This will be my 88th and final musicellanea post. The total of 88 total posts matches the total of 88 keys of a standard modern piano keyboard. Furthermore, of the 12 equal-tempered pitches within an octave, C is understood as the (mostly arbitrary) starting point. Analogously, of the 12 months in the modern Western calendar, January is understood as the (mostly arbitrary) starting point. This blog's monthly posting schedule matches this mapping from C to January, since I began on October 2013 (analogous to A0, the lowest note on the modern piano) and ended on January 2021 seven years and three months later (analogous to C8, seven octaves and three half steps higher, the highest note on the modern piano).
I will conclude this blog with a lightning round of three unrelated miscellaneous musical observations.
1. The syntonic comma is the difference between four pure 3/2 perfect fifths -- like C4 up to E6, via G4, D5, and A5 -- and two pure 2/1 octaves plus a pure 5/4 major third -- like C4 up to E6, via C5 and C6. As a fraction, and as an ascending interval, the syntonic comma is 81/80, which is (3/2)^4 / ( (2/1)^2*(5/4) ). As a decimal, this is the rather humdrum 1.0125. As a fraction, and as a descending interval, the syntonic comma is 80/81. As a decimal, this is 0.987654320 repeating, but can be rounded to 0.987654321. How fun that humans have n digits, and one of the most significant musical commas is (n-1)^2-1 / (n-1)^2.
2. In Wagner's Tannhäuser, the title character sings his "Hymn to Venus" three times in Act 1. The first time, the music is in D-flat major with a tempo indication of half note = 69. The second time, the music is in D major with a tempo indication of half note = 72. The third time, the music is in E-flat major with a tempo indication of half note = 76. Not only is this music increasing in intensity by getting slightly higher and faster with each iteration, but the ratios by which each are increasing are rather close to one another: the pitch is increasing at a ratio of around 1.059 (the equal tempered half step), and the tempo is increasing first at a ratio of 1.043 (72/69) and then around 1.056 (76/72).
3. In the one-measure piano introduction to Ives's song "Tom Sails Away," my favorite moment is near the end of the measure, the moment occupied by the sole F, encircled in the music below. If I were playing this music, I would aim to use a slight change of dynamics and/or microtiming to give this seemingly uninteresting F some special treatment.
One the one hand, this F breaks a pattern. The image below teases out some three-note motives with a melodic major third (shown with a yellow arrow) in some direction overlapping with a melodic tritone (shown with a purple arrow) -- either before or after -- in the same direction. Two of these motives occur in the first half of the measure in immediate succession, the second a major seventh lower than the first. Had the F4 of my favorite moment been an F#4, then two versions of this same motive -- one retrograded, the other inverted -- would have occurred in the second half of the measure in immediate succession.
Furthermore, by using F natural instead of F# at this moment, the first measure contains all 12 pitch classes (the 12 pitches in the equal-tempered chromatic scale without attention to register) besides F#, as shown in the graphic below. (One octave of keyboard with the standard white-black key coloration is shown on the left. The image below also replaces the major-third yellow arrow with a minor-third green arrow.) The second measure completes the set of 12 pitch classes, with F#s prominently placed as the lowest note and the first note of the treble melody. Therefore, by breaking from a pattern, attention is drawn toward something that is incomplete that later music makes complete.
On the other hand, the F continues a pattern. The top voice in the first measure plays a mi-re-do dotted-rhythm motive that overlaps with its transposition down a major third -- again, this interval is shown with a yellow arrow -- producing five of the six notes of a whole-tone scale. As shown below, the F at my favorite moment completes this whole-tone scale, albeit down the octave. A continuation of this sequential pattern would transpose the motive down another major third to G to F to E flat. The vocalist enters with something very close to this -- G# to F# to E -- which is down a minor third (plus an octave) instead of a major third. Once again, the green arrow replaces the yellow arrow.
However, G to F to E flat does occur later, to begin the last vocal line, which occurs at the end of the song, as shown below. When the first vocal entry in the second measure breaks from a pattern, once again attention is drawn toward something that is incomplete that later music makes complete.
OK, that's it for this blog. I hope that you got something out of it. Thanks to Stephen Soderberg for suggesting this idea to me in the first place.
I have one last thing to share: hidden among some of the post titles of this blog is a message.
