History of Music Theory, Part Eight

The subject of mathematics as it relates to music might seem boring, tiresome, and confusing - which is how I originaly viewed the subject - until one sees a graphical representation that illustrates the elegant simplicity of it.



Returning to the natural harmonic overtone series, if we simply number the harmonics sequentially starting with the numeral one for the fundamental, the series actually calculates all the necessary ratios for us! Above I have included the series in its sine and cosine forms, which will be necessary for following discussions. First, here is a list of the perfect and imperfect consonances.


1) Between C1 and C2 the ratio is 2:1, or the perfect consonance of an octave.

2) Between C2 and G3 the ratio is 3:2, or the perfect consonance of a fifth.

3) Between G3 and C4 the ratio is 4:3, or the perfect consonance of a fourth.

4) Between C4 and E5 the ratio is 5:4, or the imperfect consonance of a major third.

5) Between E5 and G6 the ratio is 6:5, or the imperfect consonance of a minor third.

6) Between G3 and E5 the ratio is 5:3, or the imperfect consonance of a major sixth.

7) Between E5 and C8 the ratio is 8:5, or the imperfect consonance of a minor sixth.


Note that the first five ratios are superparticular in nature (The numerals in the ratio are adjacent numbers differing by only 1), but that the sixths are not superparticular ratios. The numerals in the ratio that describes the major sixth differ by 2, and the numerals in the ratio that descibes the minor sixth differ by 3. Theorists down to and including Zarlino struggled with classifying consonances. Zarlino proposed that all superparticular ratios be considered perfect consonances, which makes only the sixths imperfect, but includes the thirds as perfect. This is obviously not right.

Just as the idea of octave inversion helped me to break through the "rules" of counterpoint - rules that previous theorists had deduced from analyzing a particular composer or school of composers - and get to the underlying laws that govern musical motion in counterpoint, octave inversion serves the same purpose here.


1) Perfect consonances maintain a superparticular ratio when inverted at the octave (Octave, fifth, and fourth).

2) Imperfect consonances are superparticulars in only one of the two possible octave inversions (Thirds and sixths).


This properly describes the difference between perfect and imperfect consonances, and furthermore displays for all to see why imperfect consonances are imperfect: They are not superparticular ratios in both positions of an octave inversion, but rather are superparticulars in only in one of the two positions.

In the previous post I mentioned that theorists wrestled with the syntonic comma, which is the difference of 81:80 between two 9:8 wholetones and a 5:4 major third. We can see where the problem arises in the natural harmonic overtone series above. The area in question is from C8 to E10.


1) Between C8 and E10 the ratio is 10:8.

2) (10:8)/2= 5:4, or the imperfect consonance of a major third.

3) There are two different sized wholetones dividing the major third in the natural series.

4) The lower and larger wholetone between C8 and D9 is 9:8, and is called a major wholetone.

5) The upper and smaller wholetone between D9 and E10 is 10:9, and is called a minor wholetone.


So, there are no commas in nature, there are rather two different sized wholetones that divide the major third. In point of fact, there are four different wholetones in the natural series: 8:7, 9:8. 10:9, and 11:10, just as there are two different minor thirds at 6:5 and 7:6. The point is, trying to apply a just tuning scheme to a chromatic idiom where you want all twenty-four major and minor keys available is frought with headaches to the point of being practicably impossible. For that reason, a means of tempering either the Pythagorean or the Just tuning system was needed. The struggle wouldn't be pretty.

Remember, all art is a reflection of nature. It is almost time to address immitation, canon, and fugue, which describe in music the phenomenon of reflected sound known as echo. I leave you with some reflected light to enjoy which mirrors the sine/cosine harmonic series above.

History of Music Theory, Part Seven

During the period of the 14th and 15th centuries the practice of musica ficta - adding sharps or flats to notes to increase the contrapuntal effect of directionality and to accent closings (Which can properly be called cadences by this point) - had resulted in the theoretical completion of the chromatic octave. The result was not the twelve note chromatic octave with enharmonic equivalents whith which we are familiar, however. To the contrary, there were no enharmonic equivalents at all originally, which resulted in a seventeen note chromatic octave:


C

d-flat
c-sharp

D

e-flat
d-sharp

E
F

g-flat
f-sharp

G

a-flat
g-sharp

A

b-flat
a-sharp

B


This corresponds to natural vocal practice in which leading tones are raised and leaning tones are lowered slightly to add tension penultimate to the resolution effect. In fact, this practice can quite easily be heard today by listening to any good performance by a string quartet, so it's not just a phenomenon unique to the vocal idiom. Keep in mind though that theorists were still using their monochords and that they were dutifully marking these intervals off on them, and then they would have to rationalize the results to the Pythagorean tuning system's intervals that they were using. Obvously, this caused a lot of head scratching and endless mathematical calculations. One result of this was the discovery of the syntonic comma and it's precise enumeration as 81/80. This is the difference between two wholetones of 9/8 and the major third of 5/4. The point is, theorists were beginning to sense that another system of tuning or a system of temperament would be required for fixed-pitch instruments, since they accompanied singers with increasing frequency over the course of this period. Splitting all the black keys on a keyboard is possible, and that was tried on a few occasions in a limited fashion, but obviously that would not really be practical due to the resulting mechanical complexity and expense, not to mention performance difficulties.

Though credit for finding the theoretical derivation of major and minor thirds as imperfect consonances with the ratios of 5/4 and 6/5 as present in the natural harmonic overtone series was given to Bartolomeo Ramos de Pareja as written in his musica tractatus of 1482 for many years, they were actually properly described by Walter Odington circa two-hundred years earlier. Odington was very much a man of his age and thouroughly devout with respects to the Pythagorean tuning system however, so he basically noted that thirds sounded consonant when sung because singers adjusted them to conform to the simpler natural ratios and left it at that. In any event, it was not until the end of the fifteenth century that theorists generally started to recognize the rising to primacy of thirds and the resulting necessity for a corresponding change to the Pythagorean tuning system that had been in use for about a millennium.

The late fifteenth century was a kind of golden age for music theory, and theorists would often engage in public scholarly debates on the subject. They also circulated polemical pamphlets to defend their positions. Some of these debates were evidently quite heated exchanges, and the pamphlets no less so. I would have loved that age in that one respect, but since some of those men died of the bubonic plague, I think I'll accept my particular place in history with a cheerful disposition. Some of the players in the late fifteenth century era's hubbub were Bartolomeo Ramis, Giovanni Spataro, Nikolaus Bertius, Philippus de Caserta, Franchino Gafori, John Hothby, and of course, Johannes Tinctoris. Awesome names all, and fun to say as well.

Miscelaneous Musings, Three

With the Ars Nova movement, music enters the Renaissance, and a riotous explosion of compositional activity lead by some very talented men occurs. The revolution in music around 1430 was so profound and so complete that Johannes Tinctoris was inspired to write in his Liber de arte contrapuncti. of 1477 that only compositions written "in the last 40 years" are worthy to be listened to. He then goes on to list a veritable Who's Who of famous early Renaissance composers: John Dunstable (c.1390-1453), Gilles Binchois (c. 1400-1460), Guillaume Dufay (1397-1474), and their brilliant pupils Johannes Ockeghem (1420-1497), and the estimable Antoine Busnois (c.1430-1492), among other luminaries of no less stature. This music proved to be so seminal and so enduring that we have preserved for us to this day copies of some of Ockehgem's works in J.S. Bach's hand, which he undoubtedly studied in great detail as was his habit with composers whom he revered.

It should be noted that practice was still ahead of theory at this time, and in fact theory would ever after be in a reactionary mode struggling to explain the genius of the men who were leading the Secular Humanist movement in the arts throughout the Renaissance and beyond right down until modern times. In the last century, for example, the rules governing the music of Palestrina were so well understood and precisely enumerated that computer programs were written that could create quite convincing Palestrina-style compositions. This illustrates a point that I think is essential if a musician wants to really understand composition: There are rules that theorists retroactively formulate to explain the style of a certain composer or school of composers, and there are underlying laws that govern musical motion that are the irreducible essence of a certain compositional approach, such as counterpoint, which I am considering now.

In my previous posts I have presented the first of the underlying laws governing musical motion in counterpoint: 1) No parallel perfect consonances, and 2) Unrestricted parallel imperfect consonances. I was also careful to show the descent of those laws from implications present in the Natural Harmonic Overtone Series. This is important to understand, because in my musico-religious philosophy, Everything Musical is a Reflection of Nature. Conversely, everything I object to and am critical of in music can be proven by these methods to be unnatural, and in the kindest possible terminology, the creators of those works of anti-music were at the very least passively ignorant of the implications of nature in art. It is my hope that I will be able to prove well enough to satisfy the open minded that these men were in actual point of fact willfully ignorant and dispensed with natural considerations with malice.

One last item: In my previous discussion concerning problems with the perfect fourth, I mentioned that per the rules of harmonic voice leading, parallel perfect fourths were acceptable in three or more voices so long as the bass was not involved, and this is true. The laws governing musical motion in harmony are subtly different than contrapuntal rules, so when a composer uses parallel perfect fourths in counterpoint, what he is doing is borrowing from another set of rules governing a different kind of musical motion. I arrived at the irreducible rules one and two above by regarding as an inherant property of correctly written counterpoint that it is invertible at the octave. In so doing, I arrived at the ultimate in elegant solutions, and one that dispenses with needless confusion in teaching counterpoint when harmonic motion laws are confused with contrapuntal laws. Now, I don't claim to have come up with this on my own, but many different theorists that I have studied over the past twenty-seven years contributed to these conclusions. As we shall see later when we address the laws governing harmonic motion, teaching that subject via voice leading rules is not the correct way to reach an understanding of the subject, because the real law that governs harmonic motion is that of circular permutations of the chord's member tones: Clockwise or counterclockwise circular permutations for triads, and clockwise, counterclockwise or crosswise permutations for seventh chords. So far as I am aware, nobody except Joseph Schillinger has ever described this phenomenon in a theoretical treatise.

History of Music Theory, Part Six

Immediately after the first great flowering of diaphonic and polyphonic Western Art Music at Notre Dame with Leonin and Perotin, music theory starts working it's way from the Ars Antigua to the Ars Nova and counterpoint as we are accustomed to thinking of it today, versus the older organum. Counterpoint is an anglicisation of the word Contrapunctus, which is itself a contraction of the term Punctus Contra Punctum meaning note-against-note. These terms appear with increasing frequency in treatises of the 13th century until the terminology is mostly standardized by the end of the 14th century into the 15th. The break with older practice is in no way a clean one, and the striving for terminology and "rules" is actually quite a messy affair.

Another term that appears is Puncti Contra Punctum - notes-against-note - and in association with that the term cantu fractibili, which refers to a melody added to the cantus firmus that moves faster in smaller note values. Here we find some of the first attemps at handling dissonance, but it is not treated in a rigorously syatematic way. Rather, it is mentioned more or less in passing that dissonances are allowed after strong-beat consonances on subsequent beats before the next required strong-beat consonance. No doubt this had been a reality of discant practice for many years, as the final musical example I presented earlier shows.

What is of primary interest to me here is the evolution of contrapuntal rules during the 14th and 15th centuries. Previously, I had shown how the evolution of the concept of consonance worked it's way up the natural harmonic overtone series, and the rules of counterpoint relate to the overtone series as well. Prohibitions against parallel movement of unisons and octaves are quite easy to understand, as the effect is that of the cantus firmus simply being doubled: No melodic independence is maintained. The effect of parallel perfect fifths is very close to that of parallel perfect octaves because the low 3/2 ratio of perfect fifths results in a sound second only to the octave in hollowness, and many of the upper partials of a note a perfect fifth above a fundamental are simply absorbed by that fundamental's overtones.

The perfect fourth could possibly be considered the most problematic interval in all of music theory. There are several reasons for this. One is that, of the perfect consonances, it has the highest harmonic density (The thickest sound), and for that reason shares more characteristics with the imperfect consonances and even the dissonance of a major second than it does with the perfect fifth. Another relates to that, and it is that many vocal formants hover around the fourth, and so the voice at the fourth below has it's syllables interfered with and is less than perfectly intelligible. This wasn't a problem in 1:1 ratio early organum where both voices were singing the same text, but it became a consideration as the voices and texts gained independence. The final reason is that the perfect fourth implies the harmony of a 6/4 chord or a 4-3 suspension to us, and those sonorities require a resolution to our ears. Lest you think that is a modern conditioning that the ancients would have been immune to, in some of the earliest treatises dealing with three voice writing, the 6/4 arrangement was singled out as something that was expressly forbidden. If you are familiar with modern voice leading rules as they apply to harmony, you know that parallel perfect fourths are OK in upper voices in three or more parts, but if you are trying to write fully invertible counterpoint in three or more voices, they must be avoided. For our purposes here, that's the way we are going to treat the perfect fourth: Parallels are not allowed. This simplifies the rule to no parallel perfect consonances, which are those found between the fundamental and the first three overtones.

Once the imperfect consonances had been worked out, which I went through in an earlier example, the second rule could be formulated: Parallel imperfect consonances have no restrictions, except those governed by taste. This is very easy to grasp. The interval density of thirds and sixths is higher than octaves and fifths, and no more than two diatonic thirds or sixths in a row are of the same size: Though momentarily coupled, the melodies retain their independence. So, the imperfect consonances found above the third harmonic are allowed to move in parallel, but the perfect consonances below the third harmonic are not. Simple really.

Way down the road from here, when I tackle "The Schillinger System of Musical Composition" again, I will show how these early formulations of rules for the handling of consonance and dissonance in countrapuntal writing were strivings toward the ultimate goal, which is the concept of combining two or more complimentary melodic trajectories into a cohesive whole. It could be years.

History of Music Theory, Part Five

The first flowering of polyphony in Western Art Music came around the end of the 12th century with the Ars Antigua as represented by the Notre Dame school of composers. Magister Leoninus and his brilliant student Perotinus Magnus - commonly known as Leonin and Perotin - were the right men, with the right talent, in the right place, at the right time. In order for this first great leap to occur, mensural notation and diaphonic theoretical and proceedural concepts had to be worked out to a high level of sophistication. Everything in Reimann's "History of Music Theory" so far has been leading up to this: A highly developed and fascile form of organum.

There really is no substitute for actually listening to this music. If you would like more information about the Notre Dame school and Leonin and Perotin there is a great article on that subject posted at Goldberg that also has a discography that can lead you to some excellent recordings.

Though there is no arguing the fact that the Notre Dame composers made a fantastic contribution to the art of music, there is also evidence that there was a robust and vibrant popular music movement at the same time that was in some ways much more modern, or at least would seem so to our ears anyway. The earliest theoretical treatise that deals with popular forms in detail is Johannes de Grocheo's Theoria, which is dated around 1300 AD. Grocheo is a man after my own heart, as it seems he broke all the rules of his time and aimed his treatise not at the scholarly classes, but at the young, educated members of the secular upper classes. Grocheo only mentions as much as he feels he has to about the Church's musica ecclesiastica, but treats the contemporaneous musica vulgaris in great detail. Theoria gives the distinct impression that there was more than just a little cross pollenization going on between Church and popular music, and he actually has the audacity to dispense with the rigors of theory where it fails to describe popular practices. He describes everything from ballad-like songs to dances that evidently had vibrant rhythmic aspects to them. I found it interesting that most of these songs and dances were not in anything like the Church modes, but were rather in the keys (!) of A minor and C major, which was know as modus vulgaris at that time. This is the year 1300 we are talking about! I can't think of any better example of how far behind practice theory had gotten by this time. In some aspects, it would be centuries before theory would fully catch up.

Miscelaneous Musings, Two

Now that I have made the points I wanted to make concerning the early evolution of consonance and dissonance and the relationship that evolution had with the natural harmonic overtone series and the Pythagorean tuning system, I thought some musical examples would be in order to graphically demonstrate the evolution of early diaphony.

I took these examples directly out of "History of Music Theory", but keep in mind that they are nevertheless generalizations and their relationship with actual practice seems somewhat tenuous. The examples do, however, represent what the ancient theorists seem to be describing in their treatises. In that light, I simply accept them as useful tools to get an idea of the evolution of musical thought represented in standard notation that is easy to visualize.

Note also that the idea of a defined meter with a definite metric pulse had not developed in these early centuries and so the rhythm, such as it is, should be considered to be very elastic and subject to the singers' interpretation. I actually added note values to the final example of discant so that it was easier for me to visualize, but that should in no way mislead you into thinking that the singers performed that kind of music to anything other than the most plastic conception of meter.



In the earliest form of diaphony, called Organum, the chant or Cantus was the upper voice. Initially, the theorists of the 9th century describe a practice wherein the entire chant is "harmonized" at the perfect fourth below by the organal voice. This practice corresponds to the top example above labelled "9th Century".

The next step in the evolution of organum - represented by example 2 above labelled "10th Century" - was for the voices to open and close on a unison. Some examples from this period show a third as the penultimate interval, but others have a major second indicated. In any event, these seem to be the earliest seeds of contrary motion, but I believe it would be premature to consider the closes to be "cadential", except perhaps in the most primitive possible sense.

By the eleventh century at least two advancements seem to have been made more or less simultaneously: 1) The cantus is now written in the lower voice, and 2) The organal voice is now gaining some independence at points other than the opening and the close. The term Organum is still being used to describe this practice, but it is not what most of us probably think of as organum proper, but rather some intermediate stage of evolution toward discant and counterpoint. Notice how all of the perfect intervals are used in this example: Perfect unison, perfect fifth, perfect fourth, and perfect octave, but that there are no imperfect consonances or dissonances. By this time it is possible, perhaps even probable, that these 1:1 ratio compositions were embellished by the singers on the organal voice, but there seems to be no absolute certainty about that.

The final example of twelveth century Discant practice is from a rather didactic treatise and is not a "musical" example, strictly speaking, because the "cantus" is not a real chant. It was the best I could do, however, as I didn't want to get into making up my own examples. Here I was figuring that authenticity was the only way to go, and would be infinitely better than any wild-assed interpretation of discant that I might come up with. Notice that even though the example starts out with an octave, the progression is nothing more than an ornamented progression of parallel perfect fifths. Not only that, but there is also no attempt to hide this progression with contrary motion: The parallel nature of the progression is actually accented. I was also struck by the fact that the neighboring and passing major seventh intervals at the beginning are quite spicy, and there is no attempt to rectify the neighboring augmented fourths at the end.

I decided to skip over the fauxbordon/faberdon/gymel practices of the thirteenth century, even though they employ the imperfect consonances of thirds and sixths, because there were no good examples and the description of the practice seems to be one of an organum-type parallel motion practice in thirds or sixths with octave doublings or transpositions. Asside from the use of imperfect consonances, I didn't really see how much was contributed to the evolution of counterpoint by these schools of composition, and counterpoint's evolution is the thread I want to pick up here.

History of Music Theory, Part Four

Last night I labored my way through the first chapter of Book II, which is chapter VIII, and it concerns itself with mensural notation to the beginning of the 14th century. I must admit to having less than no interest at all in this particular area of study, and I found myself nodding off from time to time. As a practical musician, I simply use the notational practices handed down to me, and I could hardly tell a neume from a plume of mustard gas if my life depended on it. In any event, Riemann made several incorrect assumptions and there seem to be so many areas of dissagreement and conjecture that I found myself hoplessly confused concerning several points about how neumes were actually employed in the earliest eras in which they appear.

About the only thing that really kept my interest was the continued unfolding pantheon of magnificent names. Hieronymus de Moravia is a cool name, and I didn't previously know it translated to "Jerome". There is a lady I met once with the last name of Hieronymous who is a prominent supporter of the arts, and now I'm wondering if I should address her as Ms. Jerome next time I run into her. Probably not. I also ran into the name of the earliest of the composers I really like, Perotinus Magnus, as well as Petrus de Cruce, the optimus notator. Aribo Scholasticus has to be an envious name among the scholarly class, but my favorite of all so far is without doubt, Prosdocimus de Beldemandis. Say that a few times. It really is delicious.

Fortunately, I have a couple of more points to make about the evolution of consonance and dissonance I'd like to get across. I mentioned yesterday that theorists were having a hard time rationally coming to terms with imperfect consonances because they were trying to relate them to the Pythagorean tuning system, and that practice had gotten ahead of them for this reason. Be reminded that the evolution of consonance was still an incramental process. After the unison, octave, perfect fifth, and perfect fourth that were the original consonances, thirds were next to appear. Sixths were initially treated as dissonances, but then they were admitted as consonances; first the major sixth and then the minor sixth. Note how this progression goes up the natural harmonic overtone series.



If we label the pitches as C1, C2, G2, C3, E3, G3, B-flat3, C4 - which are all the pitches we will need for this demonstration - the point can easily be made. In the beginning of diaphonic music, only the perfect unison C1-C1, the perfect octave C1-C2, the perfect fifth C2-G2, and the perfect fourth G2-C3 were admitted as consonances. Note that these are all found in that part of the series that encompasses the fundamental and the first three overtones. Note also that these intervals retain the appelation of "perfect" to this day.

Next, the thirds were admitted: The major third C3-E3 and the minor third E3-G3. At this point the fundamental and the first five overtones had been explored, but not completely. It may seem strange that the sixths were admitted incramentally until you consider that the major sixth first appears between G2-E3 - within the range of the fundamental and the first five overtones - but the minor sixth not until E3-C4, which requires that the range be extended up to the seventh overtone above the fundamental. Notice that all of the basic elements for invertible counterpoint are now in place, and that these intervals all come down to us with the appelation "imperfect consonances" still attached to them. As for the sixth overtone at B-flat3, the implications that it would have would have to wait for the development of cadential and harmonic concepts. That would take... er... a while.

History of Music Theory, Part Three

Last night I finished Book I of Riemann's "History of Music Theory", which took me through Chapter VII and to the beginning of Book II. Book II starts with mensural notation theory and goes through systematic countrapuntal theory. I had gotten to the point of wanting to make some more general observations, and to draw some conclusions from those observations, but I wanted to review a little more about Pythagorean Tuning versus the Just Intonation that is derived from the natural harmonic overtone series first. This lead me on a vast transcendental, transcontinental surfing session that gave me many insights into the nature of music and it's evolution as influenced by tuning and temperament that I had not percieved before. It wore my little brain out so completely that I had to take a nap, which explains the relative lateness of today's post.

But first, from The Department of Interesting Names: I particularly enjoy saying Guilemus Monachus, Giraldus Cambrensis, Petrus Picardus, and - last but not least - Magister Philippotus Andreas. I like things that are fun to say. Honorable mentions for Johannes Tinctoris and Franchinus Gafurius, and pity for poor Anonymous IV.

After my previous readings about the tenth and eleventh centuries, I pointed out that Western music theory started out as an investigation into the nature of sound as revealed by the natural harmonic overtone series, and that this information was borrowed from the Classical Greek culture. The early Western theorists also adopted the Pythagorean tuning system, which was a seven note diatonic scale built via a series of seven justly intonated perfect fifths starting on F: F, C, G, D, A, E, and B. This may have been partly because the perfectly just fifth expressed by a ratio as 3:2 has theological significance in the Christian religion. Or not. To avoid tritone relationships, the stack o' fifths was added to at each end to get the B-flat and F-sharp respectively.

In any event, by the 12th and 13th centuries, musical practice had gotten ahead of theory because of the conservative nature of the theorists connected with the Church and their reluctance to abandon the Pythagorean tuning system. If we compare diatonic scales built on the natural harmonic overtone series - so-called Just Intonation - with the "same" diatonic scale that results from the stack of Pythagorean fifths, we can easily see where the problem is.


Pythagorean:

C= 1:1
> 204 cent Wholetone
D= 9:8 (+204 cents)
> 204 cent Wholetone
E= 81:64 (+408 cents)
> 90 cent Semitone
F= 4:3 (+498 cents)
> 204 cent Wholetone
G= 3:2 (+702 cents)
> 204 cent Wholetone
A= 27:16 (+906 cents)
> 204 cent Wholetone
B= 243:128 (+1,110 cents)
> 90 cent Semitone
C= 2:1 (+1,200 cents)


Just: (Natural Harmonic Series)

C= 1:1
> 204 cent Wholetone
D= 9:8 (+204 cents)
> 182 cent Minor Wholetone
E= 5:4 (+386 cents)
> 112 cent Semitone
F= 4:3 (+498 cents)
> 204 cent Major Wholetone
G= 3:2 (+702 cents)
> 182 cent Minor Wholetone
A= 5:3 (+884 cents)
> 204 cent Major Wholetone
B= 15:8 (+1,088 cents)
> 112 cent Semitone
C= 2:1 (+1,200 cents)


At first blush the Pythagorean system would seem to have a lot in it's favor: There is only one size of Wholetone, there is only one size of Semitone, and the Perfect Fourths and Perfect Fifths agree with Just Intonation, as does the size of all of the Wholetones (The 9:8 Wholetone, that is). But when you look at the thirds and sixths, you see the problem right away: The major Third from C to E is 22 cents sharp (Nearly 1/4 of an Equally tempered Semitone!), and the Major Sixth from C to A is the same 22 cents sharp compared to the justly intonated intervals. Obviously, thirds and sixths would sound awful in this tuning on the monochord. But, since the medium of the time was primarily a vocal one, and singers tend to make their thirds and sixths conform to the natural justly intonated versions, composers and singers started using them anyway. So, a method of tempering one or the other of these tunings was needed so that theory could rationally explain practice. It would take a while.

Miscelaneous Musings, One

A friend's band played last night and I attended the show, so it was relaxation with beer and buddies instead of study yesterday. All work and no play and all of that. I did manage to run into a local hotel manager though, and he was interested in a classical guitarist for Saturday nights, so I landed myself a new weekly gig. There's nothing wrong with a little mixing of business with pleasure.

As I have been considering the state of our understanding of music of the earliest periods of the Western tradition a few things seem unlikely to be cleared up by modern scholarship. Performance practice bfore the advent of mensural notation seems likely to remain in the realm of conjecture. So too then does the true nature of the musical examples presented alongside or along with the earliest theoretical manuscripts. Since there really is no way to answer some of these questions, and I don't wish to get bogged down in such minutae anyway, I have decided to make the evolution of the theoretical concept of consonance and dissonance and that concept's relationship to the natural harmonic overtone series one of the main thrusts of my investigation, at least in the short term. Later, I'll pick up the thread of the evolution of tuning (By that I mean concert pitch level), and temperament (The various ways devised to account for the diatonic and syntonic commas). As you will learn later, these concepts and their evolution have everything to do with my views on the nature of music and my... ahem... strongly critical stance against much of the music of the twentieth century.

Well, I have a gig to prepare for. I plan on studying a few more chapters of Riemann when I get home tonight, so I'll see you tomorrow.

History of Music Theory, Part Two

I studied the first four chapters of "History of Music Theory" yesterday. I love the old names one encounters in early music history: Flaccus Alcuin, Boethius, Odo of Cluny, and of course Hucbald were among those I encountered. Parents have definately lost the art of naming their clildren, and you combine that with modern scholarship's abandonment of latin and people just don't end up with impressive monikers like that anymore. Somehow "Tyler of Denver" just doesn't have the same gravity.


Anyway, a few observations:

1) Western music theory started out as a scientific investigation into
the nature of sound around the middle of the first millenium AD.

2) The results of that investigation were pre-existing and were borrowed
from the Classical Greek culture of the middle of the first millenium BC.

3) The medium used for experimentation and observation was the vibrating string.

4) The phenomenon that was discovered was the natural harmonic overtone series.

5) The intervals first used harmonically were the perfect consonances: Octave, fifth, and fourth (And, of course, unison).

6) These intervals correspond to the first three overtones above the fundamental in nature's harmonic overtone series.


One thing I finally understand now is my modern reaction to the interval of the perfect fourth, which was the original organal voice beneath the cantus in early organum. We tend to want to hear that interval resolve downward to a third, as in a 4-3 suspension resolution, because of our ingraned exposure to modern temperaments and their allowances for thirds as consonances. Though early common practice folk singing likely evidenced to at least some of these ancient theorists that thirds were usable consonances in practice, the Pythagorean tuning of their monochords told them that thirds were dissonances (In Pythagorean tuning major thirds at 81/64= 408 cents and minor thirds at 32/27=294 cents and are quite unstable) , and so thirds were theoretically and practically disallowed, at least as consonances, in what was the beginning of Western Art Music proper.

In fact, since the Pythagorean tuning system was codified as the only one allowed to be used, a prima facia case could be made that early music theory acted as a brake that slowed musical progress. As late as the fourteenth century the French Academy at Notre Dame decreed that only a series of perfect fifths could make up a scale. But, I'm jumping ahead.

History of Music Theory, Part One

This will be my third and final time through Hugo Riemann's "History of Music Theory" (Da Capo 1974, ISBN 0-306-70637-7 - If I find a book that I mention here for sale via Google, I'll link to it. I found no copies of this book on the market, but any decent University Music Library ought to have this one. If you find a copy of it for sale, feel free to provide the link in comments). To prepare, I did all the "pre-reading" of the Preface to the Second Printing, Translator's Preface, Acknowledgments, Indicies, Appendicies, Bibliography, and Commentary yesterday. I was again reminded that there are more than a few problems with Riemann and that modern scholarship has moved on, making significant progress since this seminal work first appeared. Not surprising, considering that the original German edition of this book is now nearly a century old. Since I am not a musicologist and my aim is to extract the most significant threads from music history to answer the questions, "How did we get to where we are in music today?", "What is and what isn't music?" and the like, this will do fine as a survey. Besides, many problems with Riemann's original edition are noted and corrected in the Commentary and in updated footnotes.

I am not interested in the details of the hurly-burley of any particular era, but rather want to understand what is the essence of western art music's compositional heritage: I want to understand the gist of the tradition since I am ostensibly a part of it. Having already done study in this area on and off for over fifteen years, I already have a very good idea of the points I want to get across, so I won't be taking too much time with Riemann, but I will point out interesting tidbits as they come up and as they support the broader points I will be making.

Back to my study.

A Monk's Musical Musings

This blog will be used as a forum for the expression of my thoughts about music, which is the love of my life. More speciffically, my thoughts about the evolution of western art music and music theory. I have decided to re-study my way through my library of music theory books starting with Hugo Reimann's "The History of Music Theory" in order to bring my compositional technique up a notch or two, and as I go, I will be sharing my observations and drawing conclusions from those observations. If you happen to find this blog and are a musician - performer, teacher, or composer - I welcome your comments, whether you agree with me or not and whether you are in the traditional, jazz or popular generas.

Since my undergraduate degree is a jazz composition degree, and my masters degree is in traditional composition, I hope these discussions can be broad and wide ranging. The point here is for me to formulate my thoughts and increase my understanding of the art and science of music (And music was classified as a science in the Quadrivium). Perhaps we can all learn from each other in this endeavor.