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

Line 54:

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. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>)

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>'''</big>)

=== Helmholtz and Ellis ===

@@ Line 54: / Line 54: @@
 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. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''
-In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>)
+In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>'''</big>)
 === Helmholtz and Ellis ===