Extended-diatonic interval names: Difference between revisions

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Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. Nonetheless, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

Such mathematical and musical ideas are attributed to [[Pythagoras of Samos|Pythagoras]], who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. Nonetheless, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

== Ancient Greek interval names ==

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Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, [[40/27]], flat of 3/2 by [[81/80]]. This was also the interval by which four 3/2 fifths missed [[5/1]] (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, [[10/9]] and [[16/15]], where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to [[2/7-comma meantone]], but also described [[1/3-comma meantone|1/3-comma]] and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The ~~interval for which 6/5 is considered a perfect tuning was~~ referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''~~. This may seem odd~~ to ~~us now~~, ~~but in Latin 'semi' referred~~ not ~~to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'~~. Additionally 'semitone~~' referred to the interval smaller than the 'tone'. Like the tone, this interval~~ possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono ~~maggiore~~'').

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be [[5/4]], rather than [[81/64]]. The minor third, referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)'' was considered to be [[6/5]], and not [[32/27]]. Additionally the semitone possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono minore'').

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3. Chapter 16 of Part 1, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' be tuned to 8/5.

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== Ups and Downs ==

One final interval naming system, associated with the [[Ups and ~~Downs Notation~~]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.

One final interval naming system, associated with the [[Ups and downs notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'.

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 Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).
-Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. Nonetheless, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.
+Such mathematical and musical ideas are attributed to [[Pythagoras of Samos|Pythagoras]], who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. Nonetheless, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.
 == Ancient Greek interval names ==
@@ Line 41: / Line 41: @@
 Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, [[40/27]], flat of 3/2 by [[81/80]]. This was also the interval by which four 3/2 fifths missed [[5/1]] (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, [[10/9]] and [[16/15]], where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to [[2/7-comma meantone]], but also described [[1/3-comma meantone|1/3-comma]] and 1/4-comma Meantone as usable (Zarlino, 1558).
-The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').
+The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be [[5/4]], rather than [[81/64]]. The minor third, referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)'' was considered to be [[6/5]], and not [[32/27]]. Additionally the semitone possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and [[25/24]], the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono minore'').
 In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3. Chapter 16 of Part 1, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' be tuned to 8/5.
@@ Line 454: / Line 454: @@
 == Ups and Downs ==
-One final interval naming system, associated with the [[Ups and Downs Notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.
+One final interval naming system, associated with the [[Ups and downs notation]] system, belonging to microtonal theorist and musician [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from pythagorean intervals. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo|12-tET]], 19-tET or 31-tET for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo|15-tET]], [[22edo|22-tET]], 41-tET, 72-tET), or even an up-major 3rd (e.g. [[21edo|21-tET]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). The mid-4th ~4 is halfway between P4 and A4, and ~5 is likewise half-way diminished. 'Up' and 'down' prefixes may be used before mid also, e.g. 'v~ 3' in 72-tET. P1, P4, P5 and P8 when upped or downed (or midded, in the case of P4 and P5) are simply labelled '1', '4', '5' and '8'. This system benefits from its simplicity as well as its conservation of interval arithmetic. The latter makes possible the naming of chords, e.g. downminor 7th. Rank-2 temperaments may also be described, with the possible addition of an additional pair of qualifiers, lifts and drops - '/' and '\'. A rank-2 scale, such as a MOS scale may appear different than this rank-2 notation when approximated in an equal (rank-1) tuning. Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be given different names.
 [[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'.