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

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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 tunings and rank-''n'' temperaments. Dozens of different accidentals can be used on a regular diatonic 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.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. ~~In Johnston's notation an unaltered~~ major third ~~represented~~ 5/4~~, while in all most other microtonal notation systems the major third was defined by stacked~~ fifths~~. Johnston's notation is confusing as~~ a ~~result of the way he dealt with this definition~~, but ~~it can be approached in a simpler manor~~.

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 benifits 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.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.

=== Dave Keenan's system ===

In 2016 Dave Keenan proposed an alternative generalised microtonal interval naming system for edos. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The nuetral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of 72edo that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following ~~table~~ details this process:

In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The nuetral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of 72edo that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a 6/5 minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31.

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and 10/9 often a narrow or small major second. One such system that considers 10/9 a narrow major second, is that of Aaron Hunt.

== Premise: ==

Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .

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 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 tunings and rank-''n'' temperaments. Dozens of different accidentals can be used on a regular diatonic 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.
-We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In Johnston's notation an unaltered major third represented 5/4, while in all most other microtonal notation systems the major third was defined by stacked fifths. Johnston's notation is confusing as a result of the way he dealt with this definition, but it can be approached in a simpler manor.
+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 benifits 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.
+We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.
 === Dave Keenan's system ===
-In 2016 Dave Keenan proposed an alternative generalised microtonal interval naming system for edos. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The nuetral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of 72edo that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following table details this process:
+In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The nuetral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of 72edo that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a 6/5 minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31.
+Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and 10/9 often a narrow or small major second. One such system that considers 10/9 a narrow major second, is that of Aaron Hunt.
 == Premise: ==
 Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .