A few weeks ago, I completed the second major assignment for MHST 537 (Teaching Music History): Substitute teach (or “guest lecture”) for another professor at NEC; videotape your teaching and analyze it. I had the good fortune to substitute for Larry Scripp; he had to travel out of town for the latter half of his MIE 501 (Intro to MIE), so I stepped in.

The agenda I set forth for my teaching was based on an assignment Larry wanted me to give to the class: to get his students familiarized with the CMIE NewsBlog, as readers and potential writers. I worked backwards from his assignment to plan what basic learning outcomes I hoped my students would achieve—an understanding for what makes NewsBlog writers’ postings different from the “rants” that are commonly associated with blogging; a rationale for organizing the kinds of ideas and documentation that get shared on the NewsBlog; and a sense of direction—where, beyond the NewsBlog or MIE program, does this kind of documentation and writing have use and purpose?

Where’s the Video Documentation?

Although I am not able to post my video of teaching here, due to length and filesize, any readers of the NewsBlog who are interested should read the transcription file (posted as a PDF here). In fact, anyone who reads the transcription file will notice that parts of it are highlighted and color-coded; this is a technique for analysis that we encourage MIE students to undertake.

Transcribing, Coding, and Analysis

The process I have engaged myself in—of videotaping my teaching, watching it, transcribing it, coding it for objectivity, and finally analyzing and reflecting on it is one that I have observed as being useful for emerging and experienced teachers alike. It is a method that we showcased and published in the Journal for Music-In-Education (Scripp, Keppel, Wong, eds.), and that we encourage throughout the MIE department. Its value lies in the fact that words do not lie, and it is often easier to quickly see the ‘big picture’ when scanning transcripts than from sitting and watching a videotape. The benefits of watching the videotape, and doing one’s own transcription from that tape, are obvious: Body language, tone of voice, eye contact, movement, and other physicalities of teaching are easily recognizable. From watching my own tape, I was surprised to learn that my teaching voice was not as loud or enunciable as I thought it had been. I suppose that is something to continue to work on. I didn’t do an ‘exact word’ transcript here, but what I learned from the tape is that there were multiple times that I had to re-phrase questions, transitions, and other verbiage. I already knew from past experiences that off-the-cuff presentation is not my strong suit; the introductory Ten-Minute Presentation we did at the beginning of Teaching Music History is testament to that (I scripted that presentation and practically read it). Because of the limited amount of time I had to prepare this teaching session, scripting nor rehearsing were barely possible, but I did have to time to make a short Powerpoint presentation that I used as an outline of sorts.

Connection to MHST 537 course

Although the class session I taught is not a Music History course, I believe that many of the same principles that we have been studying in Anne Hallmark’s MHST 537 Teaching Music History course still apply. The past several weeks have seen discussions in class based on readings that articulate how college classrooms are run; the pitfalls and mistakes of ‘wet behind the ears’ teachers; ways to engage students in discussion; and organizational tips for lecturers, among other things. These readings are balanced with seminar-style class sessions moderated by Hallmark, which in and of themselves serve as models for successful teaching in a graduate setting.

As is evident in my coded transcription, I tried to incorporate some of the techniques that Hallmark and others are suggesting as worthwhile ways to engage students in discussion and classroom learning. Granted, there was less discussion than I would have liked, and the majority of communication was responsorial, but I think a good effort was made.

The teaching session was also an opportunity for me to go into a situation not as well rehearsed or prepared as I usually would be. There is, as Warren Senders or Larry might say, a certain amount of improvisation that that is a part of any teaching experience, and that a seasoned teacher would need to be comfortable with; things hardly ever go ‘as planned.’

Finally, I did make it to the end-point Larry projected for me: A MIE NewsBlog blogging assignment that students would need to complete, and connect, to the knowledge they’ve so far acquired on documentation, for inclusion in their process portfolios.

Download PDF:

• Outside Teaching Transcription, Coding, and Analysis by Randy Wong.

Concentrate on a particular area of technology that interests you, and be prepared to explain to your colleagues its current state of development; where it might be in five years; and the pros and cons of its usefulness in the classroom.

—Assignment for this week’s Teaching Music History course (MHST 537), taught by Anne Hallmark

One specific adaptation of technology that interests me is the use of “blogs” as a means for after-class discussion and discourse. Blogging shares many benefits with similar technologies (such as online bulletin boards, forums, email), in that its asynchronous format allows discussants to log on at their leisure; carefully think about what they want to share, and respond in thoughtful ways. A blogging website can offer users several types of opportunities, like: reading or viewing class events passively; re-articulating what happened in class (such as, from the student’s personal perspective) by writing a “post”; and/or commenting on others’ perspectives by leaving comments at the bottom of each post. As with other Internet technologies, online blogging websites usually allow the inclusion of hyperlinked articles, multimedia (videos, audio, pictures, slideshows), rich text (bold, underline, bullets, other formats), and also function as archives. Many blogging websites (such as Blogger, Xanga, MySpace Blog, BlogSpot) already exist, and most offer free general-use blogs that include some kind of technical support for inexperienced users. For a higher level of customization, “open-source” (non proprietary) software like Wordpress (which this blog is run on) or Movable Type are also popular, though these often require a more sophisticated sense of technical expertise. 

The major hurdle that I’ve observed is not with the blogging technology itself, but rather it’s use and how it is supported by the instructor, and included in the classroom: A class with access to a blog is a very different story from a class whose members post regularly to the blog, and whose instructor actively moderates the students’ posts and comments. Also, the kind and style of writing that is posted to the blog will make a significant difference in the level of engagement students have with the blog:

• What will draw them into reading the blog? 
• What type of discourse is the instructor hoping to achieve via the blog? 
• To what extent will the blog be able to help students make connections beyond what is discussed in class? 
• How can learning on the blog make the jump, back to classroom learning?

It’s my inclination that these types of issues and ideas will be with us in 5 years, 10 years, even 50 years—that it’s not the technology that poses questions like this, but the ways that educators structure and vet their own teaching processes, when working with and engaging students of multiple, or varied, learning styles. 

As parts of my professional roles (Information Architect for the Music-In-Education National Consortium, and Program Coordinator for the  MIE Concentration here at NEC), I have spent the last few years researching and developing educational communities that support blog technology. The CMIE NewsBlog (http://centerformie.org/blog) is one example of my work. It is contributed to, on a weekly basis, by a selection of students from currently-running MIE courses and Guided Internships. These students are designated as “Documentation Specialists“—they each are charged with the responsibility of collecting evidence and examples of classroom teaching/learning n their respective MIE classes or internships, and reporting/sharing/articulating what’s going on in those classes. I have designed a number of post types that students can use as springboards for writing. I also regularly meet with students to mentor them on what kinds of documentation they should collect, and how that documentation can be used in a portfolio or NewsBlog post. We try to steer our writers so that they have an uninformed audience in mind; the premise is that a thoughtfully-written NewsBlog post can also be used in a teaching portfolio, or as the basis for academic writing of some kind. Finally, each MIE instructor incorporates the NewsBlog into his classroom in  a course-appropriate way: the MIE Intro class, for example, uses the NewsBlog as practice for students learning to collect and reflect on documentation. As MIE Program Coordinator, I use the NewsBlog to show a birds-eye view of how each part of the department works in relation to the whole. 

The readership of the CMIE NewsBlog is large and varied: Not only do the Documentation Specialists’ classmates read the NewsBlog; but also MIE faculty, students at the Harvard Graduate School of Education, members of the MIE National Consortium, and the general public. NewsBlog posts are moderated by myself and other MIE faculty members, and posters often receive extraordinarily deep feedback on their writing. In fact, since the NewsBlog begun, the MIE Department has seen a marked increase in the quality and maturity of written portfolio work. I am sure that this is not unrelated. 

As long as the teacher is supportive, aware, and comfortable with the use of blogging websites, the inclusion of blog technology is generally un-intrusive and can be a welcome complement to synchronous classroom discussion. I welcome my colleagues to visit the CMIE NewsBlog, read and comment on our students’ work, and contact me should they have any questions or suggestions on its use.

-Randy Wong

The activity: Rhythm rulers

My perceived initial question: How can students use a ruler to understand rhythm and to create interesting rhythmic patterns?

A Rhythm Ruler is a representation of a conventional ruler that students can use, not to measure distance or length, but to observe divisions of the beat in a measure. Rhythm Rulers are a great tool for composition and rhythmic analysis which can encourage students to create complex and interesting rhythmic compositions before ever looking at a sixteenth note. For our class activity, Professor Scripp provided us with Rhythm Rulers made out of construction paper. Instead of a standard ruler which would begin at 0, the Rhythm Rulers began with 1, so as to represent an eight bar phrase. Each unit (ex. 1 to 2,  2 to 3) was divided into quarters, with a larger hash mark at the half as a conventional or metric ruler may have.

Blank rhythm ruler

Before I continue, I would like to point out how difficult it is to give a description of the rhythm ruler without using too much musical terminology. While Rhythm Rulers provide an excellent lesson in quarter notes, eighth notes, sixteenth notes etc., Professor Scripp made it a point to steer our discussion of our rhythm patterns away from standard rhythmic vocabulary.

Partner work: In groups of two, we composed eight-bar rhythmic compositions, notating the pattern with X markings on the Rhythm Ruler. Professor Scripp encouraged us to be creative with our rhythms and to consider dividing the rhythmic pattern into two four-bar phrases, or to have a sort of question and answer feeling to the phrase. After ten minutes of partner work, we were to demonstrate our rhythms to the class.

Group 1: Samuel and Leslie – Hatikva

Inspired by the first eight measures of Hatikva, the Israeli national anthem, Samuel and Leslie composed a very thorough rhythm. As a class, we listened to Hatikva, and then Samuel and Leslie demonstrated their rhythm to us.

Click here to see Sam and Leslie’s composition based on Hatikva

Several interesting points and questions arose from this composition based on fairly straight forward anthem:

• How does “Hatikva” fit the rhythm ruler?
• How can one best divide up the ruler to fit the anthem in the best way? (ex. on the full beat, the half, or the quarter)
• Does the beat emphasis affect the sound of the anthem?
• What characteristics define an anthem?
• Does the success of clapping this rhythm depend on a previous knowledge of the national anthem?

To address the last question, Professor Scripp had Sam and Leslie perform a “retrograde-inversion” of their rhythm–(*Hint* turn the ruler upside down). Did they exhibit true mastery of their rhythm?

Watch Hatikva…backwards…Rhythmic Ruler masters!

Group 2: Justin and Sarah – Rhythmic Ruler grooving jam session

Unlike Sam and Leslie, Justin and I did not have a specific theme or idea in mind for our composition. We composed very much from off the top of our head and ended up with some fun, off-the-beat elements that were syncopated, but also felt very natural.

An interesting phenomenon that happened while we improvised was the unintentional pitch that we added to our rhythms. Personally, I found that while I sang our rhythm, I had more success in clapping it. Watch the videos below as Justin explains our use of pitch and our performance of the rhythm.

MIE’s Behind the Music – The Rhythmic Ruler Jam Session

Jam Session – Part the Second

Thankfully, Professor Scripp did not express an interest in hearing the retrograde-inversion of the Rhythm Ruler Jam Session,  but he did ask for a performance at double time and half time. When asking that of us, I found that I had to go less on “feeling” and had to rely much more on the ruler.

As I watch our group’s videos again, a few other questions (inspired from those presented in Group 1) have come to mind:

• Although we composed our rhythm without a specific melodic theme in mind, did we continue to rely on our unintentional pitch rhythm for succesful performance?
• How would another group, who had not heard our performance, perform our piece?
• What would happen if students composed on the Rhythm Ruler not for their own performance, but for the other students in the class to perform?
• If we were to have tried clapping this piece fully realized with notes and rests, would we have had as much success as with the rhythm ruler?

Group 3: Michael and Hao Bing: Silence and the reverse of silence

Watch below as Michael and Hao Bing describe the “raison d’etre” behind their composition. In this excerpt, you will see how the musician side of Michael and Hao Bing comes out and how Professor Scripp encourages them to rely more on the Rhythm Ruler and less on counting “1 and 2 and 3 and 4 and…”

“What’s our inspiration?”

Let the Rhythm Ruler do its job

We found as a group that their composition was deceptively tricky given its emphasis on empty space. Michael and Hao Bing tried an experiment in which instead of singing or clapping their rhythm, they performed the space between the X’s–they performed the silence.

The sounds of silence

Teamwork: After a completely unscripted, yet genius (…just kidding) comment that I made about the difficulty of Michael and Hao Bing’s piece and how it may be a helpful exercise for playing an excerpted orchestral part, Professor Scripp segued brilliantly and placed our Rhythm Rulers one on top of one other into a “score.” Here we perform our individual compositions together as an ensemble.

Playing off of the Rhythm Ruler score

And then we added drums…

For me, this was the most gratifying component to this exercise. I was quite surprised at how great our individual rhythms sounded as an ensemble piece. If I had not participated in this activity and had simply come upon this video, I could not have imagined that each line was painstakingly realized and individual in its own nature…the piece was very enjoyable as a whole!

Reflection: Rhythm Rulers provide an infinite amount of opportunities for music and music integration. Here are a few follow up questions that could be considered for integration in other subjects.

• What makes a national anthem, and how does it relate to a nation’s identity?
• To what extent does rhythm determine pitch–AND–to what extent does pitch determine rhythm?
• How do units of measure relate to divisions of the beat?
• What fractions are discussed in this exercise?

While watching our videos and thinking back on our comments and discoveries, I am thinking of a myriad of possible variations on this exercise and I am surprised at the possibilities that exist for creativity and improvisation in a construction paper ruler.

Difficult and syncopated rhythms can be made very accessible with Rhythm Rulers. While clapping Justin’s and my rhythm, I was brought back to Freshman year ear-training and trying to clap out impossible rhythms from Hindemith’s Elementary training for musicians. Perhaps if I had had a Rhythm Ruler, I would have been able to see the divisions of the beat more precisely than with “1-e-and-a-2-e-and-a-3-e-and-a…” Maybe instead of one “inch” being equal to one beat, four inches could be equal to one beat, and I could successfully clap out 32nd and 64th notes! Could the Rhythm Ruler conquer even Hindemith?

Even more than understanding and mastery of rhythmic patterns, I found that Rhythm Rulers present a beautiful lesson in how individuality can exist within a group, and how each person’s contribution is an important part of group creativity and success. Within our final composition, it is difficult to hear each of the three group’s lines (even Hatikva, which we were all humming for the remainder of class!), but the end result is truly a remarkable piece of music, and any number of variables would have brought about a completely different final piece. Everyone’s voice is special and unique,  and when blended together, even the simplest offering can contribute to a masterpiece.

We began this session by learning a few folk rounds. The first was a bluesy MoonDog tune that reads as follows:

Nero’s expedition up the Nile
Failed
Because the water hyacinths
Had clogged the river
Denying Nero’s vessel’s passage
Through the Sud of Nubia

(I found a transcription online if anyone is interested in joining along!)

The second was a curious tune that compared a frog to a strange form of bird. This piece reminded me of a showtune more than a folk melody. Its text read:

What a queer bird a frog are
When he sit he stand almost
When he jump he fly almost
When he sing he cry almost
And he ain’t got no tail
Hardly he ain’t got no tail
And he sit on what he ain’t got almost…
What a queer bird a frog are!

When we broke into canon (I believe at one point we were in 4 or 5 parts) I began to get so excited that I couldn’t help but sing at the top of my lungs and by the end I had definitly broken a sweat. At times I stopped (just briefly) in order to experience the composite harmonies and rhythms as they flew by. This is the music that shaped me into the man (and musician) that I am today. My mother sang folk songs from her youth to me when I was an infant (and in fact even before I was born). When I was older she bestowed unto me her collection of LPs that contained her favored renditions of these tunes (they are still in frequent rotation on my record player). There is a strange sense of community that is lost when these songs and their traditions are ignored and undermined.

Later that night, after our class, Joanna and I took a couple friends up onto a friend’s rooftop deck. We taught them the songs that we had learned, and while a number of intoxicated college students stumbled home on the streets below, we rounded off the roof above them. This type of community is profoundly valuable. The sense of sharing is so unique and there is such a strong energy of liberation from all things ego. It is as though the music soaks into your skin and you feel such ecstasy from its internal resonance. These moments are rare (and increasingly so). We speculated in class on what life might be like if the price of gas escalated to an unreasonable and unaffordable height. Luxury as we now know it would cease to be an option. Touring musicians and ensembles would become rare and would force our community into a tighter knit microcosm. I believe that our scope of perception would narrow and we would begin to look inwards for entertainment. It is likely that our culture would return to an orally driven tradition with a focus on sharing, trading, familial communities, and the immediate experience of being. It is possible to speculate that within our current global community these things become insignificant and tend to disappear.

I would also like the mention the great attention devoted within our current cultural machine to the standardized process of test taking. This continues a conversation we began in class that Jenny also spoke of in her previous blog. As Jenny mentioned, learning to take a test, such as the SAT, targets strengths that you might use when balancing a check book. The strategies you learn when training for these exams in no way helps one to develop into a better artist, writer, musician, or creative thinker. Much of the way that high school English classes are conducted reflects this. Conversations surrounding literature and higher art are more often than not guided towards a specific end result or “correct answer” as defined by a teacher’s handbook. These so called ‘guided conversations’ are not real conversations and do not express anything but a prescribed formula and its subsequently derived answer. It leaves little room for creative thinking and no room for a student to learn freely. In tighter knit societies, we want people to sing with, to talk to, to be part of a community with.

We discussed in depth how much of what we learned as students was not what our teacher had been attempting to teach. For many of us, we learn in a variety of free-associative ways. We make connections and draw conclusions based upon previous experiences and our current understandings. It seems obvious, with this in mind, that our system of education can many times create a barrier for the minds of learners. In my view, education is something that cannot be prescribed. It is something that, when most effective, is coordinated with the specific needs and current situations of each student group (and in the most ideal situations, for each student individually). We have a long way to go in fulfilling the needs of our students, but I believe that in-depth speculation on the unique qualities of folk based communities will yield positive and provocative results.

Before I formally introduce myself to the MIE NewsBlog, I would like to first say “Hello. My name is Sarah Rogevich and I am a Blogophobe.” I was inspired to write this after reading Jenny Giardina’s excellent introduction to the NewsBlog. Like Jenny, I have also witnessed the growth of the “blog bubble” throughout my education, both in undergraduate and graduate study, but MIEatNEC is providing me with my first opportunity to participate in it directly. I hope you will all forgive my initial trepidation as I have now immersed myself into the world of intelligent and academic blogging.

What began as an initial interest in K-5 music education has quickly become an exciting research project as I have taken on the role of Documentation Specialist for MIE 501 – Introduction to Music-in-Education. In addition to providing the NewsBlog with video and photos of our own activities and discussions from MIE 501, I hope to also be posting my documentation of my observations from the music classes taught by Michael Glicksman at the Atrium School in Watertown, MA. I am currently in the process of obtaining permission from the Atrium School administration to post videos from the classroom.

What goes on in MIE 501?

In our weekly, 2 hour class, my classmates and I cover a myriad of topics under the direction of Professor Larry Scripp. At the beginning of the semester, Prof. Scripp introduced us to the double or triple entry journal and how that format may lead us to organize our research and to answer the question “What is music’s essential role in education?”.

Each week, our class strives to answer “the big question” through guided classroom activity examples, readings provided by Prof. Scripp, and general discussion of our own experiences in music and music education. Since each student in the class brings his or her own experiences, strengths, weaknesses, and interests to this question, it is safe to say that our individual answers to “the big question” may differ a bit. Through my documentation of this class, I hope to provide the MIEatNEC community with examples of my and my colleagues’ quest to discover how music contributes to education. In a small class and an equally small amount of time, we have all begun to focus our attention on our personal interests in music-in-education such as music in speech therapy, music and adult education, and Leonard Bernstein’s Young people’s concerts.

My goals

During the remainder of the Fall 2008 semester, and throughout the Spring 2009 semester, I aspire to accomplish the following goals:

1. To provide multi-media documentation of the activities covered and topics discussed in NEC’s Introduction to Music-in-Education (MIE 501) and in Music-in-Education seminar (MIE 511)

2. To provide multi-media documentation of the music lessons taught by Michael Glicksman at the Atrium School and how they relate to our class experiences and discussions in MIE 501/511

3. and to develop the official framework for formal documentation of the music curriculum at the Atrium School to be used by teachers, interns, administrators, and families to meet the educational needs of their students as outlined in the Atrium School philosophy.

An inspiration

In addition to MIE 501, I am also enrolled in Professor Warren Senders’ class Cross-cultural approaches to MIE (MIE 547). After a particular class in which we explored ratios and harmonics with the use of a monochord, I was struck by the infinite ways which music can be utilized in education and learning. By the end of this two hour class, we had covered topics not just related to music, but to arithmetic, geometry, algebra, physics, and many other mathematical concepts to which I tend to have a strong aversion. If I, Sarah Rogevich, “not-a-math-person” can become inspired to read more about Pythagoras after one music class, how many other students can benefit from music in learning?

After pondering this question, I formulated the following three inquiries that I hope to explore through my work as Documentation Specialist and as a student in NEC’s MIE program:

• What axioms exist in music education and how do these compare to those in other subjects?
• Considering those axioms, could students benefit from a music-centered curriculum?
• In a music-centered curriculum, how would all subjects be addressed and how could this inter-curricular model enhance a student’s total education and love for learning?

Within the existing framework of the music program at the Atrium School in addition to class discussions in MIE 501, I hope to explore music and it’s influence in elementary education. These are the inquiries that I will use to attempt to answer “the big question,” but as Documentation Specialist, I look forward to bringing you insight from my classmates’ diverse perspectives. Stay tuned for videos and posts from my classmates in the near future!

Many of us can remember having to take the SAT. Since 1934 when James Conant, the president of Harvard at the time began administering the test to scholarship applicants, taking the test has become increasingly standard procedure for a person moving from high school to college. But what does the Scholastic Aptitude Test really measure?

We discussed this question in class and came up with the following: LANGUAGE AND NUMBERS.

Effective? Comprehensive? Fair? We didn’t think so either. All the test tells me is that if I score between a certain range I can balance my checkbook correctly and read a newspaper article really quickly. Lyle Davidson shared that it’s really nothing more than a test to see who will pass their first year of college. I sure wish it had been presented as such instead of a test to see “who’s smartest.” I would’ve spent a lot less time worrying.

Unfortunately, because the SAT is still the standard measuring tool we are led to believe that the educators who have the last word must believe that the wisdom of priests and rabbis, the intuition of psychologists and the sheer genius of Mozart are not examples of intelligence.

In the 1980s a man by the name of Howard Gardner came forward with some new ideas on what “intelligence” really is. He presented the concept of “Multiple Intelligences,” saying that different areas of the brain support different types of expression, cultural differences and necessary awareness. For example, different cultures require individual and acute behavioral skills to survive in a specific location and environment.

We brainstormed what these “multiple intelligences” might be.

Here is the list we came up with:

• Language and Numbers (not only to be fair, but because they are important, too.)
• Spatial intelligence (2D and 3D awareness)
• Musical
• Kinesthetic
• Interpersonal/Intrapersonal
• Naturalistic

We decided that the list could be titled “Spatial Intelligences”.

As can be expected when you have a group of future educators talking about education there were many unsatisfied voices. Their statements could be passed off as hopeless complaints, but I think that would be a huge mistake. In the simplest of terms, it’s important that more of the focus in education be placed on spatial intelligences. NEC has already made the change to not requiring applicants to submit an SAT score. In my opinion, that was a smart move.

I’ll leave you with this:

Because we are individuals whose intelligences are clearly made up of more than just languages and numbers, should we not be approached as such by our educators?

In Wednesday’s class we began by pairing up and experimenting with long ropes in order to visualize the vibration of a string. One person stood holding their end in front of them while the opposite person swung the rope at different speeds. We attempted to create, at first, one broad swing of the rope (like you might see in a game of jump-rope). Then we doubled the speed so that the rope was divided into two equal parts, each rotating conversely (while one side swung upwards, the other rotated downwards). This increase in speed was continued until it wasn’t possible to divide the rope into any smaller sections (usually occuring around five divisions of the rope). Each dividing point between rotating sections is considered a ‘node’, or a place where the vibration is zero.

We then gathered into a circle in the classroom and used a monochord (an instrument consisting of a single string) to discover the specific ratios that create each interval above the tonic pitch. We began by splitting the chord in half (done by lightly touching in the center of the vibrating string) so that each section of the string was vibrating at twice its original speed. This is the same as what we had just experienced with the rope when we doubled our initial speed in order to create two vibrating sections. This time with the string of the monochord, an octave occured above the original pitch (shown by the ratio 2:1, where the higher pitch is vibrating two times for each one vibration in the lower note). We continued to use this same method to achieve the 5th (ratio of 3:2), the 4th (4:3), and so on through each of the twelve intervals. We discussed that frequency ratios always come in pairs that add up to an octave. For instance, the ratio 3:2 will be paired with the ratio 4:3 (a 5th plus a 4th equaling an octave).

The class reminded me of Stuart Isacoff’s book “Temperament” which addresses the history, problems, and evolution of tempering the Western scale. After the class, I went back and read the section concerning Pythagoras and his original discovery of the geometry of music. Pythagoras, who invented the monochord, stated that “music’s rules are simply the geometry governing things in motion: not only vibrating strings but also celestial bodies and the human soul.” Pythagoras believed that the most pleasing of harmonies arose from the simplest of proportions and that complexity would insight chaos. What is fascinating about this is that behind his discoveries of pure musical geometry there lies a forbidden and volatile darkness. He found that pure octaves and fifths, according to his ratios, are incommensurate (also referred to in Greek as ‘alogon’ meaning ‘the unutterable’). Fifths will never complete a perfect circle (as suggested by the widely accepted circle-of-fifths), but will reach toward infinity in an unending spiral. This essentially boils down to the fact that octaves are based upon multiples of 2 (2:1) while fifths are based upon multiples of 3 (3:2). In this case, no multiple of 2 will ever meet a multiple of 3. If one were to compare the pitch achieved by an octave and that achieved from the completion of a circle of fifths, they would be very similar yet “out of tune”. This spiraling phenomenon hints at a more complex mathematic sequence, that of the golden ratio. Even so, these simple ratios were believed to be an expression of the divine. It is easy to find similar ratios present within nature. Saint Augustine, in fact, believed that churches and cathedrals were to be more than just shrines, and instructed that proper proportions were to be used in their construction. Thus the heights, lengths, and depths of the structures formed the proportions of Pythagoras’s “celestial harmonies” (1:1, 2:1, 2:3, and 3:4).

So what difference does this make to us, as musicians and as people? What effect does this really have on our performance? I think it is crucial to understand the fundamentals of the creation of sound, of pitch, especially when such things are taken for granted everyday. I remember the feeling I had when I first discovered the ratios involved in music. Once I got past the initial migraine acquired from my first lecture on equal temperament, I began to look a bit into proportions. It made perfect sense (and also supplied an interesting and practical perspective to my high school math classes). This is the real foundation of what I do every day, of each note I play. It is a fundamental that comes before technique, before fingerings and musicality. In a sense it is the DNA of music (more specifically of pitch). Yet as crucial as these fundamentals are, an understanding of them is not essential for the enjoyment of music. Recently, Warren mentioned a workshop that he was conducting years ago. During the course of the class, he plucked two notes on a string, the second a fifth higher than the first. Soon after, a young boy came running into the room exclaiming “What was that beautiful music?!”. Like the young boy, a single, simple fifth can produce a level of joy bordering on ecstasy. Warren also noted that infants are particularly drawn to simple intervals. This has been quite a meal for my thoughts (even just thinking back to our class sends my head spinning!). Every time I try to find a solution to these musical systems I find that I develop more and more questions. It is truely amazing how much chaos lies within order!