Some Nuts And Bolts Of Music Theory
When I started learning music seriously I was a teenager. I’ll turn 54 in a couple of weeks, and I’m still figuring out all this stuff, despite (or perhaps because of) being a professional musician and music teacher for three decades. Having a 7-year-old daughter is an enormous help.
In high school I took my first music theory class. The teacher’s name was Mr. March, which should have been a clue. The first day, he said to the class, “I’m going to test your musical ears.” He told us to take out a piece of paper. Then he said, “I’m going to play two intervals on the piano. You write down which is bigger, the first or the second.”
Then he turned his back to us and pressed some keys on the piano.
I did not have a freakin’ clue what was going on.
I did not recognize that he was hitting two keys simultaneously. What I heard was a series of sounds. What did he mean by “which one is bigger”? I’m pretty sure I just gave up on the exam.
Mr. March was operating under some default assumptions that were never stated. This is not uncommon in teaching, and it’s practically a given in music teaching, where teachers are distressingly likely to start where they are, rather than where their students are.
Here’s what I tell students who want to learn about music theory.
Musical sound concerns itself with vibration within the frequency range that our ears can perceive. Vibrations outside that range don’t get picked up by our ears, so we won’t talk about them.
Some vibrations have periodicity. Others do not. An example of the first kind is a tone played on a flute; an example of the second kind is crumpling a sheet of paper.
While musical performance uses both types of sounds, the study of harmonic relationships is only concerned with periodic sounds — the ones with identifiable frequencies, usually measured in cycles-per-second. Sounds with identifiable frequencies are called tones. If you take a series of rhythmic impulses and speed them up, they will turn into tones.
If you have two tones with the same frequency, they are in a very specific relationship. Their numbers match; they are in a 1-to-1 ratio. The musical term for this relationship is unison.
If you and I sing the exact same note, our vocal chords are vibrating at the exact same speed, and we are singing in unison. If we’re almost but not quite at the exact same speed, the frequency ratio between our voices changes from 1:1 to something more complicated. 189.235147 : 193.772121 is almost the same as 190:190 (which reduces to 1:1) but it’s a more complex relationship — and it’s perceived by our ears as “out of tune.” Obviously there are a lot more ways to be out of tune than to be in tune!
If you have two tones in the frequency ratio 2:1, their numbers no longer match, but their relationship is still simple. One vibration moves twice as fast as the other. The musical term for this relationship (in Western musical tradition) is octave.
Notice that the term “octave” means “eight,” which has absolutely nothing to do with the actual mathematics involved.
To our ears, the frequency of any power of 2 seems to have the same “quality” as any other. Notes an octave apart are given the same name in nearly every world musical system that goes so far as to name the notes in the first place. This means that experientially, 2:1, 4:1, 8:1, 16:1… are all identical 1:1.
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Musical intervals can be quantified in various ways.
Keyboard or melodic distance simply measures how far you have to move your finger to get from one member of an interval pair to the other. From the lowest A on the piano to the highest is a finger distance of about a meter and a half. From “middle C” to the C-sharp immediately above it is a finger distance of about a centimeter. By this measure, the first interval is significantly “bigger.”
Ratio size just addresses the distance between the two numbers, and it maps nicely onto the melodic distance measure. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 (n.b., if you know this already, you also know that on the piano, thanks to the baffling miracle of equal temperament, this statement is untrue. Bear with me for the purposes of discussion, ‘k?). 128 to 1 is a bigger jump than 16 to 15, so the first interval is significantly “bigger.”
Harmonic distance, on the other hand, measures the complexity of the ratio involved. From the lowest A to the highest is a ratio of 128:1; from middle C to the adjacent C# is a ratio of 16:15 — but 128:1 reduces to 1:1, and 16:15 doesn’t reduce. An eight-octave jump has a harmonic distance of zero, while a “semitone” has a much greater harmonic distance. So when we use this measuring system, the second interval is “bigger.”
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All harmonic intervals can be described as frequency ratios. Here are some of the ones we use most often:
3:2 is described in Western musical terms as a “fifth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol / 1-2-3-4-5), which has nothing to do with the actual mathematics involved.
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4:3 is described in Western musical terms as a “fourth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa / 1-2-3-4), which has nothing to do with the actual mathematics involved.
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5:4 is described in Western musical terms as a “Major Third.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi / 1-2-3), which has nothing to do with the actual mathematics involved.
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5:3 is described in Western musical terms as a “Major Sixth.”
Notice that the Western term describes the scalar or melodic distance (Do-Re-Mi-Fa-Sol-La / 1-2-3-4-5-6), which has nothing to do with the actual mathematics involved.
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As my little videos demonstrate, rhythmic impulses turn into pitch when you accelerate them. If you record yourself tapping 2-against-3 for an hour, then accelerate the recording by multiple orders of magnitude, you’ll wind up with two tones a fifth apart.
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You don’t need to know about frequency ratios to use them effectively (just listen to the Beatles and you’ll hear some dynamite frequency ratios rendered with exquisite fidelity by people who never gave the math a moment’s thought). Most composers don’t know. Most musicians don’t know.
So why bother?
Speaking personally, I can say that learning all this has transformed my experience of music. I can spend a long time perfecting the tuning of a single interval — precisely because I have learned to perceive it as a source of deep experiential insight into simple mathematical relationships. Why bother? Because it’s cool; because it’s beautiful; because it’s universal.
Okay, that’s all for today.
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Music at home…
…Daughter and I have exchanges about music theory. She calls them “wacky questions,” and enjoys it when I give her puzzles about harmonic relationships. “If A is ONE, then what is the TWO chord? The FIVE chord?” “Spell a G major triad.” Etc., etc.
Recently we began moving into questions about harmonic sequences. “In the key of C, what is a I-IV-VI-V-I progression?”
She’s seven. I don’t have any huge expectations about this; it’s just a fun game we play. This is way out of her league.
Or is it?
At tonight’s guitar practice I was coaching her into a D-minor chord (the standard one at the bottom of the neck). She started playing a sequence, not too adroitly…and when I tried to steer her in the direction of something I had planned, she said, “Stop! I want to play my own progression!”
Then she dictated: “D minor, A minor, C, A minor, D major, G, A major, D.”
I did a little on-the-spot voice-leading to make two harmony parts and we sang through them. Cool. My daughter’s composing her own chord patterns.
Then she told me to “write it down, so we don’t forget it.”
I think it’s time to show her more about notation.
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The Tony Schwartz Music Exchange Tape
In the mid-to-late 1970s, I lived in group houses with a broad assortment of interesting people. One of them was Seth Deitch, who had as part of his vast array of stuff an assortment of reel-to-reel tapes inherited from his father, the brilliant animator Gene Deitch.
Eventually we acquired a reel-to-reel machine and began the process of dubbing all these tapes onto cassette. They were in poor condition, so this amounted to a rescue operation.
Some of the material was old jazz, some of it was old radio commercials; one reel contained a set of 1949 performances by John Lee Hooker that many years later got released as “Jack Of Diamonds.”
And one reel held this extraordinary document:
Tony Schwartz, master of electronic media, created more than 20,000 radio and television spots for products, political candidates and non-profit public interest groups. Featured on programs by Bill Moyers, Phil Donahue and Sixty Minutes, among others, Schwartz has been described as a “media guru,” a “media genius” and a “media muscleman.” The tobacco industry even voluntarily stopped their advertising on radio and television after Schwartz’s produced the first anti-smoking ad to ever appear (children dressing in their parents’ clothing, in front of a mirror). The American Cancer Society credits this ad, and others that followed, with the tobacco industry’s decision to go off the air, rather than compete with Schwartz’s ad campaign.
Born in midtown Manhattan in 1923, a graduate of Peekskill High School (1941) and Pratt Institute (1944), Tony Schwartz had a unique philosophy of work: He only worked on projects that interested him, for whatever they could afford to pay.
{snip}
For many years he was a Visiting Electronic professor at Harvard University’s School of Public Health, teaching physicians how to use media to deal with public health problems. He also taught at New York University and Columbia and Emerson colleges. Because Schwartz was unable to travel distances, he delivered all out of town talks remotely. Schwartz was a frequent lecturer at universities and conferences, and gave presentations on six of the seven continents (not Antarctica). He was awarded honorary doctorates from John Jay, Emerson and Stonehill Colleges.
{snip}
“Documenting life in sound and pictures” is something Tony Schwartz begin in 1945, when he bought his first Webcor wire recorder and began to record the people and sounds around him. From this hobby developed one of the world’s largest and most diverse collections of voices, both prominent and unknown, street sounds and music, a collection that resulted in nineteen phonograph albums for Folkways and Columbia Records.
During the 1950s, Tony Schwartz sent this recording out into the world, presumably under an early version of a Creative Commons license.
While I was already getting interested in what was then called “Ethnic Music,” this recording was something completely different — dozens of different songs from all over the planet, each introduced by the same voice. I must have listened to the Tony Schwartz Exchange Tape a couple of hundred times over the next few years, but time marched on and the dubbed version of the Tony Tape came to rest in my collection alongside hundreds of other cassettes. In the early 2000s I duplicated it onto a CD, where it continued to lie dormant.
I bumped into Tony Schwartz’ name a few times on various Folkways lps, but never learned much about the man until I started listening to the Kitchen Sisters’ wonderful “Lost And Found Sound” series — and then I had a delightful shock of recognition. Give this audio portrait a listen.
Anyhow, I’ve been transferring all my sound files to my computer, and this one finally had its turn…and I says to myself, says I, “Well, this certainly deserves to be out in the world.”
Here you go, world.
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Pune Concert, August 20, 2011
This concert was arranged by Chaitanya Kunte, the extraordinary musicologist, composer and harmonium virtuoso.
It was a pleasant and unusual experience to have two melodic accompanists — Chaitanyaji on harmonium and Eeshan Devasthali (my Guruji’s grandson) on violin. Milind Pote provided the rock-solid and very sympathetic tabla sangat.
Ragas:
Shyam Kalyan
Puriya Dhanashri
Tilak Kamod
Kafi
Bhairavi
Here’s the concert, embedded as a single playlist:
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Nasik Concert, August 19, 2011
Finally getting around to uploading and embedding the concerts from last summer’s trip to India. Here is the concert from Nasik embedded as a single playlist, leading off with Puriya Kalyan, and including Mian ki Malhar, Kafi tappa, Tilak Kamod, Khamaj, Pahadi and Bhairavi.
I greatly enjoyed this evening. Nitin Ware’s accompaniment was extremely solid, and Dyaneshwar Sonawane gave very supportive sangat on harmonium.
Note the cascade of inaccuracies in the news clipping. I began studying khyal in 1977, went to India first in 1985. I never studied with Nana Joshi, who was my Guru’s first teacher. Etc., etc., etc.
I’m grateful to Asmita Sevekare and her father for arranging this program. With luck I’ll go back there again next year.
This review is remarkable for its near-complete inaccuracy!
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In case anyone wondered, this is the first reason I’m vehemently pro-choice:
This was uploaded to the 1in3Campaign’s website. Please go there and participate as fully as you can.
Thanks.
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Loaves Of Truth…From My Brother, The Baker
This is not an article about bread. But bread is central to what I want you to read. So just nibble briefly at the first few paragraphs, and plunge downward to find words that taste of truth.
A few years ago, my brother left his career in academia. Weary of endless tenure battles and internecine squabbles between departmental factions, he sought another path.
And in Wide Awake Bakery, he found it.
We live and bake in Mecklenburg, just a few miles from Ithaca, Trumansburg, and Watkins Glen. We love the place and we love the people–and that’s what this bakery is about. We work with local flours, grown just a few miles away in Newfield, Lansing, and Brooktondale, and we work with our local flour mill, Farmer Ground Flour.
Here’s a little snippet from their F.A.Q.:
Why should I care if my bread is made by hand?
Most of the bread, even “artisan bread,” sold nowadays is untouched by human hands. Mixed, shaped, and baked by machines, the bread is flash-frozen and warehoused before being shipped to distribution points where it is eventually warmed (“baked”) in display ovens. Modern bread factories are consistent and they produce a pretty good product, but they require huge quantities of standardized ingredients. Bread factories simply don’t have the flexibility to bake with small crop runs and local grains. Big factories, big farming.
Small local bakeries where bread is made by hand offer something else entirely. We can vary our baking quickly in response to our flours and our customers. When a farmer comes to us with a batch of fruit, for example, we can quickly make it part of your share. We can keep a close watch on the fermentation process and give it the care it requires. We can change our baking to make more of the breads that you particularly like. Hand made bread generally tastes better, and is better for our communities, than factory made bread.
There are other reasons to care about how your bread is made. We bake our bread as if we were going to eat it ourselves, and feed it to our friends and family—because that’s exactly what we do. Baking bread, good bread, is one way we try to take care of our community. It’s a great feeling to know that the pile of flour we start with in the early morning has turned into shining loaves of bread that are nourishing and pleasing our neighbors. Every week we read about another food crisis, and it’s clear that we’ve got to start making food differently. That’s what we’re trying to do.
But I didn’t invite you here to tell you about my brother’s right livelihood, wondrous though it is. Or about his bread, which is mindbendingly delicious. Or even about the fact that those fabulous loaves fed the Occupiers in Zucotti Park.
Commendable though these things are, they’re not what prompts this post, or what’s brought me to tears of pride.
I want to share what he said to the assembled crowd at Monday’s day of action against hydrofracking in Albany, NY.
After handing out over 200 loaves of bread to the assembled crowd, my brother spoke. (I am reproducing his words in full. We’re family; it’s cool.)
My name is Stefan Senders, and I am a baker. Beside me are Thor Oechsner, an organic farmer, and Neal Johnston, a miller. We work together.
Today we bring bread to Albany to intervene in the self-destruction of the great State of New York. We come, Farmers, Bakers, and Millers, to remind our state and our Governor, Andrew Cuomo, that despite the promises of industry lobbyists, the exploitation of Shale Gas in New York is bad and broken economy of the worst kind.
This bread is the product of our community and our farms. The wheat, grown, tended, and harvested by our local organic farmers, is fresh from the soil of New York. The flour, ground in our local flour mill, is as fine as concerned and caring hands can make it.
To resurrect a term long since emptied by advertisers, the wheat, the flour, and the bread are wholesome: they bring our communities together, give us work, nourish us, please our senses, and make our bodies and our land more healthy.
This is good economy. It is wise economy. It is a steady economy that nourishes the State of New York.
We know that for many New Yorkers, Fracking sounds like a good idea. We have all heard the fantastic tales: Fracking, it is said, will save our state from financial ruin, release us from our dependence on “foreign oil,” and revive our rural economy by bringing cash, if not fertility, to our once vibrant farmland.
For politicians, these stories of money and growth are hard to resist: the numbers are large, deficits are unnerving, and elections are expensive.
For many farmers and land-owners, the promises of cash are dizzying, and to risk the land’s fertility to extract gas is only one step removed from risking the land’s fertility to extract a few more bushels of corn or soybeans.
But farmers might know better.
Farming has not always been, and need not be, an extractive industry. There was a time when farmers worked with a longer view, keeping in mind their role as stewards and caretakers of the land. That long view is the farmer’s wisdom, and it is as good and wise today as it ever was.
The promises of the gas industry are demonstrably false, and they miss what farmers know well: There is no independence that does not demand care and responsibility. There is no quantity of cash that can restore fertility to a poisoned field. There is no adequate monetary “compensation” for poisoned water. There is no payment, no dollar, no loan, that can restore life and community to a broken world.
Our work and the work we provide others—on the farm, at the mill, and at the bakery—depends on fertile soil, pure water, and a viable community. All of these are put at risk by Fracking.
What happens to our land in an economy bloated by gas exploitation? Prices rise, rents rise, and good arable land becomes scarce as acres once leased to farmers are set to quick development schemes—flimsy housing, storage barns, parking lots, and man-camps.
And what happens to our water when gas exploitation takes over? Storage pools, as safe as the Titanic was unsinkable, overflow, contaminating the soil; inevitable leaks in well-casings allow gasses and Frack-fluids to pass into our aquifers, into our bodies, and into the bodies of our children.
And what happens to communities held in thrall to gas exploitation? As we have seen in other parts of the country, the boom-bust cycle of the petroleum economy fractures communities, undermining our capacity to act wisely and civilly.
With every boom, a few get rich, a few do better, but all are impoverished. For every hastily built motel there are dozens of apartments with rising rents; for every newly minted millionaire there are many dozens who see nothing but the pain of rising costs and receding resources. For every short-term dollar there are hundreds in long-term losses that can never be recouped. To go for gas is to go for broke.
With this bread we are here to remind you that there is another economy, one that works.
This bread symbolizes a commitment to the health of New York State. It embodies the knowledge that good work, not a gambler’s dream, is the basis of a sound and sustainable economy.
This bread symbolizes the farmer’s simple truth that without fertile soil, without pure water, and without strong community, we go hungry.
This bread reminds us all that the promises of gas exploitation are empty: What are we to grow in fields broken by the drill and tilled with poison? What are we to feed our children when our water and wheat are unfit? Shall we grind money to make our bread?
We do have a choice. We need not poison our land to live. We need not taint our water to drink. We need not sell our future to finance our present. These are choices, not inevitabilities.
With this bread we say: take the long view; pay attention to the health of the soil and nourish it; treasure pure water; remember the value of your community and keep it whole.
If something must be broken, let it be this bread, not shale. Break bread, not shale!
I can’t add anything more to his words. Beautifully spoken, bro. I love you. And your bread.
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So true…
While I haven’t heard all of these, I’ve definitely heard a lot of them.
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I Have Fourteen Minutes, Fifty-Eight and a Half Seconds Left In My Allotment.
A friend posted this clip on Facebook, and something about the scene rang a bell.
Ah-hah! It was the first half of the “Concert For National Integration” at Shanmukhananda Hall in Bombay on Republic Day, 1986. I was in the audience; I had traveled from Pune along with Bhimsenji, who was singing a duet with Balamurali Krishna in the second half of the event.
Why is it interesting? I mean, honestly, most of these Hindustani/Carnatic jugalbandis aren’t that satisfying. This one’s no exception; I’m including it in this post for the sake of completeness. The whole ending frenzy is IMO totally inexcusable.
The next day Bhimsenji and his accompanists flew back to Pune; I was a member of the party. I received more than the usual amount of respect from airport personnel, who seemed to go out of their way to greet me courteously. On arriving in Pune I found out why.
Doordarshan’s cameraman had found me in the crowd, and given me a full-face closeup, which is reproduced at 33:47 in the televised video:
I was famous!
On seeing this, my wife commented that I looked like someone who was forcing himself to enjoy something against his own will. That sounds about right.