Arithmetic tuning: Difference between revisions
Clarify the term; heading levels and other styles |
Cmloegcmluin (talk | contribs) →See also: remove link to AID because Shaahin's pages are below the level of pedagogical clarity and conceptual coherency that this page should link to; and the link to the arithmetic mean is outdated because I have since realized the inherent link between means and progressions |
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An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). This | An '''arithmetic tuning''' is one which has equal step sizes ''of any kind of quantity'', whether that be pitch, frequency, or length (of the resonating entity producing the sound). | ||
This sense of the word arithmetic comes from the mathematical concept of an [[Wikipedia:Arithmetic sequence|arithmetic sequence]], which means a sequence with a constant difference between successive terms. | |||
All arithmetic tunings are [[harmonotonic tunings]]. | All arithmetic tunings are [[harmonotonic tunings]]. | ||
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Equal frequency steps: | Equal frequency steps: | ||
* [[OD|OD, otonal division]] | * [[OD|OD, otonal division]] (such as an [[ODO]]) | ||
* [[EFD|EFD, equal frequency division]] | * [[EFD|EFD, equal frequency division]] | ||
* [[OS|OS, otonal sequence]] | * [[OS|OS, otonal sequence]] | ||
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Equal pitch steps: | Equal pitch steps: | ||
* [[EPD|EPD, or ED, equal (pitch) division]] (such as an [[EDO]]) | * [[EPD|EPD, or ED, equal (pitch) division]] (such as an [[EDO]]) | ||
* [[AS|AS, ambitonal sequence]] | |||
* [[APS|APS, arithmetic pitch sequence]] | * [[APS|APS, arithmetic pitch sequence]] | ||
Equal length steps: | Equal length steps: | ||
* [[UD|UD, utonal division]] | * [[UD|UD, utonal division]] (such as a [[UDO]]) | ||
* [[ELD|ELD, equal length division]] | * [[ELD|ELD, equal length division]] | ||
* [[US|US, utonal sequence]] | |||
* [[ALS|ALS, arithmetic length sequence]] | * [[ALS|ALS, arithmetic length sequence]] | ||
== Basic examples == | == Basic examples == | ||
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Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4. | Other arithmetic tunings can be found by changing the step size. For example, if you vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math><span>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4. | ||
If the new step size is irrational, the tuning is no longer JI, so we | If the new step size is irrational, the tuning is no longer JI, so we fall back to a more general acronym to distinguish it: AFS, for arithmetic frequency sequence. For example, if we wanted to move by steps of φ, like this: <math>1, 1+φ, 1+2φ, 1+3φ...</math> etc. we could have the AFSφ. Because different acronyms are used to distinguish rational (JI) tunings from general tunings which include irrational (non-JI) tunings, while the acronyms used for general tunings technically include the JI tunings, these general acronyms are more useful when reserved for non-JI tunings, and this is what is typically done. So when "irrational" is used on this page, it more accurately means "probably irrational". | ||
OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation of OS]]. | OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency. By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation of OS]]. | ||