{"id":572,"date":"2015-02-10T21:24:39","date_gmt":"2015-02-10T21:24:39","guid":{"rendered":"http:\/\/casfaculty.case.edu\/ross-duffin\/?page_id=572"},"modified":"2017-02-02T19:52:59","modified_gmt":"2017-02-02T19:52:59","slug":"theoretical-background","status":"publish","type":"page","link":"https:\/\/casfaculty.case.edu\/ross-duffin\/just-intonation-in-renaissance-theory-practice\/theoretical-background\/","title":{"rendered":"Theoretical Background"},"content":{"rendered":"

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\u00a0 \u00a0 \u00a0Beginning with Ramos de Pareja in 1482,Fn11<\/sub><\/a><\/span> the simple ratios of Just intonation began to be cited by theorists and used in their divisions of the monochord. Before that, the 3:2 ratio of the pure fifth and its resulting musical system\u2014a whole tone derived from two pure fifths and a major third derived from four pure fifths (in other words, Pythagorean tuning)\u2014comprised the almost universal theoretical model. The ratios of the Pythagorean system, after the 2:1 octave and the 3:2 fifth, were mathematically complex, however, so Ramos\u2019s identification of intervals with simple harmonic ratios was greeted with near-unanimous acceptance. Just intonation was easy to support from both a scientific and practical standpoint because the intervals not only had simple mathematical ratios, they sounded demonstrably better than those derived from the Pythagorean system.<\/p>\n

\u00a0 \u00a0 \u00a0Table 1<\/a><\/span> shows the interval ratios promulgated by a selection of prominent Renaissance musical theorists. The one famous \u201chold-out\u201d who continued to favor the old Pythagorean system was Gafurius, although he did acknowledge that the simple ratios were used by practical musicians.Fn12<\/sub><\/a><\/span> As may be seen, there is not much disagreement as to what constituted the correct ratios for intervals: for each theorist I have given those ratios that are either mentioned explicitly in the text, or can be deduced by the writer\u2019s recommended combination of ratios to create other intervals.<\/p>\n

\u00a0 \u00a0 \u00a0Many of the basic interval ratios are superparticular, where the first number exceeds the second by one. Thus, an octave is 2:1, a fifth is 3:2, a fourth is 4:3, a pure major third is 5:4, a pure minor third is 6:5. These \u201csix-limit\u201d ratios\u2014the senarius<\/em>, or senario,<\/em> as Zarlino calls them\u2014are the basic harmonic intervals. Zarlino attributed almost mystical significance to the senario<\/em> in the same way that Pythagoras apparently did to the tetractys<\/em>\u2014the first four numbers. But the intervals represented in the senario<\/em> cannot be the only ones in the system because they do not fully account for stepwise motion. The 9:8 whole tone is in common with the Pythagorean system, but Just intonation also possesses a 10:9 \u201cminor tone\u201d which is the difference between a 5:4 (that is, 10:8) major third and a 9:8 major tone. Like the common harmonic intervals, those smaller intervals are also part of the harmonic series (the series of overtones of any naturally produced sound), so the 10:9 ratio, for example, is simply the distance from the 9th to the 10th harmonic in the overtone series and thus, in itself, a natural acoustical phenomenon.<\/p>\n

\u00a0 \u00a0 \u00a0Other intervals can be deduced by inversion: the 5:4 pure major third inverts to an 8:5 pure minor sixth. The 6:5 pure minor third similarly inverts to a 5:3 pure major sixth, the 9:8 major tone to the 16:9 lesser minor seventh, the 10:9 minor tone to the 9:5 greater minor seventh.<\/p>\n

\u00a0 \u00a0 \u00a0It is perhaps surprising to see ratios for the tritone and the diminished fifth given by Renaissance theorists. Some of these are given explicitly as numerical ratios; others were derived by specified combinations of intervals, such as the tritone as a pure major third plus a major tone (5:4 + 9:8 = 45:32), or a pure fifth minus a major semitone (3:2 \u2013 15:16 = 45:32). The diminished fifth (or semi-diapente) is the inversion of that at 64:45, but was also derived as a pure fourth plus a major semitone (4:3 + 16:15).<\/p>\n

\u00a0 \u00a0 \u00a0The semitone with the lowest ratio in the system, 16:15, can be derived as the difference between the 4:3 fourth and the 5:4 major third, but of course, it also exists as that interval in the harmonic series, as the distance between the ascending overtones gets smaller and smaller. Regarding the other semitone sizes, two of them consist of differences between the 16:15 major semitone and the two different sizes of whole tone: Subtract a 16:15 major semitone from a 10:9 minor tone and the 25:24 semitone remains (10:9 \u2013 15:16 = 25:24); subtract it from a 9:8 major tone and the 135:128 semitone remains (9:8 \u2013 15:16 = 135:128).<\/p>\n

\u00a0 \u00a0 \u00a0That accounts for most of the ratios in Table 1, with the exception of the semitones given by Vicentino, and the 10:7 ratio given for the tritone by Cardanus. Vicentino\u2019s semitones seem a simple arithmetical way to divide the 9:8 (18:16) major tone, and the 18:17 ratio, in particular, found some use, at least, in lute fretting schemes in the Renaissance. The idea of dividing the whole tone in this way stems from Boethius’s early sixth-century De Institutione Musica<\/em> I.16.Fn13<\/sub><\/a><\/span> So-called septimal ratios\u2014those using the seventh harmonic, like 10:7\u2014are often considered by modern theorists as part of the Just scale, but with the exception of Cardan\u2019s citation, they were clearly avoided by Renaissance theorists and, indeed, I believe they were not normally part of the Renaissance harmonic and melodic vocabulary.<\/p>\n

\u00a0 \u00a0 \u00a0Table 2<\/a><\/span> gives the ratios and cents values for Just intervals, along with a selection of other intervals for comparison. One characteristic of the differences between Pythagorean-derived notes and notes in Just intonation is that they are separated by a syntonic comma. The syntonic comma is a microtonal interval which is the difference between the Pythagorean major third, or ditone, at 81:64\u2014that is, two consecutive 9:8 whole tones\u2014and the pure major third at 5:4, or 80:64. Thus, the syntonic comma has the ratio 81:80. This small interval can thus be recognized as the difference between a major tone and a minor tone, and between Just intervals and their Pythagorean equivalents. Modern theorist and composer Easley Blackwood\u2019s comma-variant notation of C0<\/sub>, C+1<\/sub>, and so on, is extremely useful, both for discussions and charts of intervals, as in Tables 1 & 2, as well as in annotating musical scores.Fn14<\/sub><\/a><\/span> <\/a> Briefly, octaves, fifths, fourths, and major tones that are pure to one another have the same annotation; major thirds, major sixths, and minor tones are all narrow by one comma; minor thirds and minor sixths are wide by one comma. I will use Blackwood\u2019s system for analyzing Benedetti\u2019s problemmatic progressions, as well as the difficult passages he himself culled from the works of Orlando di Lasso.<\/p>\n

\u00a0 \u00a0 \u00a0First, I need to mention some other very small intervals in Table 2. One is the Pythagorean comma, which is the difference between 12 pure fifths and 7 pure octaves. It is larger than the syntonic comma by a miniscule interval called a schisma. This is also the amount by which a pure fifth exceeds an equal-tempered fifth, as well as the amount by which a syntonic comma exceeds the diesis in the Just system\u2014the diesis being the discrepancy between diatonic versions of C# and Db, for example. There are thus three closely related microtonal intervals of about the same size\u2014the syntonic comma and intervals one schisma on either side of it\u2014and these sometimes all get loosely referred to as commas without further definition. Vicente Lusitano, for example, refers to the discrepancy between the major and minor semitone\u2014the diesis\u2014as a comma, suggesting that for practical purposes in Renaissance performance, they were probably indistinguishable.Fn15<\/a><\/sub><\/span> These three microtonal intervals are given in Table 3, with buttons for hearing their sound produced by electronic means. As with all of the sound examples in this article, these respresentations are correct to within one cent (1\/100 of an equal-tempered semitone)\u2014typically much closer than that.<\/p>\n

Table 3. Common Microtonal intervals of Just intonation<\/strong><\/p>\n\n\n\n\n\n\n
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Interval<\/strong><\/p>\n<\/td>\n

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Example<\/strong><\/p>\n<\/td>\n

\n

Ratio<\/strong><\/p>\n<\/td>\n

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Cents<\/strong><\/p>\n<\/td>\n

\n

Sound<\/strong><\/p>\n<\/td>\n<\/tr>\n

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diesis<\/p>\n<\/td>\n

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C#-1 <\/sub>\u2013 Db+1<\/sub><\/p>\n<\/td>\n

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2048:2025<\/p>\n<\/td>\n

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19.55<\/p>\n<\/td>\n

\n\n
\n

syntonic comma<\/p>\n<\/td>\n

\n

C0 <\/sub>\u2013 C+1<\/sub><\/p>\n<\/td>\n

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81:80<\/p>\n<\/td>\n

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21.51<\/p>\n<\/td>\n

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Pythagorean comma<\/p>\n<\/td>\n

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C0 <\/sub>\u2013 B#0<\/sub><\/p>\n<\/td>\n

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531441:524288<\/p>\n<\/td>\n

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23.46<\/p>\n<\/td>\n

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