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This article was based on http://napalm.firest0rm.org/issue4.txt, and was used here with the original author's (ajax) permission. |
Theory of Musical Scales |
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Theory of Musical Scales The variation of air pressure against the inner ear is what gives rise to the experience we call "sound". Most sound that we recognize as "musical" is dominated by periodic variations rather than random ones, and we call the transmission mechanism as a "sound wave". In a very simple case, listening to the sound of a "sine wave", the air pressure increases and decreases in a regular fashion, and we hear it as a very "pure" tone. The rate at which the air pressure varies governs the "pitch" of the tone, and can be measured in oscillations per second, or Hertz. Whenever two different pitches are played at the same time, their sound waves interfere with each other - the highs and lows in the air pressure get mixed together to produce a different sound wave. As a result, any given sound wave can contain many different oscillation frequencies; the human ear and brain are capable of isolating these frequencies and hearing them distinctly. When two notes are played, there is a single variation of air pressure at your ear that "contains" the pitches of both voices, and your ear and brain isolates them into two distinct notes. |
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The variation of air pressure against the inner ear? gives rise to the experience we call "sound". Most sound that people recognize as "musical" is dominated by periodic variations rather than random ones, and we call the transmission mechanism as a "sound wave". In a very simple case, listening to the sound of a "[sine wave]?", the air pressure increases and decreases in a regular fashion, and we hear it as a very "pure" tone. Pure tones can be produced by tuning forks. The rate at which the air pressure varies governs the "pitch" of the tone, and can be measured in oscillations per second, or Hertz. Whenever two different pitches are played at the same time, their sound waves interfere with each other - the highs and lows in the air pressure get mixed together to produce a different sound wave. As a result, any given sound wave can contain many different oscillation frequencies; the human hearing apparatus (composed of the ears and brain) can isolating these frequencies and hear them distinctly. When two notes are played, a single variation of air pressure at your ear "contains" the pitches of both voices, and your ear and brain isolates them into two distinct notes. |
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have a "musical" relationship to the originals. However, whenever one pitch is a simple multiple of the other (1,2,3,4 times the oscillation frequency) the interfence does not generate these any new pitches. Thus, any two pitches related like this sound perfectly "in tune" in that you hear those pitches, and nothing else. |
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have a "musical" relationship to the originals. However, whenever one pitch is a simple integer multiple of the other (1, 2, 3, 4 times the oscillation frequency), the interfence does not generate any new pitches. Thus, any two pitches related like this sound perfectly "in tune" in that you hear those pitches, and nothing else. |
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The simplest ratio is clearly 1:1, but this is a trivial case of the same note being played twice. More interesting is the 2:1 ratio. Any |
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Musicians call the trivial case of a 1:1 ratio a "unison." More interesting is the 2:1 ratio. Any |
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together, they sound perfectly "in tune". The human ear can perceive from about 20Hz at the low end to around 20,000Hz at the high end. By starting at 20 and doubling up to 20,000, you can see that the human ear has a range of a little under ten octaves. There are clearly many other integer ratios, and even though they do not all avoid the generation of additional pitches, when the ear hears any two notes with such a ratio (or close to it), they are perceived to be "in tune". So, we can now move to the definition of a scale: a way of defining the intervals between each of a set of notes within an given distance. The distance and number of notes is variable, but in the majority of the western classical and popular tradition, twelve notes are used to span a single octave. The intervals between them are: |
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together, they sound perfectly "in tune". The average human ear can perceive tones from about 20Hz at the low end to around 20,000Hz at the high end. Starting at 20 and doubling up to 20,000 shows that the human ear has a range of about ten octaves. There are clearly many other integer ratios, and even though they do not all avoid the generation of additional pitches, the hearing apparatus perceives any two notes with such a ratio (or close to it) to be "in tune". We can now define a scale as a set of intervals between the lowest note in a set of notes within a given distance and each other note in the set. The distance and number of notes varies, but in the majority of the western classical and popular tradition, twelve notes span a single octave. |
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For purposes of tuning we need a reference pitch, something all the instruments can agree on. Usually a 440hz sine wave is used as the reference pitch, as an A natural. Now, according to our table above, we can calculate the pitch of any other note by setting up a simple ratio relationship. For example, if I wanted to calculate the pitch of a perfect fifth from an A440, I would write: |
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For purposes of tuning we need a reference pitch, something all the instruments can agree on. Usually a 440hz sine wave is used as the reference pitch, as an A natural. Now, according to our table above, we can calculate the pitch of any other note by setting up a simple ratio relationship. For example, if I wanted to calculate the pitch of a perfect fifth from an A440, I would write: |
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and solve for X. Simple algebra, right? In the above, X comes to 660. |
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and solve for X. Simple algebra, right? In the above, X comes to 660. |
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440.000 A 462.000 A# 495.000 B 528.000 C 550.000 C# 586.667 D 616.000 D# 660.000 E 704.000 F 733.333 F# 792.000 G 831.111 G# 880.000 A Any scale in which the ratio of any note to the tonic is an integer ratio is called a scale of just intonation. These scales have a very natural-sounding quality to them. The problem is, they're very difficult to achieve in any stopped or fretted instrument. The difficulty is subtle, but it means big headaches. It's hard to explain, though, so I'll give an example. The interval of a major second is the "whole step" so common in the western tradition. It defines the distance between A and B, |
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440.000 A 462.000 A# 495.000 B 528.000 C 550.000 C# 586.667 D 616.000 D# 660.000 E 704.000 F 733.333 F# 792.000 G 831.111 G# 880.000 A Any scale in which the ratio of any note to the tonic is an integer ratio is called a scale of just intonation. These scales have a very natural-sounding quality to them. This is the common western scale of just intonation; other scales of just intonation exist, such as Indian raga scales. The problem with just intonation is that it is very difficult to achieve in any stopped or fretted instrument. The difficulty is subtle, but it means big headache?s. For example, the interval of a major second is the "whole step" so common in the western tradition. It defines the distance between A and B, |
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If this is true, then the major second of a major second (that is, two whole steps from a given note) should be the same as the major third (two whole steps from a given note), or: |
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If this is true, then the major second of a major second (that is, two whole steps from a given note) should equal the major third (two whole steps from a given note), or: |
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note in tune, and space all the other notes equally (logarithmically |
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note in tune, and space all the other notes equally (logarithmically |
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major sixth below A440). This is called a scale of even (or equal) |
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major sixth below A440). We call this approximation a scale of even (or equal) |
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Now, waitasec, doesn't this put everything out of tune? Well, yes it does, but not by very much. Observe: |
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This throws everything very slightly out of tune. Observe: |
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In this system the fifth tone ratio is about 1.4983 instead of 1.5, and the half-tone ratio is 1.059463 instead of 1.05. Only the octave is still 2:1. It was not easy when they first learned to tune the "well tempered clavier" to interpolate between tuning suggested by different keys. Insofar as tuning is still done by ear we are not likely to achieve a half-tone ratio that matches the twelfth root of two in six or seven digits! |
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In this system the fifth tone ratio is about 1.4983 instead of 1.5, and the half-tone ratio is 1.059463 instead of 1.05. Only the octave is still 2:1. It was not easy when they first learned to tune the "well tempered clavier?" to interpolate between tuning suggested by different keys. Tuning done by ear cannot achieve a semitone ratio that matches the twelfth root of two in six or seven digits. Many classical composers wrote compositions for just-intonated instruments (wind instruments in particular). However, since these instruments couldn't re-tune to a new tonic, modulating the key of the piece created a tension; it sounded like you were still playing in the original key and wanted to return to it. Some people insist that playing the piece on a JI instrument is the only way to truly hear what the composer intended. Other music fans disagree. |
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( There are also Indian raga scales and other non-standard scales that are "just intonation" scales, but I wanted to present something understandable.) --- [ one thing i neglected to mention in the original article was that many classical composers wrote compositions for just-intonated instruments (wind instruments in particular). however, since these instruments couldn't re-tune to a new tonic, modulating the key of the piece created a tension; it sounded like you were still playing in the original key and wanted to return to it. at least, that's what some people say, and they're usually quite insistent that playing the piece on a JI instrument is the only way to truly hear what the composer intended. i personally don't believe that, but judge for yourself. { ajax } ] |
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This article (or a previous version thereof) was based on http://napalm.firest0rm.org/issue4.txt, used by permission of the original author (ajax). |
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