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

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=== Ancient Greek interval names ===

Intervals in Ancient Greek music were written either as ~~frequency~~ ratios, after Pythagoras, or as positions in a [[tetrachord]]. ~~Some ratios/intervals were also given names:~~

Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].

[[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

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=== Zarlino and Meantone ===

Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|556.962x556.962px|Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, ''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15, pg. 25.]]

Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

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'' and ''minor tone'', 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 and 1/4-comma Meantone as usable (Zarlino, 1558).

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 and 1/4-comma Meantone as usable (Zarlino, 1558).

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 by ~~Zarlino~~ 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'' and 25/24 as the ''minor semitone''.

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'').

In ~~Zarlino~~'~~s 1558 treatise~~ ''~~Le istitutioni harmoniche~~,'' the ~~first diagram associates many intervals with their tuning~~ as ~~perfect consonances~~.

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, which we are tempted to translate to 'major sixth'. Chapter 16, ''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,'' or perhaps 'minor sixth' be tuned to 8/5.

=== English interval names in the Baroque ===

After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the following table in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds:''

2/1, 3/2 and 4/3 were labelled 'perfect' because they were seen to be the perfect consonances. The two sizes of second, third, sixth and seventh in the diatonic scale are labelled ''major'' or ''minor'', the larger sizes of each labelled major. Any perfect or major interval raised by a chromatic semitone (the difference, for e.g. between C and C#) is referred to as 'Augmented' and any minor or perfect interval lowered by a chromatic semitone is referred to as 'diminished'.

@@ Line 12: / Line 12: @@
 === Ancient Greek interval names ===
-Intervals in Ancient Greek music were written either as frequency ratios, after Pythagoras, or as positions in a [[tetrachord]]. Some ratios/intervals were also given names:
+Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].
 [[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'
@@ Line 33: / Line 33: @@
 === Zarlino and Meantone ===
-Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':
+[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|556.962x556.962px|Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, ''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15, pg. 25.]]
+Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':
 /1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
-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'' and ''minor tone'', 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 and 1/4-comma Meantone as usable (Zarlino, 1558).
+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 and 1/4-comma Meantone as usable (Zarlino, 1558).
-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 by Zarlino 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'' and 25/24 as the ''minor semitone''.
+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'').
-In Zarlino's 1558 treatise ''Le istitutioni harmoniche,'' the first diagram associates many intervals with their tuning as perfect consonances.
+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, which we are tempted to translate to 'major sixth'. Chapter 16, ''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,'' or perhaps 'minor sixth' be tuned to 8/5.
+=== English interval names in the Baroque ===
+After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.
+Music theorist and mathematician Robert Smith provides the following table in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds:''
 /1, 3/2 and 4/3 were labelled 'perfect' because they were seen to be the perfect consonances. The two sizes of second, third, sixth and seventh in the diatonic scale are labelled ''major'' or ''minor'', the larger sizes of each labelled major. Any perfect or major interval raised by a chromatic semitone (the difference, for e.g. between C and C#) is referred to as 'Augmented' and any minor or perfect interval lowered by a chromatic semitone is referred to as 'diminished'.