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Theory Thursday: Musical Homophones Part 4
I sort of dropped the ball on this series, but I’m back. Anyway…when I last left this discussion, we had defined 2 of our 4 answers to the question ofhow many unique triads there are (48 and 68). Today, as promised, we’ll talk about the next answer, which is 84.
To recap, we get 48 by simply multiplying the 12 unique tones of the octave by the 4 types of triads to get how many different sounding triads there are.
68 we got by adding in enharmonic spellings of black keys (e.g. F# and Gb are two different spellings of the same tone). This gave 17 possible roots x4 possible chord types. And this set is the set of chord spellings for every chord which cannot be spelled in a simpler way (but may have 2 equally simple spellings).
So, let’s talk 84. As it turns out, black keys aren’t the only ones that have multiple names. When you first start learning music theory you might have learned that there is no such thing as B#. But that’s not exactly true. B# does exist, it’s just enharmonic to C. To sharp a note really just means to raise it by half a step (the opposite is true of flatting). This can be done even if there is no adjacent white key to raise or lower it to. So the note B# is actually the white key we would normally call C. Like any note, though, we can build triads off of it. In this case, B# diminished would be (B#, D#, F#). C diminished on the other hand is (C, Eb, Gb). The same keys. Just as the two notes sound identical but are spelled differently, so too the chords we build off of them. So, to our list of possible roots, we can now add B# (enharmonic to C), E# (enharmonic to F), Cb (enharmonic to B), and Fb (enharmonic to E). 4 more roots x 4 triad types yields 16 more possible triads, which gets us to 84.
But the weirdest is yet to come. Next time we’ll talk about how there are actually an infinite number of triads.
I Wish People Would Stop Saying Music Helps You With Math...Even Though It's True
You know what nobody ever says?
”Learning math is important. It can really help you become a better musician.”
It’s an undeniable fact. Music theory is mostly mathematics. Tone frequencies are spaced logarithmically. Intervals, and the chords they form, are mathematical structures. Scales, keys, and the functional relationships contained within them are set theory. Understanding complex rhythmic subdivisions requires intuitively understanding the process of multiplying fractions. And on a more abstract level, music theory is about symbol manipulation in the same way mathematics is symbol manipulation.
Being good at math will definitely make you a better (or at least a more intuitive, more thoughtful) musician. So why doesn’t that ever come up? Its opposite does. I can’t tell you how many times parents of new students have told me they want their child to learn an instrument because they’d heard it helps with math.
It does, of course. Study after study has demonstrated this, even once confounders like selection bias are controlled for. But why does this motivation always seem to go one way and not the other. The difference highlights an important discrepancy in the way we assign value. Increasingly over the past 70 years (hint: it has something to do with the Cold War, but this is my music blog not my history blog) STEM (Science, Technology, Engineering, and Mathematics) has largely become the area of study by which the utility of all others is judged. Today, we find ourselves in a position where only the STEM fields, and to a lesser extent the skilled trades, are seen as having intrinsic value. The humanities, and even more so the arts, are seen as largely trivial. And so, to avoid being relegated to complete obsolescence, they have learned to defend themselves as instrumental (pardon the pun) to STEM in some way.
I saw this in my other life as a college history professor as well. Learning about the past is seen almost as a curiosity, certainly not as a social necessity. “Enriching” at best, but certainly not “essential.” History is the purview of majors (who presumably will go on to teach it because collectively we have no other frame of reference for what else history might be good for), and electives (which non-majors take for fun or to satisfy curiosity). And music is no different.
Here’s a thing my adult students DO say, and frequently. They tell me “I wish I’d learned the piano when I was younger.” Or the guitar. Or whatever. And I think that’s telling.
Music is universal. It’s one of the few things that EVERY human civilization we know of has in some form. It is a language for conveying ideas and emotions that is millennia older than the written word and, depending on your definition of each, at least the same age as spoken language. It’s one of the first things babies recognize, and one of the first ways that they express themselves (I know from personal experience as a new father). It expresses and translates feelings that words often fail to capture. It accompanies every major milestone in life, and is often the most visceral and evocative part of what we remember about those times. It can move us to tears either all by itself or in combination with such memories. Music is one of a very short list of things that is universally human. If mathematics is the language of the universe, music is the language of the soul. Or of consciousness, or of humanness, or whatever. Surely that has value.
But of course, that value isn’t monetary. I mean, sometimes it is. It pays my bills. And some people are able to translate it into significant wealth. But this is the exception. Music is not a safe route to financial stability. We intuitively understand that the STEM fields are marketable, whereas music is not. Math is a safe(ish) route to material comfort. But so what?
The number of adult students I have who express regret at having not learned an instrument earlier is indicative of the intrinsic value of the joy and self-actualization playing music could have brought to their lives (and still can. It’s NEVER too late. I promise). Surely giving your children the tools to be happy is at least as important as giving them the tools to be comfortable.
And none of this is to say that the physical world that the STEM disciplines describe doesn’t have its own inherent beauty. In many ways, I relate to pure math the same way many of my adult students talk about music. I wish I’d put in the effort to really get it when I was younger. It reveals, and describes such wondrous, transcendent, beautiful things about existence. And I’ve learned to love it as an adult, not just because learning about mathematics and training my mind to be more mathematical has made me a better musician, but also for all those other reasons. One thing I believe very strongly is that all knowledge is beautiful on its own terms, and for its own sake.
But when you defend music or music education on the grounds that it helps with math, you are NOT defending music. You are ghettoizing it and instrumentalizing it, and alienating it from its own most important reason for being, which is that it is beautiful and wondrous and reveals truth and describes the contours of the human experience in ways that absolutely NOTHING else can. When you defend it on any other grounds, you are conceding the most important point. If the study of music (or any other of the arts or humanities) doesn’t have intrinsic value, then it doesn’t have any value at all.
I guess what I’m trying to say is…you should study math. It will make you a better musician. I’m serious.
Theory Thursday: Musical Homophones Part 3
Theory Level: Intermediate
Ok, so let’s put our Bb Augmented chord aside. We’ll come back to it. Two weeks ago, I posed the question: How many different triads are there, and I proposed 4 different answers. Those answers were 48, 68, 84, and infinitely many. So let’s look at each of those answers in turn.
Answer #1: 48.
This one is simple. As we discussed last week, there are 4 different triad qualities, diminished, minor, major, and augmented, which each come from stacking a combination of two thirds on top of one another, (min+min), (min+maj), (maj+min) and (maj+maj) respectively. These are the only possible ways to stack two thirds. There are 12 tones in the octave, 4x12=48. So there are only 48 possible combinations of tones that create a triad (again excluding inversions, doublings and spread voicings but that was part of the original rules).
Answer #2: 68
The first answer is good enough if what you want to know is how many combinations of tones can be combined to create triads. But that isn’t the full story because sometimes 1 set of tones can have 2 different names, and what you call it matters to how you understand the role of a particular chord in a particular piece of music (I’m going to come back to this concept but for now just trust me. It matters). For instance, if I build a major chord (4+3) from an F# root I get F#,A#,C#. But that root note can also be called Gb, and if I spell it that way I get Gb,Bb,Db. These chords sound exactly the same because they contain the same tones (sounds) but are spelled differently.
This is what we mean by musical homophones. In linguistics we can have two different words that sound the same but are spelled differently and have different meanings (e.g. right and write). In speech they are interchangeable, but if you write(wink) the wrong one, you are wrong. Music works the same way. F# Major and Gb Major sound the same but are spelled differently and have different meanings (function).
So, if we treat each black key as having 2 common names, we get 17 possible roots (5 black keys with 2 names each and 7 white keys with 1 name each). 17x4=68.
Those other two answers get even stranger. But I’ll get to that next week.
Theory Thursday: Musical Homophones Part 2
Theory Level: Intermediate
It’s Bb Augmented. That’s the answer to last week’s question. The only triad that is properly spelled with both a sharp and a flat. In this case Bb, D, F#.
But why is such a seemingly normal chord the one exception? Well, right off the bat, if you know a little theory, you probably should have guessed it would be an augmented chord. After all, every major, minor or diminished chord is diatonic in at least one major key, and since no major key signature contains both sharps and flats, it stands to reason that no major, minor, or diminished chord could possibly satisfy the requirement. And that only leaves augmented chords as possible answers. But I’ve already lost half of you, so let’s back up.
The building block chords of tertian harmony (the most common system of western harmony) are triads, made by stacking two thirds on top of one another, that is starting on a root note, going up two letters, then going up two more letters (e.g. C, E, G, which is C Major). However, knowing that we go up two scale degrees (letters) doesn’t tell us everything we need to know about a chord, because notes can be sharp, natural, or flat. And thirds can be either 3 half steps (a minor third) or 4 half steps (a major third). Half steps are the smallest interval in standard western tuning, equal to a single piano key (white or black) or a single fret on the guitar. The closest two notes can be to one another in standard tuning is one half step, and a third can contain EITHER 3 OR 4 half steps. So if I go up a minor third (3 half steps) from A I’ll get C but if I go up a major third (4 half steps) I’ll get C#. Both are thirds and both are 2 letters apart. This is an important concept for later.
So, if we go up a third and then another third, there are 4 possible combinations of distances between our 3 notes. 3+3 (which we call a diminished chord), 3+4 (minor), 4+3 (major), or 4+4 (augmented). Using the example root C we can get C diminished (C, Eb, Gb), C minor (C, Eb, G), C major (C, E, G), or C augmented (C, E, G#). But notice that no matter which triad we build off the root C we will end up with the same 3 letters: C, E, and G. That means that since there are only 7 letters in our musical alphabet there are only 7 combinations of letters that can form triads, and memorizing them is pretty easy. This makes tertian harmony very intuitive if you understand the basic theory, and is really quite a brilliant way to organize pitches and the chords formed from them.
It also means that even though two notes might be enharmonic to one another (the same piano key), for instance C# and Db, the two are not interchangeable when spelling chords. Hence, in our Bb augmented, we cannot spell the chord Bb, D, Gb, because even though F# and Gb are the same key on the keyboard, from a spelling standpoint, Gb is three letters above D, not 2 like F#. No G note can ever be part of a triad with Bb as the root. So F# is the only correct spelling. But that still doesn’t tell us why this one chord is so special.
Next week…
Theory Thursday: Musical Homophones Part 1
Theory Level: Intermediate
Here’s a question with an ambiguous answer. Ignoring inversions and doubled voicings and multi-octave range, how many distinct triads are there?
There are at least four legitimate answers to this question, and maybe more (if you come up with a different answer, I’d love to hear about it in the comments). The four possible answers are: 48, 68, 84, and infinitely many. The difference depends on what you count as a legitimate chord root, and whether you care about your answer’s real-world usefulness. BTW "There are multiple answers and one of them is infinity but that answer is a philosophical thought experiment with no practical application” is a commonly recurring theme in music theory. We’ll get to the why of each of those answers a little later in this series.
But first, I want to ask another question that only has one right (and very odd) answer. How many triads have both a sharp and a flat in them?
No matter what your answer to the first question is, the answer to this one is 1. There is exactly 1 triad that is correctly spelled with both a sharp and a flat. And that’s weird. Music is math and math generally hates rules that are universal except for a single exception. How can it be that “triads may contain either sharps OR flats but not both” is a maxim that holds in all cases except 1? And how can it be that that single exception exists whether the whole set is 48 or infinity? What is going on here?
To answer that question, we’re going to have to dive deep into the internal structure of tertian harmony, and talk not just about music and math but also about language and symbolic structures, and we’re going to make pit stops at a number of other weird, anomalous musical roadside attractions, before finally winding up talking about one of my favorite all-time theoretical concepts, and how to use it to make your composition/songwriting better. See you next week.
Oh, but before I go…any guesses what it is? What is the one and only triad that is correctly spelled with both a flat and a sharp?
Comment with the answer. You have seven days.