Extended-diatonic interval names: Difference between revisions

[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''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.|link=https://en.xen.wiki/w/File:Della_propriet%C3%A0_del_numero_Senario_&_della_sue_parti;_&_come_in_esse_si_ritroua_ogni_consonanze_musicale,_figura_1.png]]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 ''[[Zarlino|intense diatonic scale]]'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

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.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to [[31edo|31-tET]], which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''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|link=https://en.xen.wiki/w/File:Harmonics,_or_The_Philosophy_of_Musical_Sounds,_Section_2_figure_3.png]]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 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.

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.''

[[File:Helmholtz consonances table.png|thumb|617x469px|link=https://en.xen.wiki/w/File:Helmholtz_consonances_table.png|Table detailing the influence of the different consonances on each other, from ''On the Sensations of Tone as a Psychological Basis for the Theory of Music'', Helmholtz, 1863, Translated by Ellis, 1875, Chap. X, pg. 187.]]Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the [[5-limit]]. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as [[16/9|9:16]] seemingly because of the 9 partial limit imposed on the table. It is also worth noting that [[7/5|5:7]] is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by [[36/35|35:36]], the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond [[8/7|7:8]], which lies [[64/63|63:64]], the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be [[25/18|18:25]], 80:81 above Smith's [[64/45|45:64]] minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, [[32/25|25:32]], an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse [[45/32|32:45]] a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.

In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many [[7-limit interval names|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|9:7]], Sub-minor Sixth 14:9, Sub-minor Third [[7/6|7:6]], Super-major Sixth 12:7, Sub-minor or Harmonic Seventh [[7/4|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 and at odds with Smith's interval names. 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 referred to by Smith.