosell published the now-popular version with the 700 Hz corner frequency.
As Ganchev et al. have observed, "The formulae [with 700], when compared to [Fant's with 1000], provide a
closer approximation of the Mel scale for frequencies below 1000 Hz, at the price of higher inaccuracy for frequencies higher than 1000 Hz." Above 7 kHz, however, the situation is reversed, and the 700 Hz version again fits better.
Data by which some of these formulas are motivated are tabulated in Beranek (1949), as measured from the curves of Stevens and Volkman:
A formula with a break frequency of 625 Hz is given by Lindsay & Norman (1977); the formula doesn't appear in their 1972 first edition:
Most mel-scale formulas give exactly 1000 mels at 1000 Hz. The break frequency (e.g. 700 Hz, 1000 Hz, or 625 Hz) is the only free parameter in the usual form of the formula. Some non-mel auditory-frequency-scale formulas use the same form but with much lower break frequency, not necessarily mapping to 1000 at 1000 Hz; for example the ERB-rate
scale of Glasberg & Moore (1990) uses a break point of 228.8 Hz, and the cochlear frequency–place map of Greenwood (1990) uses 165.3 Hz.
Other functional forms for the mel scale have been explored by Umesh et al.; they point out that the traditional formulas with a logarithmic region and a linear region do not fit the data from Stevens and Volkman's curves as well as some other forms, based on the following data table of measurements that they made from those curves:
Stanley Smith Stevens
Stanley Smith Stevens was an American psychologist who founded Harvard's Psycho-Acoustic Laboratory and is credited with the introduction of Stevens' power law. Stevens authored a milestone textbook, the 1400+ page "Handbook of Experimental Psychology" . He was also one of the founding organizers...
, Volkman
John Volkman
John E. Volkman was an American industrial scientist, spent his career at RCA, designing studios and auditoria, and sound reinforcement components....
and Newman
Edwin Newman
Edwin Harold Newman was an American newscaster, journalist and author.-Early life and education:Newman was born on January 25, 1919 in New York City to Myron and Rose Newman. His older brother was M. W. Newman, a longtime reporter for the Chicago Daily News. Newman married Rigel Grell on August...
in 1937
is a perceptual scale of pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
es judged by listeners to be equal in distance from one another. The reference point between this scale and normal frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
measurement is defined by assigning a perceptual pitch of 1000 mels to a 1000 Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
tone, 40 dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
above the listener's threshold. Above about 500 Hz, larger and larger interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
s are judged by listeners to produce equal pitch increments. As a result, four octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
s on the hertz scale above 500 Hz are judged to comprise about two octaves on the mel scale. The name mel comes from the word melody to indicate that the scale is based on pitch comparisons.
A popular formula to convert
hertz into mel is:History and other formulas
There is no single mel-scale formula. The popular formula from O'Shaugnessy's book can be expressed with different log bases:The corresponding inverse expressions are:
There were published curves and tables on psychophysical pitch scales since Steinberg's 1937
curves based on just-noticeable differences of pitch. More curves soon followed in Fletcher and Munson's 1937
and Fletcher's 1938
and Steven's 1937 and Stevens and Volkmann's 1940
papers using a variety of experimental methods and analysis approaches.
In 1949 Koenig published an approximation based on separate linear and logarithmic segments, with a break at 1000 Hz.
Gunnar Fant proposed the current popular linear/log formula in 1949, but with the 1000 Hz corner frequency.
An alternate expression of the formula, not depending on choice of log base, is noted in Fant (1968):
In 1976, Makhoul and Cosell published the now-popular version with the 700 Hz corner frequency.
As Ganchev et al. have observed, "The formulae [with 700], when compared to [Fant's with 1000], provide a
closer approximation of the Mel scale for frequencies below 1000 Hz, at the price of higher inaccuracy for frequencies higher than 1000 Hz." Above 7 kHz, however, the situation is reversed, and the 700 Hz version again fits better.
Data by which some of these formulas are motivated are tabulated in Beranek (1949), as measured from the curves of Stevens and Volkman:
A formula with a break frequency of 625 Hz is given by Lindsay & Norman (1977); the formula doesn't appear in their 1972 first edition:
Most mel-scale formulas give exactly 1000 mels at 1000 Hz. The break frequency (e.g. 700 Hz, 1000 Hz, or 625 Hz) is the only free parameter in the usual form of the formula. Some non-mel auditory-frequency-scale formulas use the same form but with much lower break frequency, not necessarily mapping to 1000 at 1000 Hz; for example the ERB-rate
Equivalent rectangular bandwidth
The equivalent rectangular bandwidth or ERB is a measure used in psychoacoustics, which gives an approximation to the bandwidths of the filters in human hearing, using the unrealistic but convenient simplification of modeling the filters as rectangular band-pass filters.Researchers have derived a...
scale of Glasberg & Moore (1990) uses a break point of 228.8 Hz, and the cochlear frequency–place map of Greenwood (1990) uses 165.3 Hz.
Other functional forms for the mel scale have been explored by Umesh et al.; they point out that the traditional formulas with a logarithmic region and a linear region do not fit the data from Stevens and Volkman's curves as well as some other forms, based on the following data table of measurements that they made from those curves:
External links
- Hz–mel, mel–Hz conversion (uses the O'Shaughnessy equation)
- J. Acoust. Soc. Am. table of contents for Stevens et al. paper
- Handbook for Acoustic Ecology