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"Like ",
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ButtonStyle->"Hyperlink"],
" tuning, Just Intonation is based on intervals with simple ",
ButtonBox["ratios",
ButtonData:>{"intro.nb", "ratios"},
ButtonStyle->"Hyperlink"],
". Where Pythagorean tuning uses simple ratios between consecutive notes in \
the process of tuning, Just Intonation places the simplest possible ratios \
between C and the other notes in the scale. These are:\n\tC: 1, D: ",
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", A: ",
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", B",
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".\nExtending the tuning to include sharps and flats we have:\n\tC\[Sharp]: \
",
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", E\[Flat]: ",
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".\nJust Intonation has been known since Ptolemy of Alexandria \
(70\[Dash]147 AD). It has the advantage over Pythagorean tuning that it is \
more suitable for harmonic music, due to the fact that most of the fifths are \
pure, as well as the thirds, given that the chosen key has no more than two \
flats or three sharps. This corresponds to the ratios of the intervals in the \
harmonic series."
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"This tuning is not usable in many keys, C major is the best. Those keys \
which use chords containing the bad fifths D\[Dash]A, \
E\[Flat]\[Dash]B\[Flat], and particularly F\[Sharp]\[Dash]C\[Sharp] are quite \
sour. Chords such as C major, F major, G major and surprisingly C\[Sharp] \
major and G\[Sharp] major are pure (ie. they are as the major triad appears \
in the ",
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fifth.\nThis tuning is too early to have been used extensively for keyboard \
music, and gave way to the more sophisticated meantone temperaments in the \
16th century. The music at the time did not contain the complex harmony of \
later music, and did not require the ability to modulate extensively. \
Originally only the natural notes were used (ie. \"white\" notes on the \
modern keyboard), the sharps were eventually added in order to allow \
transposition of the scale. Music was written in twelve modes, which have \
condensed into our major and minor scales, and the different flavour of each \
mode was characterized by the relationships between the intervals as regards \
tuning, as well as the intervals themselves."
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unlike some of the ",
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