Extended-diatonic interval names: Difference between revisions

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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)''. 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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 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)''. 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.