Introduction
Chapter one is presentation on foundational material, necessary for our more advanced study this semester. It is designed to serve as a reference place for you throughout this semester, and will also be useful to you when you prepare for your theory comps in January. It is essentially a distillation of the material that is covered in the first and second semesters, with an additional page that deals with the concept of Harmonic Polarity.
Musical Ratios
Theoretical music knowledge begins with an understanding of harmonic ratios. Knowing the ratios of Pythagorean and Just Intonation helps you to understand certain concepts of how tones should relate to each other in an idealized world, which in turn better help you to understand the various compromises of temperament and the special compromise of equal temperament, which has changed the sound of our music in fundamental ways.
When you look at theory texts from a few generations ago, they all started with integer ratios. American text books have since chosen to consider this information irrelevant, but recently I see a shift occurring outside the mainstream of academic textbooks. There is clearly a renewed interest in this topic. I saw a prominently displayed book at a bookstore this summer in D.C. called How Equal Temperament Ruined Harmony (and why you should care). Throughout my teaching at Earlham, I have been slowly incorporating the work of Allaudin Mathieu, whose book Harmonic Experience changed my thinking in the year 2000. No doubt, I resonate with Mathieu's book because he is a jazz musician as well as a classical player and composer, and he also studied north Indian classical music with the great vocalist Pandit Pran Nath (with whom one of the founders of minimalism, Terry Riley, also studied). It all connects.
In general, there is an exceptional amount of new information coming out these days on this topic, and clearly there is a renewed interest in temperament and the prototype tuning systems, with their pure integer ratios, amongst contemporary musicians. However, this material does not appear in the current college music theory textbooks, which in my view are woefully behind the times. (Of course, many academic theorists would look askance at some of my presentations, but then that's why I'm at Earlham!) As a consequence of all this, I am forced to create all my own course materials, and now get them up on line. It's time consuming, but it's rewarding as well.
To begin, it is helpful, I think, to remember that musical ratios are not simply about mathematical tone relationships, but also about principles of ethics, cosmology and spirituality. This is not the place to review the philosophy in depth, but I will remind you that concepts equating harmony on earth with harmony in the cosmos go back to the earliest of philosophical inquiries, and that the quest to understand musical intonation was also a quest to understand universal harmony, "the music of the spheres." Musical harmony occupied some of the greatest minds of Western Europe, including Ptolemy, Boethius, Galileo, Kepler and Newton. 19th Century German philosophers Goethe, Kant and Schopenhauer were all involved with gaining an understanding of musical theories—Schopenhauer was consumed with music, and his philosophy was consumed by Wagner—and there is a profound relationship between socalled esoteric philosophy and 19th Century French and German speculative music theories, many of which were directly influential on such composers as Debussy and Bartok. We will be looking at these influences on Wagner, Debussy and Bartok in the main part of the course.
Pythagorean Thought
The teachings of Pythagoras are difficult to summarize in a few words, and scholarly views of Pythagoras are always filtered though an individual's predilections. Contemporary scholarship tends to praise his mathematical prowess while looking away with embarrassment at his perceived "mysticism." But from the standpoint of musical theory, it is difficult to separate the esoteric from the practical, since for Pythagoras and his followers they were one and the same. Pythagoras was both a mystic and a scientist. 20th century philosopher Betrand Russell characterized him as a combination of Mary Baker Eddy and Albert Einstein, which is a great image! Although with no ill respect to Mary Baker Eddy, his "mysticism" was probably more genuine—certainly it was connected to a greater, more ancient tradition.
Personally, the reason I continually come back to Pythagorean thought is the understanding that what we call "Pythagorean" is in fact a convenient label for a tradition of "speculative music theory" that reaches back into the dawn of human thought. What Pythagoras represents is a knowledge base of harmonic ratios that were known to the earliest civilizations, most especially the Egyptians, the Babylonians and the tradition of the Vedas in India. This allows me to connect emotionally and philosophically to his teachings, understanding them to be central to our human perception of reality.
For example, the scholar Ernest McClain presents an amazing "underground" tradition of research that can be summed up by a title of one of his books: The Myth of Invariance: the Origin of the Gods, Mathematics and Music from the Rg Veda to Plato. Here, he posits and goes to great effort to show that the cosmologies of all the great ancient civilizations—Egyptian, Babylonian, Hindu (Vedic), Hebraic, and Greek— drew correspondences between number and tone, and that these correspondences shaped their understanding of the heavenly spheres and formed the basis for such fundamental practical applications as the development of the calendar and the architecture of sacred temples. Here is a link to one his essays on line, should you wish to read more. This relates universal constants of harmonic ratios to ancient mythologies, especially that of the Babylonians and Egyptians.
Musical Theory and Ancient Cosmology
Naturally, these ideas are met with a healthy degree of scholarly skepticism, especially in today's academic climate, where metaphysical concepts are often discounted or left to the "dustbin of history." However, as a practicing musician interested in a broad range of spiritual matters, I have found Mr. McClain's work most illuminating.
At the very least, it is undeniably true that musical theories grew out of mathematical ratios, and that these ratios proved to contain important truths related to our relationship to the cosmos as it was understood through the great part of human history.
In coming to understand how scales can be derived from lowinteger relationships, we participate in one of the oldest constructs in human thought. At the same time, we help tune our ears to subtleties of intonation, especially as regards the perfect fifth and the major and minor thirds. In addition, as Pythagorean ideas were brought forward into modern civilizations, the theoretical aspects of these tuning systems were brought into conflict with the practical aspect of singing and creating musical instruments, which in turn led to the entire history of intonation and practical tuning systems, an issue that continues to resonate today and, in fact, is becoming increasingly popular. Part of this popularity comes from the early music movement, the practitioners of which want to get as close as possible to the historical tunings. Another reason for the rise of interest in tuning systems is the "mainstreaming" of the minimalist movement, which since the 1960s has been involved with seeking pure intonation. (Minimalist founder La Monte Young made a famous recording on a Bösendorfer concert grand piano tuned to pure just intonation. I looked for this online and could only find a used CD that was for sale for $1000!).
So in this brief introduction and review, let's look at the concept of Pythagorean and Just intonation systems, and see what they tell us about the way musical tones have been organized theoretically, and the conflicts that arise from trying to reconcile these systems with the practical aspect of musical performance and temperament.
3Limit (Pythagorean) and 5limit (Just) Systems
Pythagorean thought was not simply an ancient Greek idea that died along with Greek culture. The thought process was carried on through the Dark and Middle ages in the writings of Boethius in the 8th Century, whose work was preserved in monasteries and centers of learning, and a great many musical scholars that followed him (I would draw your attention to Zarlino should you wish to pursue this topic further). As the Renaissance artists rediscovered Greek philosophy in general (largely by its having been preserved in Arabic culture), this system of thought was restored and reinvented for "modern" times (that is, the 14th through the 18th Centuries). The Pythagorean understanding of the Music of the Spheres inspired such great scientific minds as Kepler, Galileo and Newton, for whom the exploration of things astronomical were intertwined with things musical and mathematical. Gaining a very basic understanding of these musical ratios not only is important for your musical studies, it also helps us to connect this study of "music theory" with the larger history of ideas. This is what a "liberal arts education" is all about. And anyway, as the great experimental composer Iannis Xenakis said in the 1960s: "We are all Pythagoreans."
In brief, Pythagoras (connecting as he was to an even more ancient tradition from the Sumerians, Babylonians and Egyptians) both experienced and taught a harmony of the spheres wherein the heavenly bodies were said to resonate at frequencies that equaled those of pure integer relationships. Thus, a correlation was posited between the harmony of the heavens and the harmony of tones on earth. As the (apocryphal) legend told it, Pythagoras heard the beautiful resonance of blacksmith hammers, weighed them, and discovered that they were in perfect relationships of 1:2:3:4 to each other. This grouping of numbers, called the tetractys, are a fundamental symbol in Pythagorean thought, and are usually shown this way, as if laying out pebbles on the ground:
X
X X
X X X
X X X X
It's a picture of the universe. No, really, it is. The One, the unity behind all diversity, divides, as yin/yang, into the two. But with the two, we have only one dimension, just a line between two points (yes, geometry, cosmology, music and psychology are all interconnected here). Thus the line and the unity gives birth to the triangle, representing the plane, and finally from this emerges a tetrahedron or pyramid, representing the fullydimensional universe. The sum of these figures, 1 + 2 + 3 + 4 is 10, ten thus standing in as the symbol for the known universe.
In this way, Pythagoras' tuning system (as all tuning systems before him) were limited to the number 4, and thus are called a 3limit system, since the highest prime number in the system is 3. This notion of threefold division took on deep mystical qualities, since the mere division of something into 2 equal parts was likened to "mere proliferation and reproduction," that is, increasing octaves. It is only through the introduction of the number 3 that true creation and harmony is made possible, since once the interval of the perfect fifth is introduced, all manner of musical scales and materials are made possible, as we will see below.
As I'm sure you remember, the basic foundation of Pythagorean tuning is the relationship of 3:2, what we today call the perfect fifth (the Greeks called it the Diapente). Legend again tells us that Pythagoras invented a monochord with a movable bridge and a carefully ruled surface, wherein he was able to measure the tone relationships of string division, using only the numbers of the tetractys.
2:1 is the octave, 3:2 is the perfect fifth and 4:3 is the perfect fourth. The perfect fifth and the perfect fourth are separated by a tone, and the integer relationship of the tone is found to be 9:8. Why? Because to find the difference between two ratios, we divide them, and when you divide fractions, you flip one, then multiply. Thus 3/2 divided by 4/3 is 3/2 X 3/4 = 9/8.
During the Renaissance, this teaching was traditionally diagrammed by taking a string and dividing it into 12 equal parts. so as to end up with integer results. The first octave (2:1) reduces to the distance between 6 and 12, and then the fifth (3:2) lies at the point of the 9, while the fourth (4:3) lies at the point of the 8. Visually, it is then easy to perceive the whole tone between the 9 and the 8, and the ratio formed between them.
Mathematical terms were used to describe the various midpoints between the second octave: the arithmetic, the harmonic and the geometric mean. As the formulas are simple, they are worth understanding and memorizing:

The arithmetic mean creates the diapente or the fifth. The formula is (a + b) / 2. Thus (12 + 6) / 2 = 9. The ratio 9:6 reduces down to 3:2. It is interesting that what we consider the simple process of "averaging" two numbers creates the perfect fifth, even though the fifth is not actually at the midpoint between octaves.

The harmonic mean creates the diatesseron or fourth. The formula is 2ab / (a + b). Thus 2(12X6)/(12+6) > 144/18 = 8. The ratio 8:6 reduces down to 4:3.

The geometric mean was not considered a valid ratio in Pythagorean tuning, as it was expressed by the formula square root of 2, which is an irrational number. Interestingly, this is the interval of the tritone, which of course was called the devil in music during the period of vocal polyphony, due to the impossibility of tuning the interval according to the standards of just intonation.
It is traditionally diagrammed as:
There is a counterintuitive aspect to this drawing that music be understood, then it will be clear to you. Even though the numbers are moving up from 6 to 8 to 9 to 12, the actual pitches are moving downward. So the 6 to 8 ratio is the perfect fourth down, the 6 to 9 ratio is the perfect fifth down. This is best understood if we play this on a cello low C string and mark the cello fingerboard as divided into 12 equal lengths (this happens to conveniently work out to be 2.25 inches, since the cello string is 27 inches long). Take the 1 as being at the bridge and the 12 as being at the nut, so you could visualize it this way, with the top of the diagram being the top of the cello's string, and the bottom of the diagram being where the cello bridge stops the strings vibration, represented by a curved line:
These measurements are based on the monocord, which was a stringed instrument with a movable bridge. On the monochord, as you moved the bridge along the string from 1 to 12, you first hit the octave at 6th, then you continued to lengthen the string, reaching the fourth at 8 and the fifth at 9. It's the opposite of how we actually play the cello with the fingers, so it's a little counterintuitive at first.
Demonstrating this on the cello (you could also do it on the guitar):
 The 6 represents the octave above the low C string. It's fairly high up on the body of the cello, and also is the place where you can touch the string to get the octave harmonic (same goes for the guitar).

The 8 corresponds to the fourth below the C, the G, as in the diagram above.

The 9, being a step lower in pitch, represents the fifth below the string, or the F.

6:9 is 2:3 which is the fifth down.

6:8 is 3:4 which is the fourth down.

8:9 is the whole tone between them.
Creating the "Dorian scale" using Pythagorean tuning
Based on these principles, creating a full diatonic scale is simple for trained musicians like yourselves to understand. We'll begin with D as the central tone for purposes of notation. This both allows us to create a perfect symmetry of fifths above and below the central note, and also allows us to create what we today call the Dorian mode. This mode is one of the urscales of human history, and scholars have identified something very simi liar to this mode (with the particular arrangement of whole steps and half steps) as resembling the basic mode of the Vedas and other ancient philosophies. (See Music and Musical Thought in Early India by Lewis Rowell, U Chicago Press, 1992.).
The first principle is that we take a central tone—let's call it the tone center—and build (and tune!) perfect fifths in precise 3:2 intonation above and below the tone center. Let's start with just the fundamental pentatonic scale, which for our purposes begins as:
From this first principle, a variety of pentatonic scales can be realized, depending on which tone is taken to be the resting tone. Thus, using the central, the high and the low tones as the actual tonic or resting tone, we can create these five note patterns (with the resting tone being the low tone), using octave reduction/expansion to bring the other notes within an octave:

The first version takes the central tone as the tonic or resting note. We could think of this as the Dorian pentatonic, due to its relationship to the Dorian mode.

The second one uses the lowest tone and builds up from the bottom, pulling all the notes down into the octave. This is the most "tonal" version of the pentatonic and is the scale we are most used to in the West (Amazing Grace, O Freedom, lots of other "spirituals," huge numbers of English and Continental folk songs).

Number 3, with A as the resting point, keeps the original D as the central tone and pulls the other pitches closer in with octave reduction, resulting in a scale beginning on A. It is the second most common pentatonic in the West and we usually refer to it is as the minor pentatonic. Since we will be more modally based in our class, we'll call it the aeolian pentatonic.

The last version is the reverse of number two, as it builds down from the top and could be considered the reciprocal version of the pentatonic, which we call also call the Phrygian pentatonic.
These contrasts encapsulate some fundamental principles and polarities that are worth pointing out:

Scales often have their origin around a symmetrical center point.

The central point can then also become the root tone of a complete ascending scale (# 1).

The central point can also remain in the center, with other tones brought into the octave by octave reduction (#3), with the scale range being within a fifth above and a fifth below.

These symmetrical arrangements are also reenvisioned from the top up or the bottom down, so that the lowest tone in the symmetry can assume the root function (# 2), or...

More radically, the top tone in the symmetry can also assume the root function (#4). As we will be exploring, this forms the reciprocal relative to the ascending scale.

Finally, it is important to notice the presence or absence of the perfect fourth and fifth in each of these pentatonics:

The Dorian and Aeolian pentatonics contain both the perfect fifth and perfect fourth. Another way to think of it is that they contain both the harmonic and arithmetic means. In this way, they are perhaps the most balanced of the pentatonics, being centered (or built upon) the D tonality.

The tonal pentatonic, on the other hand, contains only the perfect fifth but not the perfect fourth. This is significant, and I'm only realizing this as I write, because it resembles the situation with the harmonic series, which does at any harmonic contain the perfect fouth (see the next link in the series for more on this).

The Phrygian pentatonic, in contrast, contains the perfect fourth but not the perfect fifth. Another way to put this is that is has the reciprical dominant, the fifth below (octave expanded into the range of the scale), but not the overtonal dominant. This is keeping with the Phrygian mode, which as we'll explore in more detail in the next link, is the reciprocal of the Ionian mode.

If we now look at the ratios of these notes, we can observe the following. These ratios will hold, regardless of the type of pentatonic we are creating, because the intervals are all based on 3:2 relationships. This is beauty and simplicity of 3limit systems, which work great for melodic music but create problems with harmonic music, as we will see.
The Ratios of the Pentatonic in the 3limit system
The ratios are:

D to D is 1:1 or 1, the unison central tone.

D to A is 3:2, the arithmetic mean

D to G is 4:3, the harmonic mean

D to E is 9:8. This is derived by taking two fifths above the D, which 3/2 X 3/2 = 9/4. Then you reduce it down the octave by subracting the ratio of the octave, which is 2/1. Subtracting intervals means dividing them, so 9/4 divided by 2/1 is the same as 9/4 X 1/2 = 9/8. This is what we would expect, since the E is also a pure Pythagorean whole tone above the E, just as the G and the A are a whole tone apart, or 9:8.

D to C is 16/9. It is not surprising that this ratio is the most complex, as it results in the sounding of a minor seventh. Note how it is just "off" from being a pure octave (2:1). It can be derived by going up two perfect fourths, which is mathematically equivalent to going down two perfect fifths. Thus 4/3 X 4/3 = 16/9.
Please work this out for yourself with a pencil and piece of paper if it doesn't immediately make sense to you.
Make sure you can clearly sing and hear these in your imagination, and that you can internalize the sound and feeling of them. Try very hard to hear these as a relationship of pure perfect fifths (in other words, don't just play them at the piano but internalize them as pure harmonic ratios  sing them above a centrol tone). Pentatonic scales become important in Western classical music when we get into the music of Debussy, Stravinsky and Bartok.
The Pentatonic Scale and the Children's Chant
It's important to stress that, while this is the way music theory conceives of the pentatonic scale, the actual pentatonic scale was probably born from more direct and innate experience. This is sometimes referred to as the Children's Chant.
Anthropologists and biologists have various theories concerning the origins of music, going back to the most primitive utterances of homo sapien.
Such investigation, while fascinating, is beyond the scope of this course. As we begin to explore melody, however, there is one place where we can begin, which is intimate with our own experience. That is what we could think of as the Children's Chant.
If you think back to your own childhood, or if you observe the behavior of very young children, such as a young sibling, you will observe a very basic melodic pattern, that of the descending minor third. When children around the world vocalise, often in a manner of taunting or speaking in a singsong way to one another, they use a very similar interval, that of the descending minor third.
Curt Sachs, as quoted by Peter Van der Merwe in Roots of the Classical, notes the following:
"The earliest attempts of children less than three years old resulted in onetone litanies and in melodies of two notes a narrow minor third apart, the lower of which was stressed and frequently repeated. At the age of three, children produced melodies of two notes a second apart, and even threetone melodies. Children three and a half years old sang in descending tretrachords."
This phenomenon appears to be universal across cultures, as if embedded in our human DNA. The basic melodic shape, if notated in the treble clef in the key of C, comprises a core interval of a descending third (G down to E), with a slight hint of an additional note above the center tone (the A):
I think of this as the mocking "nyah, nyah, nyahnhay, nyah" chant that children sing to each other as a method for taunting another child and showing superiority. It is also sung to such charming words as "Johnny is a sissy" or "I can see your underwear!" Go to another culture and language, and you'll get the same tune in the local vernacular. It's really quite amazing! If you were lucky enough to have gone to a school that offered Kodaly or Orff training in elementary school, you would have begun your training by learning to sing and recognize, in a formalized way, this descending pattern. Using the syllables of the solege (which we will study by and by), the syllables are "Solmi La Solmi." Here's the tune if we were to transcribe it off the playground:
Thus, while it is useful to see that the Pentatonic scale was absolutely derived by philosophers extracted integer ratios into an abstract tonal space, the pentatonic also seems to stem from more innate human faculties. Like many truths, there are many ways of looking at the same phenomenon (like the 13 ways of viewing a blackbird or the many blind men touching and describing an elephant).
Building the Dorian Scale
The full Dorian scale can be conceived as being created by an additional pairing of fifths above and below the pentatonic set. This give us:
Taking the central tone as tonic (method # 1, above), it is easy to see the Dorian mode emerge, using octave reduction/expansion to pull the notes within the octave:
It is important to note the beautiful symmetry of the resultant Dorian mode. The Dorian mode has the same order of half steps and whole steps whether one ascends or descends. No doubt this has much to do with its rather calm and noble character. To our ears, it is that characterstic "minor third with a major sixth and minor seventh" that is the Dorian modes distinguishing characteristic. The minor third is there whether you go from D up to F or from D down to B. Similarly, the major sixth is there whether you go from D up to B or D down to F. Minor seventh? D to C or D down to E. And of course the perfect fourth and fifth are inversions of each other. Thus the scale is TSTTTST whether ascending of descending.
The Remaining Ratios
What is also important to notice is that we have added in the phenomenon of the semitone, which is of course what distinquishes the septatonic (7note) scale from the pentatonic. The ratios of D to E, G, A, and C were derived above. Now we need to add the ratios of D to B and D to F, and then find the intervals between them.

D to B is 27/16. This can be found in two ways. Either take another fifth above the E, so 9:8 X 3:2 = 27/16 or find the pure tone above the A, which is 3/2 X 9/8 = 27/16. You could also find the real B that is four fifths above the D, which would 3/2 X 3/2 X 3/2 X 3/2  81/16. Bring the high B down by octave reduction (divide by 3 ) to give you 27/16.

D to F is 32/27. This can be found by various reciprocal processes. Since we've found C to be the ratio of 16/9 (see above), be can find the fifth below by dividing by 3/2 (ie. 16X2 / 9X3 = 32/27.) Another way is to find the F that is three fifths below the D, which is 2/3 (invert the ration when descending) X 2/3 X 2/3 or 8/27. Then bring the numerator into the proper octave by multiplying by 4 (two octaves is 2 to the second). This again gives 32/37.
What remains is to find the ratios that exist between each step of the scale. These ratios are found by dividing the various fractions and reducing down as needed. For example,
When this is accomplished, the following result is obtained. All whole tones work out to be the ratio of 9:8, while all semitones work out to be the ratio of 256/243 (!):
1

2 
3 
4 
5 
6 
7 
8 
1/1 
9/8 
32/27 
4/3 
3/2 
27/16 
16/9 
2/1 
D 
E 
F 
G 
A 
B 
C 
D 
9/8 
256/243 
9/8 
9/8 
9/8 
254/543 
9/8 
The 9:8 ratios are obtained by dividing the various ratios and then reducing down the fraction. For example, the interval between the F and G can be shown to be 9:8 by dividing the two ratios (27X4/32/3 = 108/96) and dividing both by 12, which gives 9/8. A similar process occur between the A and the B, using 6 as the reduction factor. Do this yourself!
You are now grasping a fundamental principle of music theory, known to every trained musician for millennia, except for the last few generations. Tuning septatonic scales by Pythagorean, 3limit intonation results in scales with identical whole tones and semitones (9/8 and 256/243, respectively).
As neat and tidy as this system seems, a fundamental problem occurs. In this system, two semitones do not add up to a full tone! Thus we can see that this is not an equally tempered scale, because the ratio 256/243 is not exactly a half of a tone. If you do the math and add two Pythagorean half steps, you get the wonderful ratio of 65536/59049! This does not reduce down to a 9/8 ratio, is it slightly smaller. (It's easier here to express it in decimal notation:

9/8 = 1.125

65536/59049 = 1.1099087
This discrepancy between ratios is but the first of many important ones we will encounter in understanding intonation.
Hearing the Dorian Mode
It is important to spend some time getting very comfortable with the sound of the Dorian mode in its purest form. I know of now better way to do this but to listen to the Miles Davis Quartet recording of So What from Kind of Blue.Here it is. The form is AABA, with the A section in D dorian and the B section, for contrast, bumped up a half step to Eb dorian. Bill Evans lays down very hip chords in Dorian mode, based on stacked fourths and thirds. Miles Davis, John Coltrave and Cannonball Adderly pretty much stay within the notes of the mode when they improvise. Paul Chambers does likewise on bass—not too many chromatic passing tones. James Cobb is the drummer. This is one of central recordings of "modal jazz." Indeed, it pretty much was the genesis of the entire modal approach to jazz music. When you've heard it awhile, go to a piano and just improvise on the white keys, holding D A D as a drone in your left hand. Better yet, do this while the recording is playing and see what you can pick out of their improvisations.
So What, Miles Davis Quintet
Comparing Dorian with C major (Ionian) ratios
By this principle, we can also find the ratios between any notes of a mode. For purposes of reference, let us also create the ratios for the tonal scale of C major. Note that the ratio intervals between the notes will remain the same. All that shifts are the ratios that relate to the tonic note of C, since the intervals have shifted. (But note also that interval types: perfect fourths/fifths, major thirds, minor thirds, etc, are identical ratios):
1

2 
3 
4 
5 
6 
7 
8 
1/1 
9/8 
81/64* 
4/3 
3/2 
27/16 
243/128+ 
2/1 
C 
D 
E 
F 
G 
A 
B 
C 
9/8 
9/8 
256/243 
9/8 
9/8 
9/8 
256/243 
*Note that the major third is found to be the ratio of 81/64, which is two tones added together (9/8 X 9/8). This means that the major third intervals found in the in the Dorian mode (between the F—A and between G—B) should be the same. Let's check:

Find the difference between the A and the F. Thus 3*27/2*32 = 81/64. Check.

Find the difference between G and B. Thus 3*27/4*16 = 81/64. Check
+Note also the interval of the major seventh. We could find this in a variety of ways

Take a whole tone above the A, thus 27*9/16*8 = 243/128.

Find the fifth above the E; 81*3/64*2 = 243/128.
Finding the Just Scale (5limit intonation)
Before we leave the world of ratios and intonation (for the present), we must become clear about Just Intonation and the 5limit system, and then discover the various commas that occur within and between these systems, as they relate to equal temperament. This is not just about dry facts and numbers. It lies at the heart of how we hear and tune our music and our music making!
While Just Intonation came out of a long historical process, and was was an accepted form of tuning long before the discovery of the Harmonic series, it is useful for our purposes to understand Just Intonation in relationship to the harmonic series.
Let's recall the Harmonic Series, which you should know like your middle name:
This is how the series is typically presented, with attention drawn to the Bb and the F#, which are lower than what we find in equal temperament. More on these pitches in later links.
The harmonic series connects to just intonation in this way. We expand our prime number limit to the number five, thereby allowing us to include the ratio of 5:4 as a usable interval. (We can also use 6:5, since 7 is the next prime). Thus, using C major for purposes of demonstration, we keep
What is important for Just Intonation is to notice the presence of the major third and the major triad within the naturally vibrating Harmonic Series, and also to notice that the perfect fifth and perfect fourth correspond precisely with the ratios of Pythagorean intonation. The central point of Just Intonation and Pythagorean tuning lies in the the tuning of that major third. This brings us to the discussion of commas.
The Pythagorean, Didymic/Syntonic Comma and the Great Diesis
Cool names for some cool and important concepts. You've been exposed to these concepts in the past. Here, they are reviewed for you for all time.
The Pythagorean Comma is most easily expressed by stating that no power of two can equal any power of 3. If take a very low tone, represented by the lowest C on the piano and assign that the value of 1, and go up to the top C on the piano, we encompass 7 octaves, and since octaves are expressed as the ratio of 2:1, we find ourselves with the number 128, which is 2 to the seventh. If we go back to that low C and play up by perfect fifths on the keyboard, we of course go up the Cycle of Fifths...
...which for the purposes of Pythagorean thought should be notated as all sharp notes, ending on B#. In equal temperament, of course, the B# becomes the enharmonic equivalent of C natural, but in Pythagorean tuning we need to think in all perfect fifths. Thus, we must go up in perfect fifths 12 times. This would be 3/2 raised to the 12 power (1.5 to the 12th). This results in the number 129.74632.
This ratio, 129.75632/128 is the Pythagorean comma. 12 perfect fifths do not equal up to 7 perfect octaves. This is the basic process in tuning equal temperament, where each fifth is reduced by a twelve of a semitone tone, thereby distributing out the comma equally among the twelve pitches of the chromatic scale.
Since symmetry is a big part of our work this semester, it's also useful to look at this cycle in both ascending and descending directions, so that the comma is seen to occur between the low Gb and the high F#:
The Syntonic/Didymic comma is the more critical to understand, as it directly effects our everyday music making and ability to tune in ensemble playing. It refers to the diffence between a major third that is tuned by perfect fifths with octave reduction, as opposed to the major third based on the ratio of 5:4, which is what also appears in the harmonic series.
The math here is simple. We've already seen that the Pythagorean major third is 81/64. Study the math above, if this is not clear. Not only was the third from C to E tuned in this way, but the major thirds between F and A and between G and B were also found to have this ratio.
Just intonation is based on the principle that these thirds sound somehow "out of tune" when performed in harmony with one another (as opposed to their appearance in a strictly melodic situation). This is because it is literally true. When a string vibrates, it generates the pure third above the fundamental, and the pure fifth. Not only does nature seem to provide for this pure interval, it also relates to the inner workings of our ear, where harmonic ratios that reflect lowinteger relationships are more easily processed within the ear. The process is to set the major triads that occur within the major scale (we think of them of course as tonic, dominant and subdominant) as possessing the ratio of 4:5:6, in this way:
C 
D 
E 
F 
G 
A 
B 
C 
D 
4 
: 
5 
: 
6 







4 
: 
5 
: 
6 





4 
: 
5 
: 
6 
Since all the major thirds become "pure", the ratios work out to be:
1 
2 
3 
4 
5 
6 
7 
8 
1/1 
9/8 
5/4* 
4/3 
3/2 
5/3** 
15/8*** 
2/1 
C 
D 
E 
F 
G 
A 
B 
C 
9/8 
10/9 
16/15 
9/8 
10/9 
9/8 
16/15 

*This is the pure 5:4, tuned above the C

**This is the third above the F, which is 4/3 * 5/4 = 20/12, which reduces to 5/3.

***This is the third above the G, which is 3/2 * 5/4 = 15/8
A glance at the above chart will show the other startling fact about Just Intonation. While it provides ensembles (especially Renaissance vocal ensembles) with the ability to tune pure thirds, it also results in two whole tones ratios (9:8 and 10:9). When singing or playing a nonfretted instrument, this is of course quite doable and professional vocal ensembles learn to create such variants to their scales. But for manufacturers of instruments, especially keyboard instruments, this proved to be an impossible situation, especially as composers wished to exploit the possibility of modulation to distant keys. The result of all this was a variety of meantone temperaments, which in essence found ways to tune the D and the A at the midpoint between the major thirds of C—E and F—A respectively.
The variety of meantone temperaments were common to organ building right up through the 19th Century, and thus in Europe this style of temperament can still be heard on any historic organ that has been preserved in the great cathedrals.
It should be clear now what the Syntonic or Didymic comma is: it is the difference between the Pythagorean third, which is 81/64, and the Just Third, which when expanded out to yield a common denominator, gives us 80/64 (5/4 * 16/16). Thie ratio 81/80 is the dydmic comma, the slight but perceptable difference in intonation between Pythagorean and Just thirds.
The Great Diesis
The last tuning concept to grasp is related to the Syntonic comma. The Great Diesis results from the fact that three major thirds, tuned in just intonation, do not add up to a perfect octave:

5/4 * 5/4 * 5/4 = 125/64.

An octave, expressed with the same denominator, would of course be 128/64.

The ratio 125/128 is the Great Diesis, the difference the pure octave and the stack of pure thirds.
Thus, we finish a summary of basic musical ratios, which should be known by all musicians, and must be known by you because I will quiz you on it!
The next link will be a brief review of the concept of mode and scale before moving on to Harmonic Polarity.