{"id":574,"date":"2015-02-10T21:26:08","date_gmt":"2015-02-10T21:26:08","guid":{"rendered":"http:\/\/casfaculty.case.edu\/ross-duffin\/?page_id=574"},"modified":"2017-02-02T19:52:59","modified_gmt":"2017-02-02T19:52:59","slug":"benedettis-puzzles","status":"publish","type":"page","link":"https:\/\/casfaculty.case.edu\/ross-duffin\/just-intonation-in-renaissance-theory-practice\/benedettis-puzzles\/","title":{"rendered":"Benedetti’s Puzzles"},"content":{"rendered":"

\u00a0<\/h1>\n

<\/a><\/p>\n

\u00a0 \u00a0 \u00a0In his second letter to Cipriano de Rore, Benedetti gave the progression shown in Ex. 1a, and demonstrated, as illustrated here in Blackwood-style numeric annotations, that using Just ratios would cause the pitch to rise by a comma for each repetition of the two-measure pattern. I use the term “puzzle” for this and Benedetti’s other progression since we are clearly left to wonder what the “solution” is if these passages are not to migrate microtonally. (The sound of all migrating examples is set to loop so that listeners can hear again where the passage started in terms of pitch.)<\/p>\n\n\n\n\n\n
Ex. 1a. Benedetti\u2019s first passage in his own version<\/strong><\/td>\n<\/tr>\n
\u00a0<\/p>\n\n
\u00a0\"Benedetti1\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u00a0 \u00a0 \u00a0This ascent results because the singer of the bottom voice tries to make the C at the beginning of bars 2, 4, and 6, pure as a minor third (or major sixth) against the A tied over from the previous bar in the top voice. This necessitates raising the C by one comma, which necessitates raising the G in the top voice in order to be pure to the sustained C in the bottom, and so forth. There is no question that the consistent application of the theoretical ratios to the notes in this passage would result in the comma ascent described by Benedetti and the presumptive conclusion that Just intonation is impractical. It did not go away, however, so the problem of comma migration continued to plague theorists for a long time. Christian Huygens, at the end of the 17th century, noted:<\/p>\n

\u00a0 \u00a0 \u00a0For if you ask any of our Musicians, why two or more perfect fifths cannot be us\u2019d regularly in composition; some say \u2019tis \u2026 because when you pass from one perfect fifth to another, there is such a change made as immediately alters your Key, you are got into a new Key before the Ear is prepared for it, and the more perfect Chords you use of the same kind in Consecution, by so much the more you offend the Ear by these abrupt Changes.\u2026<\/p>\n

\u00a0 \u00a0 \u00a0I say therefore, if any Persons strike those Sounds which the Musicians distinguish by these Letters, C, F, D, G, C, by these agreeable Intervals, altogether, perfect, interchangable, ascending and descending with the Voice: Now this latter sound C will be one Comma, or very small portion lower than the first sounding of C [as follows:]<\/p>\n\n\n\n\n
\u00a0\"Huygens.sm\"<\/a><\/td>\n<\/tr>\n
\u00a0<\/p>\n