User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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Today a small ~~under~~ of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

Today a small number of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

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We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet~~</span>~~'''</~~big~~>)

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet'''</span>)

=== Helmholtz and Ellis ===

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-Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.
+Today a small number of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.
 == Background ==
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 We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''
-In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>'''</big>)
+In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet'''</span>)
 === Helmholtz and Ellis ===