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

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=== Zarlino and Meantone ===

[[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.]]

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|538.976x538.976px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, 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'':

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=== 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After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of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.

[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|547.986x547.986px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]

After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of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:''

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

~~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'~~.

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.

=== Helmholtz and Ellis ===

Names were given also to 7 and 11-limit ratios in the 19th century. In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Alexander Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as we today understand it, especially given that it is exactly two 4:3 Fourths.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

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'.

=== Common microtonal interval names ===

Dating back at least to 1880, after Alexander Ellis and John Land, the interval [[7/6]] has been associated with the label ''subminor third''. in a generalisation of this idea, [[9/7]] is most commonly reffered to as a ''supermajor third,'' [[12/7]] a ''supermajor sixth'', [[14/9]] a ''subminor sixth,<!-- plural?!

@@ Line 33: / Line 33: @@
 === Zarlino and Meantone ===
-[[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.]]
+[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|538.976x538.976px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, 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'':
@@ Line 45: / Line 45: @@
 === 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.
+[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|547.986x547.986px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]
+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:''
+Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.
-/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'.
+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.
+=== Helmholtz and Ellis ===
+Names were given also to 7 and 11-limit ratios in the 19th century. In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Alexander Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as we today understand it, especially given that it is exactly two 4:3 Fourths.
+There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.
+/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'.
 === Common microtonal interval names ===
 Dating back at least to 1880, after Alexander Ellis and John Land, the interval [[7/6]] has been associated with the label ''subminor third''. in a generalisation of this idea, [[9/7]] is most commonly reffered to as a ''supermajor third,'' [[12/7]] a ''supermajor sixth'', [[14/9]] a ''subminor sixth,<!-- plural?!