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 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. | ||
== Background == | == Background == | ||
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''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth | ''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth | ||
''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense (Chalmers) | ''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993]) | ||
''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, was referred to as ''tonos'', resulting in [[81/64]], the Pythagorean major third, as a ''ditone''. | ''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, was referred to as ''tonos'', resulting in [[81/64]], the Pythagorean major third, as a ''ditone'' ([[Joe Monzo|Monzo]], http://www.tonalsoft.com) | ||
[[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale | [[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale | ||
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[[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale | [[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale | ||
The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth ( | The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8. | ||
=== Our current diatonic interval names === | === Our current diatonic interval names === | ||
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=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] === | === [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] === | ||
One such system, sagispeak was developed by Dave Keenan and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. Pythagorean intonation is assumed as a basis, like in previous microtonal ideas by Ellis, Helmholtz, Wolf, Monzo and others (combined into [http://tonalsoft.com/enc/h/hewm.aspx HEWM notation]). Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. Noting that the set of 'sub' and 'super' 2nds, 3rds, 4ths and 5ths above all involve ratios of 7, Sagittal features an accidental of [[64/63]], the Archytas comma, which may be used to take a Pythagorean major third, or minor third to a supermajor third or subminor third respectively, for example. The prefix 'tao' takes Pythagorean minor 2nds, 3rds, 6ths and 7ths to subminor 2nds, 3rds, 6ths and 7ths respectively, and the prefix 'tai' takes Pythagorean major 2nds, 3rds, 6ths and 7ths to supermajor 2nds, 3rds, 6ths and 7ths respectively. Other prefixes include 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]], the meantone comma, respectively, and 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11. The ratio [[5/4]], the just major third is considered almost universally to be a major third, however, so is the Pythagorean major third, 81/64. In [[Meantone]] temperament, with which we are very familiar, 81/80, the difference between these notes is tempered out, however in Sagittal, and arguably in any effective notation or interval naming system, these two major thirds need to be distinguished from each other. In Sagispeak 5/4 is a 'pao-major 3rd' (and 6/5 a 'pai-minor third), and 81/64, a major third. In meantone tunings they are both represented by the same interval, the major third. | One such system, sagispeak was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. Pythagorean intonation is assumed as a basis, like in previous microtonal ideas by Ellis, Helmholtz, Wolf, Monzo and others (combined into [http://tonalsoft.com/enc/h/hewm.aspx HEWM notation]). Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. Noting that the set of 'sub' and 'super' 2nds, 3rds, 4ths and 5ths above all involve ratios of 7, Sagittal features an accidental of [[64/63]], the Archytas comma, which may be used to take a Pythagorean major third, or minor third to a supermajor third or subminor third respectively, for example. The prefix 'tao' takes Pythagorean minor 2nds, 3rds, 6ths and 7ths to subminor 2nds, 3rds, 6ths and 7ths respectively, and the prefix 'tai' takes Pythagorean major 2nds, 3rds, 6ths and 7ths to supermajor 2nds, 3rds, 6ths and 7ths respectively. Other prefixes include 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]], the meantone comma, respectively, and 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11. The ratio [[5/4]], the just major third is considered almost universally to be a major third, however, so is the Pythagorean major third, 81/64. In [[Meantone]] temperament, with which we are very familiar, 81/80, the difference between these notes is tempered out, however in Sagittal, and arguably in any effective notation or interval naming system, these two major thirds need to be distinguished from each other. In Sagispeak 5/4 is a 'pao-major 3rd' (and 6/5 a 'pai-minor third), and 81/64, a major third. In meantone tunings they are both represented by the same interval, the major third. | ||
Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. Diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved . This is a very helpful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is the generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt. | Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. Diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved . This is a very helpful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is the generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt. | ||
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P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8. | P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8. | ||
=== 10edo, Pajara and a problem === | === [[10edo]], Pajara and a problem === | ||
The primary interval names for | The primary interval names for 10edo consist of all the neutrals and all the intermediates with all the perfects as alternatives for some of the intermediates: | ||
P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8 | P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8 | ||
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In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo. | In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo. | ||
=== 14edo and Injera === | === [[14edo]] and Injera === | ||
The primary interval names for | The primary interval names for 14edo includes all the neutrals, perfects and intermediates: | ||
P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8. | P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8. | ||
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We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes. | We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes. | ||
=== Blacksmith and further extension === | === [[Blacksmith]] and further extension === | ||
10edo also support | 10edo also support Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as: | ||
1-2 sM2 2-3 sM3 3-4 s5 5-6 sM6 6-7 sM7 7-8, or | 1-2 sM2 2-3 sM3 3-4 s5 5-6 sM6 6-7 sM7 7-8, or | ||
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which we have seen before as Tetracot[13] 6|6. | which we have seen before as Tetracot[13] 6|6. | ||
=== [[Miracle]], 11edo and 21edo === | === [[Miracle]], [[11edo]] and [[21edo]] === | ||
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS: | All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS: | ||
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Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here. | Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here. | ||
The fifth in | The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens: | ||
P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8. | P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8. | ||
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We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine. | We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine. | ||
21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations: | |||
P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8. | P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8. | ||