Architecture as a Sequence of Harmonic / Melodic Space pt1

The following is taken from a paper entitled "Frozen Music" that I wrote in my independent study with Professor Ronit Eisenbach, during the fall 2007 semester at the University of Maryland.

The famous quote “Architecture is Frozen Music” seems to be an oxymoron; music relies on the notion of time, so if music is frozen in time, it is non-audible. Some believe that the key to unravel the connection between both arts, that the quote suggests, lays in the phrase Frozen Music; if one understands the musical notes not as vibrations that resonate through time, but as a series of proportional ratios between two notes playing together in harmony.

There are twelve notes found in the traditional Western music vocabulary. Each of these notes is a semitone apart from one another, creating what is known as the Chromatic
scale. The notes in this scale are commonly known as: 

A - A# / Bb - B - C - C# / Db - D - D# / Db - E - F - F# / Gb - G - G# / Ab

A semitone (also known as a half-step) is the interval between each note and its successor in the chromatic scale, for an example; a semitone after the A note would be the A# / Bb note, or a semitone after the B note would be a C note. A wholetone is equal to two semitones, for example; a whole step from an A note would be a B note, or a wholetone from an E note would be an F# / Gb note.

Western music are not based on the Chromatic scale, but uses a variety of Diatonic scales. The difference between them is that while the Chromatic scale relies on the 12 notes being played successfully at semitone intervals, the Diatonic scale incorporates a mixture of both semitone and wholetone intervals. A Diatonic scale generally consists of seven notes and repeat at the octave. Western music in the Medieval and Renaissance periods (1100-1600) tends to use the white-note (natural major) diatonic scale C-D-E-F-G-A-B and its transpositions. Notice that this scale is made of Whole-tone and Semi-tone intervals, more specifically, two Whole-tones / one Semi-tone / three Whole-tones / one Semi-tone.

Rameau discusses that the reason for that is that these are the intervals that are easily proportionally extracted from a length of string. As documented in his Treatise on Harmony, a length of string that produces a certain pitch could be easily divided up to create different intervals of that pitch.
 
Let’s assume that we have a piece of string held at two points that produces that pitch C. If the length is divided into two equal parts, the pitch of each side of the string also produces a C pitch, except an octave higher. This pitch is the tonic degree of the scale, meaning the first note of the musical scale.
 
Now lets divide the string into three equal parts and pluck the string at the point where the resultant length is 2/3 times the original length. The pitch produced would be the 5th note in the scale, in this case the G note. This pitch is the dominant degree of the scale.
 
If the string is divided into four equal parts, and the string is plucked at the point where the resultant length is 3/4 times the original length, the pitch produced would be the 4th note in the scale, the F note. This pitch is the sub-dominant degree of the scale.

Note that while F is the 4th degree after C, C is the 5th degree after F. If you play the sequence of pitches C - F – C (octave higher), you will be moving 4 degrees to the sub-dominant and then 5 degrees to reach the octave. The same is reversed with the G; while the G is a 5th degree to the C, the C is a 4th degree to the G. This also makes sense mathematically, if you multiply 2/3 to 3/4 the answer would be 1/2. The tonic - dominant - subdominant degrees are the most vital in any musical scale; they are sometimes called perfect tones and the most commonly used chord, the 5th chord (or power chord) is derived from these notes; either using the tonic and the dominant with the tonic in the bass note, or using the subdominant and the tonic with the subdominant in the bass (both examples are 5 degrees apart from one another). Such chords are vital because the note that completes the chordal triad, distinguishing the major and minor chords (the 3rd degree) is absent, so the chord could be applied to both major and minor scales. If the string is divided further, into 5 or 6 parts, the 6th (Submediant) and 3rd (Mediant) degrees would be produced. 

Professor Eisenbach's website: www.roniteisenbach.com