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This is a different thought process than before.
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four ways of thinking about musical spaces (decoding Straus’ ch. 1) The most important idea here is that there are several different ways of understanding musical intervals. Straus’ ch. 1 offers four different approaches to conceptualizing musical space, and we will see that all can be placed on a spectrum from specific to general. To begin, we’ll take the first example in the Straus text, the opening theme from Schoenberg’s 4th string quartet.
As you can probably tell, this is not a tonal melody. There is no key signature, there is no tonic, and there isn’t even a common diatonic or referential collection that unifies all of these notes. The pitch organization of a great deal of music composed since 1900 can be described this way. It may seem frustrating that little of what you have learned in previous tonal theory courses will give you the either tools to understand such music or the vocabulary to explain it. However, you must know that musical meaning is not constrained to the tonal system. The following discussion will show you how to find meaningful intervallic correspondences in free, atonal music. Furthermore, it will show you how important it is to understand the level of abstraction (i.e., how specific or general) at which you engage a musical idea. Outside of any tonal context, intervallic names like M3, m7, M2, m6, etc. are meaningless. Without keys, spelling really doesn’t matter; for this reason, there is no difference between an augmented 2nd and a minor 3rd (or a doubly diminished 4th, for that matter). When discussing intervals in free atonal music, we will be primarily concerned with counting half steps. There are at least four ways to do so. ORDERED PITCH INTERVALS The first way of discussing the interval content of the example above is most specific. It describes the space between two successive pitches in terms of 1) direction, and 2) the number of semitones between. These kinds of intervals are called ordered pitch intervals.
opi: -‐1 -‐4 +1 +7 -‐2 +1 +8 -‐4 -‐1 -‐1 +5 As you can see, the distance from the D directly above middle C to the C# directly above middle C is 1 semitone, and since the direction is downward, the first value is negative (opi = -‐1). UNORDERED PITCH INTERVALS The second way of (re-‐)conceptualizing musical space is not too different from the first (and for this reason, many students forget to remember it—an unfortunate oversight, trust me.). This type of intervallic space still deals with distances between pitches, but it does not account for order. In reference to the ordered pitch intervals, the numbers stay the same but the positive and negative signs are gone. See the example below.
upi: 1 4 1 7 2 1 8 4 1 1 5 This might seem like a meaningless distinction, but it really isn’t. Consider, for example, the following:
upi: 1 4 1 7 2 1 8 4 1 1 5 The passage above has the same unordered pitch intervals as the passage in the example that precedes it, and although it sounds quite different, there is unarguably a similarity between the passages—one that extends beyond the vague rhythmic correspondence. Unordered pitch intervals are less specific than ordered pitch intervals, and they allow us to recognize similarities between things that are, admittedly, somewhat different.
ORDERED PITCH-‐CLASS INTERVALS Referring again to the opening passage from Schoenberg’s 4th string quartet, we can describe its intervallic content even less specifically (i.e., even more generally). This type of intervallic description (as well as the next) involves distances between pitch classes, rather than between pitches. For this reason, you will have to conceptualize intervallic relationships on the pitch-‐class clockface.
opci: 11 8 1 7 10 1 8 8 11 11 5 The values beneath the example account for the distance from one pitch class to the next on the clockface in clockwise motion. That is, starting at D (2 o’clock) one must travel 11 semitones around the pitch-‐class clockface before reaching C# (1 o’clock). Similarly, if one started at pitch class C# and travelled clockwise, one would cover 8 semitones before reaching pitch class A. This may seem complicated, but ordered pitch-‐class intervals will come in very handy in readings and discussions about twelve-‐tone music. An easy way to think about ordered pitch-‐class intervals is to put the relationships into a story (of course, you must first understand the pitch-‐class clockface from our lectures on referential collections). For example, take the first interval, which concerns the connection between pitch class 2 and pitch class 1. Think about it this way: “If I got to work at 2 o’clock and left work at 1 o’clock, how many hours did I work?” To determine the second intervallic connection between pitch class 1 and pitch class 9 you could say “If I got home at approximately 1 o’clock and played Candyland™ with my daughter until 9, how many hours did I play?” In both cases, the answer will reveal ordered pitch class intervals. UNORDERED PITCH-‐CLASS INTERVALS This final way of describing intervals is the most general, or abstract. Unordered pitch-‐class intevals reflect the closest distance between two points on the clockface. For this reason, no unordered pitch-‐class interval has a value greater than 6. Any value greater than 6 would not reflect the closest distance between two points on a clockface. (Imagine leaving Las Vegas and flying East to reach Los Angeles). While unordered pitch-‐class intervals are quite abstract, they really are rather intuitive. See the example below.
ic: 1 4 1 5 2 1 4 4 1 1 5 The term ‘unordered pitch-‐class interval’ is a mouthful. And, since we’ll be using this distinction quite a bit, we’ll shorten it to “interval class” (hence, the “ic” rather than “uopci:” below the example). SUMMARY The foregoing discusses four different ways to describe intervallic space. Ranging from most specific to most general, we defined
1. Ordered pitch intervals 2. Unordered pitch intervals 3. Ordered pitch-‐class intervals 4. Unordered pitch-‐class intervals (ic)
The figure below lists all four of the different ways for describing the opening passage to Schoenberg’s SQ #4.
opi: -‐1 -‐4 +1 +7 -‐2 +1 +8 -‐4 -‐1 -‐1 +5 upi: 1 4 1 7 2 1 8 4 1 1 5 opci: 11 8 1 7 10 1 8 8 11 11 5 ic: 1 4 1 5 2 1 4 4 1 1 5 Notice that the first way of describing note-‐to-‐note intervallic content (opi) features seven different values (-‐1, -‐4, +1, +7, -‐2, +8, +5). The second and third approaches feature six different values each (upi: 1, 4, 7, 2, 8, 5; opci: 11, 8, 1, 7, 10, 5). The fourth way of describing the note-‐to-‐note intervallic content of this passage only has four different values. While this difference is not extreme (though it certainly could be with some other excerpt), it reflects something that is essential to learning how to understand much post-‐tonal music: When we generalize, we allow for correspondences between more abstractly related elements. By doing so, we allow ourselves to recognize
more similarities and understand what gives unity to seemingly abstruse works. Let me give you one last example, in the form of a story told two ways. VERSION ONE: A VERY SPECIFIC ACCOUNT OF A TYPICAL MORNING IN MY LIFE I got up this morning at precisely 6:25 AM. I went down exactly 15 stairs and made a pot of shade-grown organic free-trade coffee, extra strong (French press). After drinking a cup and a half (with cream, no sugar) and eating a quick bowl of peaches and cream oatmeal, I put on a suit, got in my 2002 beige honda civic, drove it to my daughter’s daycare (where I dropped her off for the day) and expeditiously arrived at the parking lot to Ruebush hall where I parked in my favorite spot (usually found unoccupied before 8:30 AM), went straight up to my office, and (having prepared for lectures the night before) printed handouts and answered emails until it was time to teach at 9:00 AM. Very few of you (if any) had a morning like this. However, if I am less specific, you will find we have quite a lot in common. Consider the general version: VERSION TWO: A VERY GENERAL ACCOUNT OF A TYPICAL MORNING IN MY LIFE I got up, got dressed, and came to school. Now, it seems like we’re identical twins, doesn’t it?
