Just intonation: Difference between revisions

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(In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1):

''1. The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].''

# ''The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].''

# ''Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[Wikipedia: Harry Partch|Harry Partch]]'s [[Wikipedia: Tonality diamond|tonality diamond]]. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.''

# ''Other theorists who, in contrast to the above, advocate the use of [[Wikipedia: Hexany|products sets]] of given arrays of prime numbers, such as [[Wikipedia: Erv Wilson|Ervin Wilson]], Robert Dussaut, and others.''

# ''[[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.''

# ''Restricting the denominator to one or very few values (the [[harmonic series]]).''

# ''Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).''

''2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[Wikipedia: Harry Partch|Harry Partch]]'s [[Wikipedia: Tonality diamond|tonality diamond]]. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.''

To this may be added:

# ''The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.''

''3. Other theorists who, in contrast to the above, advocate the use of [[Wikipedia: Hexany|products sets]] of given arrays of prime numbers, such as [[Wikipedia: Erv Wilson|Ervin Wilson]], Robert Dussaut, and others.''

# ''While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [http://anaphoria.com/wilsonintroMERU.html Meru scales] are a good example.''

# ''Choosing some set of relatively high overtones (almost always higher than in prime limit-based approaches), and using each note as a root for extended harmony within the set ([[primodality]]).''

''4. [[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.''

~~''5. Restricting the denominator to one or very few values (the [[harmonic series]]).''~~

~~''6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).''~~

to this ~~can~~ be added

''7. The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.''

''8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [http://anaphoria.com/wilsonintroMERU.html Meru scales] are a good example.''

== Variations ==

@@ Line 55: / Line 55: @@
 (In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", ''1/1'' vol. 11, no. 2: p.1):
-''1. The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].''
+# ''The principle of "[[harmonic limit]]s", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See [[3-limit]], [[5-limit]], [[7-limit]], [[11-limit]], [[13-limit]].''
+# ''Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[Wikipedia: Harry Partch|Harry Partch]]'s [[Wikipedia: Tonality diamond|tonality diamond]]. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.''
+# ''Other theorists who, in contrast to the above, advocate the use of [[Wikipedia: Hexany|products sets]] of given arrays of prime numbers, such as [[Wikipedia: Erv Wilson|Ervin Wilson]], Robert Dussaut, and others.''
+# ''[[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.''
+# ''Restricting the denominator to one or very few values (the [[harmonic series]]).''
+# ''Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).''
-''2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of [[Wikipedia: Harry Partch|Harry Partch]]'s [[Wikipedia: Tonality diamond|tonality diamond]]. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.''
+To this may be added:
+# ''The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.''
-''3. Other theorists who, in contrast to the above, advocate the use of [[Wikipedia: Hexany|products sets]] of given arrays of prime numbers, such as [[Wikipedia: Erv Wilson|Ervin Wilson]], Robert Dussaut, and others.''
+# ''While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [http://anaphoria.com/wilsonintroMERU.html Meru scales] are a good example.''
+# ''Choosing some set of relatively high overtones (almost always higher than in prime limit-based approaches), and using each note as a root for extended harmony within the set ([[primodality]]).''
-''4. [[Just intonation subgroups|Restrictions on the variety of prime numbers]] used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in Just Intonation.''
-''5. Restricting the denominator to one or very few values (the [[harmonic series]]).''
-''6. Restricting the numerator to one or a very few values (the [[subharmonic series]] or [[aliquot scales]]).''
-to this can be added
-''7. The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.''
-''8. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's [http://anaphoria.com/wilsonintroMERU.html Meru scales] are a good example.''
 == Variations ==