|This is a hand out from BU choral workouts. We were practicing singing half tones.|
Answers from friends:
Jim Binkley if a system is based on just intonation, Ab and G# are different pitches. With Equal Temperament - they are the same.
Ralph Knag Jim's right - here's some more words: "They are only the same pitch in equal temperament tuning. In previous tuning systems, the black keys were a major problem in which ones to tune them to - for example, do you tune it to G# or Ab?
They are not the same note, and they do not have the same pitch, because they belong on different diatonic scales. Ab is slightly sharper than G#. It was a problem back then, because keyboard instruments back then could not play all 24 diatonic scales without having at least 1 of them having wolf intervals. For example: the key G#/Ab was tuned to G#, and the D#/Eb key was tuned to Eb, because that's what they used more often. G#-Eb (actually a diminished sixth) is supposed to be enharmonically equivalent to a perfect fifth, but it was so out of tune that it was dubbed the wolf fifth.
Today's keyboards are tuned to a compromise system called equal temperament, in which each note is the same distance from each other. The problem with this is that thirds and sixths tend to be quite out of tune, but because of its versatility, it has remained in widespread use."
David Badagnani In 12-tone equal temperament, fifths and fourths are within 2 cents of just intonation, but major thirds are 13.69 cents too large (400 cents as opposed to approximately 386.31 cents for just intonation). That means if you play a major third on a piano you hear a horrible 5 beats per second, something piano turners actually use in order to know if a piano is "in tune" or not. The 386.31-cents pure (just) major third doesn't have beats. The meantone tunings used in the Renaissance and Baroque periods have thirds that are closer to this ideal, though somewhat compromised for more distant keys.
Good string players and vocalists do slightly lower or raise their half steps depending on whether the pitch is the third of a chord or a leading tone, for example, but, unfortunately, many musicians don't pay attention to this and, as a result, some choirs sing chords with thirds that have the 5 beats per second, something I find intolerable.
A great invention from the 16th century was a keyboard with split keys that allowed for the differentiation of the enharmonic notes (such as the A-flat and G-sharp you mentioned). You can hear how it works with some actual music of that time period in this video. Chromaticism (coming from the root word "chroma," meaning "color") is really much more colorful with these gradations, and it's a shame that this original knowledge of intervals was mostly lost in the Western tradition, outside specialists in early music.