tag:blogger.com,1999:blog-20318294.post7353517946949142707..comments2017-06-12T20:29:28.122+00:00Comments on O cuniculi! Ubi lexicon Latinum posui?: Concerto in the Key of ChristWarwickensishttp://www.blogger.com/profile/01310450226153796760noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-20318294.post-40831650823859060772017-03-20T14:52:11.933+00:002017-03-20T14:52:11.933+00:00Thank you for your comments, Father! I have long b...Thank you for your comments, Father! I have long been interested in the mathematics of tuning ever since I studied organ pipes, especially the Mixtures. I've also spent many a time trying to tune virginals and harpsichords which have been neglected. <br /><br />The fascinating thing is that modern tuning of course evens out discrepancies by looking at fractional powers of 2. Raising a note by a semitone multiplies its frequency by the twelfth root of 2. Trouble with fractional powers of 2 is that they are cannot be expressed as fractions which Pythagoreans demanded. Indeed, the old story goes that a man who demonstrated that the square root of 2, which is 2 to the halfth power, cannot be expressed as a fraction was taken out to sea in a boat which returned to shore without him!Warwickensishttp://www.blogger.com/profile/01310450226153796760noreply@blogger.comtag:blogger.com,1999:blog-20318294.post-34612561364025867532017-03-20T08:40:17.970+00:002017-03-20T08:40:17.970+00:00You bring up a big subject, that of tuning tempera...You bring up a big subject, that of tuning temperaments: what to do with the Pythagorean Comma, which is the difference between the octave derived from pure fourths and fifths and the pure octave. You would be able to work out the mathematics very easily from the frequency values of the notes. Here is an article. <br /><br />https://en.wikipedia.org/wiki/Musical_temperament<br /><br />Keyboard instruments have been standardised since about the end of the 17th century. Some harpsichord makers experimented with split sharps and flats, but this was abandoned. The keyboard only has 12 notes in the chromatic scale, the octave being the 13th. This brings about the need for a temperament - dividing the Comma into more or less tolerable fractions. J.S. Bach saw in the equal temperament system (the Comma divided into 12 equal parts) and wrote the Wohltemperierte Klavier (BWV 846–893) - preludes and fugues in all keys. In mean tone temperament, you can't play anything other than in a major key with no more than one sharp or flat, assuming you tune from C.<br /><br />Violins aren't tempered but tuned pure, so the vibrato was introduced to hide the dissonance from the Comma, especially when playing in a full symphony orchestra and keyboard instruments. I'm not sure, but I think the harp also has to be tempered. This is why young string players find intonation so difficult and school orchestras can be quite painful to listen to.<br /><br />Just my tuppence. Of course you meant this question as an analogy for a spiritual point you were putting over.<br /><br />Fr Anthony Fr Anthonyhttp://www.blogger.com/profile/15521671841072661886noreply@blogger.com