(a) What is the difference between an open and closed temperament? (b) Which tem
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Question
(a) What is the difference between an open and closed temperament? (b) Which temperaments were open and which were closed? List all the ones we have studied and all the ones in Berg. (c) Which intervals were supposed to be perfect (ideal) in the Pythagorean temperament? (d) Which intervals were out of tune in the Pythagorean temperament, and which required switching to new temperaments as new harmonizing developed? (e) Why was the Pythagorean temperament a reasonable temperament to use at the time and why did it become necessary to switch to a new temperament? (f) How does the Pythagorean temperament compare with the just temperament? Explain in detail. (g) If a just temperament is tuned for C Major, show why the key of F^# Major would be out of tune by comparing the frequency ratio for the perfect fifth and major third relative to the tonic in the key of F^# Major and comparing with the ideal harmonic ratios. Use Table 9-2, remembering to get notes in the correct octaves by multiplying or dividing by factors of 2.Explanation / Answer
A.. All temperaments fall into three groups: Open
temperaments, closed unequal temperaments and one equal temperament that dominates now.
Historically earlier open temperaments use perfect intervals (fifth, forth, etc.) in basic music scales such as C. The price to pay is that the intervals in the scales remote from C (such as the C-sharp scale) are pretty far from the perfect intervals and sound horrible. As a results, if a music instrument
is tuned with an open temperament, only a limited number of music scales can be used.
The need to increase the number of usable music scales lead to the introduction of closed unequal temperaments where the imperfections of tuning are distributed all over the octave in different ways that is a too special thing to discuss here in detail.
Unequal means that there are two or more different intervals close to the corresponding perfect intervals. Within this group of temperaments, music works written in different scales whereas all of them are usable.
C. Pythagorean temperament was historically the first of temperaments using all 12 semitones within the octave. This temperament uses the fifth as the biulding block and tries to make all fifths perfect while preserving the octaves perfect, too.
Pythagorean temperament is an open temperament, as we shall see that here all fifths but one are perfect, and the last one is very bad. This the error (of fifths) concentrates in one place.
Taking C4 as the starting tone, one can tune the notes G4 (up from C4) and F3 (down from C4) requiring that both intervals F3C4 and C4G4 are perfect fifths and using the ear. Tuning is not very difficult since the perfect fifth is the second-simplest interval (after the octave).
4.The procedure of setting the Pythagorean temperament outlined above is the simplest theoretically.
Practically it is impossible to tune a 7-octaves interval, as was suggested. Combining tuning perfect fifths and octaves up and down, one can achieve the goal within only a couple of octaves. One of possible realizations
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