Book 1

Summary

After reading the treatise and going back through my notes, this is definitely a case where reading the book out of order is probably better for understanding some of the concepts in Book 1. Aristoxenus introduces a lot of concepts that won't end up getting explained entirely until Book 2 and Book 3. That being said, some of the big takeaways are:

1. Aristoxenus is noticing a trend of not giving Enharmonic and Chromatic genera the respect they deserve.
2. Aristoxenus has beef 🐄 with the Harmonists (namely Eratocles), whose theories have proven to strip music of its humanity.
3. Aristoxenus is a proponent of the voice as a metric for determining what should be theoretically relevant.
4. There's a distinction between speaking and sining, and it mainly has to do with discernable pitch.
5. Even though some intervals might be deemed irrational to Aristoxenus' predecessors, the human ear can still discern them and use them for practical music-making.
6. Tetrachords are a series of (usually) four notes (three consecutive intervals) that occupy the span of a perfect Fourth. The smallest interval is always between the lowest two pitches, and must be between 1/4 tone and 1/2 tone. The highest interval must be between 1 and 2 tones.
7. There are three families (genera) of tetrachord: Enharmonic, Chromatic, and Diatonic, and at this point in history, Diatonic tetrachords are the newest and most unfamiliar.
8. There are certain tetrachords that are more favored than others, but in theory, tetrachord intervals are an infinite spectrum, so long as their first and last intervals fall within the ranges above.

"[The enharmonicists] confin[ed] themselves to what is but the third part of that complete system, they selected for exclusive treatment a single magnitude in that third part, namely the Octave." (166)

This quote is introduced early in the book, and portrays Aristoxenus in a snarky light. At this point, I knew nothing about the enharmonicists or the complete system, but I do know that Aristoxenus took issue with how his predecessors provided a limited view of music theory.

“The truth is that of all the objects to which the five senses apply not one other is characterized by an orderliness so extensive and so perfect.” (168)

This is one of the first hints we get in the book that discusses the idea of sense perception as an important factor in finding significance in music theory. Aristoxenus suggests that the order in which we classify sounds is unique to hearing as a sense, and that instead of relying purely on the mathematical relationships between intervals, we should also account for our tendencies and intuitions when it comes to hearing and producing pitch.

"Every voice is capable of change of position, and this change may be either continuous or by intervals." (170)

Aristoxenus presents a binary of continuous and intervallic vocal delivery. Continuous refers to the "motion of speech" (171), where he's observed that the voice never stays on a single identifiable pitch for a long enough time to perceive it. Intervallic refers to the opposite, where the voice moves between stations that have discernable pitch.

"...tension takes place when the string is in motion, height of pitch when it is stationary." (173)

Here, Aristoxenus discusses 4 qualities of a string in motion: tension, relaxation, height, and depth. Tension refers to the motion from a lower frequency, to a higher frequency, and relaxation is the opposite. Somewhat confusingly, height and depth are used separately to describe the result of tension and relaxation, respectively. Aristoxenus justifies the need for both sets of terms by suggesting that one set of terms is used to describe the motion, and the other set is used to describe the result of the motion. Even though this quote translates to refer explicitly to a string, I think it's safe to assume that Aristoxenus is using the string as a metaphor for the states that any pitch producing object can be in.

"What the voice cannot produce and the ear cannot discriminate must be excluded from the available and practically possible range of musical sound." (175)

In this quote, Aristoxenus is plainly suggesting that intervals that aren't practical to produce, nor readily discernable by ear, shouldn't be focused on theoretically. In this section, he introduces the terms in parvitatem and in magnitudinem to describe 2 types of interval relationships, the former being too small to sing/hear accurately, and the latter being large enough to sing/hear.

“[A note is] the incidence of the voice upon one point of pitch. Whenever the voice is heard to remain stationary on one pitch, we have a note qualified to take a place in a melody.” (176)

Aristoxenus defines notes as remaining on a pitch for a period of time. This makes sense, but I'm curious about whether or not this brings into question embellishments like slides and scoops. Could a note not consist of a gradient of pitches?

“An interval, on the other hand, is the distance bounded by two notes which have not the same pitch.” (176)

Intervals are the distance between two unlike pitches. Another question comes up regarding whether or not unisons are intervals in Aristoxenus' theory.

“A scale, again, is to be regarded as the compound of two or more intervals.” (176)

This definition challenges our modern usage of the word scale. A "compound of two or more intervals" would more likely fit our definition of a chord, which Aristoxenus also uses somewhat interchangeably throughout his treatise. However the important things to bear in mind are:

1. Scales are a minimum of 3 notes in length (2 intervals results in 3 notes total).
2. They don't have to occupy the span of an octave.

“Here we would ask our hearers to receive these definitions in the right spirit, not with jealous scrutiny of the degree of their exactness...and to consider our definition which affords an unexceptionable and exhaustive analysis is a difficult task in the case of all fundamental definitions…” (176)

I thought this quote of Aristoxenus getting defensive about his definitions shows a bit of vulnerability (or trouble to appear vulnerable) on his part. He appears to be aware that these definitions are broad, and leave some room for criticism; however, he also suggests that making definitions that are universally true is a challenge. It reads a little like a cop-out in place of providing examples to reinforce the definitions' legitimacy.

"The first classification of intervals distinguishes them by their compass,...concordant or discordant,...simple or compound,...divi[sion] according to the musical genus,...[and] rational or irrational." (176)

Aristoxenus goes into detail about intervals here, suggesting that they have certain qualities that allow them to be classified.

1. Compass refers to the relative distance of intervals. For example, C to D would have a smaller compass than C to F.
2. Concordances are what we would call perfect: Fourths, Fifths, Octaves, and their compounds. Notice that Unisons are excluded, since Aristoxenus will go on the refer to the Fourth as the "smallest concord." Discordant intervals are all non-concordances.



3. An interval is simple if it's less than or equal to an octave. Compound intervals are simple intervals with an added octave.



4. Division by musical genus is trickier and isn't exactly clarified later. Genus refers to the three families of tetrachords, which are groups of pitches spanning a perfect fourth that serve as the building blocks for longer scales (like the ones we are familiar with). Though an interval may be in multiple tetrachords, its context differs. For example, both Diatonic and Enharmonic tetrachords contain a semitone, but in the Enharmonic genus, it's divided into two quarter tones. I think this relates to Aristoxenus’ ideas of continuity and transilience.



5. An interval's rationality, ultimately, has to do with it's ability to be expressed as a ratio of whole numbers. A perfect Fifth, for instance, can be represented as a 3:2 ratio above the reference pitch (when using Pythagorean tuning). However, Aristoxenus puts more of an emphasis on the ear's ability to perceive pitch. So, Aristoxenus makes use of intervals like the quarter tone, for instance. It can be identified by ear, and sung, but expressing it as a whole number ratio isn't practical.

“In scales will be found, with one exception, all the distinctions which we have met in intervals.” (177)

Aristoxenus says that scales follow all the same classifications as intervals, with the exception of simple and compound. While removing that quality, he adds three more in its place.

1. When you have two tetrachords that are linked together by a shared note, the scale they form is considered conjunct. If the two tetrachords are separated by a tone, the scale they form is considered disjunct. In order for a conjunct scale to complete the octave, an additional tone needs to be added above the last note of the second tetrachord.



2. Continuous scales are groups of notes that move from one to the next as they occur in the genus. Notes that move further than their neighboring note are considered transilient. The wording in the genus is important here, because some scales might be continuous in one genus, but transilient in another.



3. If a scale is made up of one or fewer tetrachords, it's considered single. If a scale is made up of 2 tetrachords, it's considered double. If a scale is made up of 3 or more tetrachords, it's considered multiple.

“Any melody we take that is harmonized on one principle is diatonic or chromatic or enharmonic...the diatonic must be granted to be the first and oldest, inasmuch as mankind lights upon it before the others...the chromatic comes next....The enharmonic is the third and most recondite…and with great labour and difficulty, that the ear becomes accustomed to it.” (178)

This quote stood out to me for a couple of reasons. First, and less important for now, is that Aristoxenus reintroduces the names of the three tetrachord genera: enharmonic, chromatic, and diatonic.

Second, and more importantly, is that Aristoxenus gives insight to the history behind them. Enharmonic is described with a sense of reverence due to it being the "first and oldest," since he suggests it was the first genus that was discovered.

Chromatic is also really interesting because all Aristoxenus says about it is that it comes second. There's probably a bunch of research that can (or has) been done about the Chromatic genus in particular, but seeing that we'll find out that the Chromatic genus is arguably the most complex of the three, it's interesting to see that Aristoxenus doesn't go into much detail about it here. The numbers in diagrams this point forward are the number of cents added or subtracted from the pitch below it due to the lack of a unique symbol to represent the division beyond a quarter tone.

Lastly, the diatonic genus, which we know to be most akin to the major scale using our modern sensibilities, is described to have little known about it in comparison to the other two. And, surprisingly to me, it's also described as needing "great labour" to get sonically accustomed to. This is fascinating because I find the other two tetrachords (you can use the calculator above along with the preset buttons to hear them right now) to be much more challenging due to the sub-semi-tone intervals. It makes me wonder about our ability to get conditioned to certain sounds, and whether or not Diatonic tetrachords were comprised of the intervals we think they were.

“The nature of melody in the abstract determines which concord has the least compass. Though many smaller intervals than the Fourth occur in melody, they are without exception discords...we find no similar determination for the greatest...concords seem capable of infinite extension just as much as discords.” (179)

This section reiterates a lot of what we already know. Aristoxenus suggests that intervals smaller than the Fourth are all discords. This would technically include the Unison, so the Fourth by definition is the smallest concord. He also suggests something I find really progressive which is the idea of infinite expansion and diminution. Concords don't appear to have a limit if you keep adding octaves to the smaller concords (Fourths and Fifths). Although our ability to hear and sing them stops at a point. Similarly, discords can continue to shrink as intervals within the Fourth are divided. I don't know whether or not this is objectively progressive, but my instinct about music from a historical perspective is that the further back in time you go, the more restrictive the music theory seems to be. Considering the history of Western European music is told through the lens of harmony becoming more and more complex, it's interesting to see writing that suggests possibilities for pitches that are smaller than the semitone, which wouldn't get a ton of attention until the 20th century.

"A tone is the difference in compass between the first two concords...may be divided by the three lowest denominators, as melody admits of half tones, thirds of tones, and quarter-tones, while undeniably rejecting any interval less than these...the smallest enharmonic diesis, the next the smallest chromatic diesis, and the greatest, a semitone.” (180)

I thought this excerpt was interesting because Aristoxenus goes into detail about defining what a tone is. Not only does it remind me that there are things we take for granted in music that we just accept as true because we were told they were true, but it also reinforces the idea that to Aristoxenus, the Fourth is the smallest concord. Subtracting the difference between the Fourth and the Fifths results in a tone.

He also goes into detail about divisions of the tone (which would result in irrational intervals, by the way) by the "three lowest denominators": 2, 3, and 4. This results in semi (half), third, and quarter tones.

Lastly, he introduces the word diesis, which is the smallest interval of a tetrachord. The quarter tone is the "smallest Enharmonic diesis," the third tone is the "smallest chromatic diesis," and the, "greatest, a semitone." Going back to reading this quote, I think phrasing is important for understanding everything Aristoxenus is trying to say here. If the quarter tone is the smallest Enharmonic diesis, and the third tone is the smallest chromatic diesis, that implies that all tetrachords with a diesis smaller than a third tone are Enharmonic, and all tetrachords with a diesis larger than a third are Chromatic. Something that's easy to glance over, but probably the most imporant takeaway is that the semi tone isn't referred to as a diesis, but as a "greatest" interval. This implies that location of the diesis (which we'll learn will always be between the 1st and 2nd notes of the tetrachord) can't be larger than a semi tone, and that the semi tone isn't assigned to a particular genus because it can be found in both Chromatic and Diatonic tetrachords (more on this later).

Differences of the genera (180)

The next section is what originally inspired the calculator at the top of the page. click here to read the translation. It'll be formatted similarly to the rest of the webpage (quotes from the translation, followed by my own comments), but this section is going to contain quotes that are all related to the 3 tetrachord genera, as opposed to earlier quotes which stood on their own.

"...the smallest of the concords, that of which the compass is usually occupied by four notes.” (180)

This is Aristoxenus' base definition of a tetrachord. It spans the range of "the smallest of the concords," which we know to be a perfect Fourth, and that it usually contains 4 notes. All the examples Aristoxenus gives contain 4 notes, but there could potentially more, or maybe even less. I always assumed the "tetra" in tetrachord referred to the number of notes, but it could also refer to the outer interval.

“An example of the order required will be found in the interval between the Mese and the Hypate…Further, while there are several groups of notes which fill this scheme of the Fourth…there is one which, as being more familiar than any other to the student of music…It consists of the Mese, Lichanus, Participate, and Hypate.” (180)

Here, Aristoxenus introduces the names of the scale degrees: Hypate, Parhypate, Lichanus, and Mese.

“The locus of the variation of the Lichanus is a tone, for this note is never nearer the Mese than the interval of a tone, and never further from it than the interval of two tones. The lesser of these extreme intervals is recognized as…the Diatonic Genus….The greater of these extreme intervals…finds no such universal acceptance. That there is a style of composition which demands a Lichanus at a distance of two tones from the Mese.” (181)

This section describes a rule for the interval between Lichanus and Mese (scale degrees 3 and 4), which is that the interval must be between 1 and 2 tones, inclusive.

When the interval between Lichanus and Mese is a tone, the tetrachord can be classified as Diatonic. And when the interval between Lichanus and Mese is 2 tones, the tetrachord can be classified as Enharmonic. Aristoxenus excludes the Chromatic, because it occupies a space in between the two extremes (we'll learn that, just like the case of dieses occupying a range of intervals, that the range between Lichanus and Mese is also a range).

“...the fact that time and attention are mostly devoted to chromatic music…that when the enharmonic is introduced, it is approximated to the chromatic, while the ethical character of the music suffers…” (181)

This is a bit of a philosophical aside that suggests some important information about Aristoxenus' perspective on the three genera. From an ethical standpoint, Aristoxenus is suggesting that during his time, the Enharmonic and Chromatic genera were being unfairly likened to one another (try them both in the calculator to compare for yourself). Because they are distinct, lumping them together would result in an ethical dilemma. Admittedly, the Enharmonic and (Soft) Chromatic tetrachords are very similar due to their sub semi-tone dieses; however Aristoxenus seems to be advocating for a theory that acknowledges them both as distinct. Perhaps he feared that the direction music was going would've resulted in a loss of any non-diatonic genera. In hindsight, he was somewhat right, as most Western music theory is based on Diatonic scales. Ironically, the Chromatic scale was reappropriated by Western theorists to describe an octave divided in equal (usually 12 semi tones, not third tones) parts.

“...the locus of the Lichanus to be a tone, and that of the Parhypate to be the smallest diesis...” (181)

Here we see Aristoxenus talk about distances between Parhypate and Lichanus (scale degrees 2 and 3). This range is a bit more nebulous than the other two intervals, since this range is a result of modifying the outer intervals. We can figure out what the extremes of this range would be by using the ranges we know from the outer intervals. It's helpful to know that a perfect Fourth is 2 and a half tones to follow along with the math here.

First, we're going to find out that the interval between Hypate and Parhypate is the smallest interval in the tetrachord, which means this interval can be equal to or larger than. So, if the smallest allowable melodic interval is a quarter tone, the interval from Parhypate to Lichanus can't be smaller than a quarter tone.

We can find the upper limit of the interval by hypothesizing about a tetrachord with a quarter tone leftmost interval, and a tone rightmost interval. If a tetrachord has a total of 2 and a half tones, that means the middle interval can't be larger than 1 and a quarter tones. This is confusing, since Aristoxenus seems to be implying that the middle interval is between a quarter tone and a tone, but the phrasing might just be awkward. Instead, he might be saying that the total range is a tone plus the smallest diesis.

Long story short, the range of the middle interval is a quarter tone to 1 and a quarter tones (inclusive).

"...as the [Parhypate] is never nearer to the Hypate than a diesis, and never further from it than a semitone." (181-182)

This section talks about the ranges we were already able to infer from Aristoxenus' notes ealier. The interval between Hypate and Parhypate (scale degrees 1 and 2), will be between a quarter tone and a semi tone (inclusive).

“[The] Pycnum [is] two intervals, the sum of which is less than the complement that makes up the Fourth.” (182)

Aristoxenus introduces a new term here, pycnum, which refers to a special case where the lowest and middle intervals of the tetrachord add up to an interval that's less than the highest interval in the tetrachord. For example, the Enharmonic tetrachord might have a pycnum of 1/2 tone, leaving 2 tones for the interval from Lichanus to Mese. A Diatonic tetrachord won't have a pycnum because the interval from Hypate to Lichanus is 1 and 1/2 tones, which is larger than the rest of the tetrachord (1 tone).

“...starting from the lower of the two fixed notes, take the least Pycnum...the two least enharmonic diesis...while a second Pycnum…will consist of two of the least chromatic diesis. This gives the two lowest Lichani of two genera the—enharmonic and the chromatic.” (182)

This quote and the following quote provides us with preliminary information about not only the 3 tetrachord genera, but also foreshadows the concept of shading, which refers to different flavors of tetrachords that belong to the same genus.

The first scenario involves making an Enharmonic tetrachord with "the least pycnum." In other words, this would be a interval that's the sum of two of the smallest Enharmonic dieses, or quarter tones. This would result in a tetrachord that looks like: 1/4 tone > 1/4 tone > 2 tones.

The second scenario describes the Soft Chromatic tetrachord, and it involves another pycnum. This time the pycnum is a sum of the two smallest Chromatic dieses, or 1/3 tones. This would result in a tetrachord that looks like: 1/3 tone + 1/3 tone + 1 5/6 tones. This fraction looks a bit weird, seing that we're only dealt with 1/2s, 1/3s, and 1/4s so far. However, the need for 1/6s comes from having to do math with 1/2 tones and 1/3 tones, because their lowest common denominator is 6.

“Again, let a third Pycnum be taken, still from the same note; then a fourth, which is equal to a tone; then fifthly, from the same note… a tone and a quarter, then a sixth scale… a tone and a half.” (182)

Like I mentioned in the above commentary, this quote is set up to explain Aristoxenus' process of deriving 6 shades of the 3 tetrachord genera.

The third Pycnum being "taken, still from the same note" is by far the most unintuitive part of the six part process. In fact, I'm suspicious that this might even be a mistranslation given how Aristoxenus elaborates on the structure of this third shade of tetrachord. Even though Aristoxenus says to take a pycnum from the same note, which would imply a 2/3 tone interval, the third shade he refers to is called the Hemiolic Chromatic (more on this later). This tetrachord is actually made up of 3/8 tone + 3/8 tone + 1 3/4 tones. The sum of 3/8 and 3/8 is 6/8 or 3/4s, not 2/3s, and serves as another division of the tone before we get to the 1/2 tone. In summary, I don't understand why Aristoxenus says to take a pycnum from the same note, when this doesn't end up being the case.

The fourth case describes what Aristoxenus will call the Intense Chromatic tetrachord. The pycnum is equal to a tone, which implies that it's the result of adding two 1/2 tones together. This would result in a tetrachord that looks like: 1/2 tone + 1/2 tone + 1 1/2 tones.

The fifth case describes the a Tonic (or Soft) Diatonic tetrachord. Technically, it doesn't involve a pycnum because the sum of the lower two intervals is 1 1/4 tones. And if a perfect Fourth is 2 1/2 tones, the highest interval would also be 1 1/4 tones. This particular scenario involves 1/2 tone + 3/4 tone + 1 1/4 tones.

The sixth and last case describes a tetrachord with the largest possible combination of lower intervals: a diesis of 1/2 tone and a middle interval of 1 tone. This is the only shade that Aristoxenus describes in this 6-part process that has a high interval that is smaller than the sum of the two lower intervals.

“Continuity in melody seems in its nature to correspond to that continuity in speech which is observable in the collocation of the letters. This is done in no random order: rather, the growth of the whole from the parts follows a natural law.” (184-185)

Before wrapping up book 1, and getting into more details about the tetrachords at the beginning of book 2, we get another philosophical nugget that connects to Aristoxenus' focus on practice informed theory. The notion of notes having a sense of trajectory from one to the next is something that, according to Aristoxenus, is partially natural and intuitive.

“...we must avoid the example set by the Haromnists in their condensed diagrams, where they mark as consecutive notes those that are separated from one another by the smallest interval. For so far is the voice from being able to produce twenty-eight consecutive dieses, that it can by no effort produce three dieses in succession.” (185)

Another aside is a snark from Aristoxenus to his predecessors. This quote suggests that the Haromnists (whoever they were), implied through their teachings that an octave could be divided into 28 quarter tones. In theory, this is true because the octave is a series of 8 unique tones, 6 tones and 2 semi tones, which adds up to 28 quarter tones. However, singing more than a tone (let alone an octave) in consecutive quarter tones isn't practical.

"If ascending after two dieses, [the voice] can produce nothing less than the complement of the Fourth, and that is either eight times the smallest dieses, or falls short of it only by a minute and unemlodic interval. If descending, it cannot after the two dieses introduce any interval less than a tone.”" (185)

This quote goes into more detail about what Aristoxenus was setting up in the two previous quotes. He suggests that, when ascending, the voice naturally wants to complete the Fourth after a series of 2 dieses. Conversely, when descending, the voice doesn't want to produce intervals any less than a tone. This is convenient because it aligns with cases of conjunct and disjunct scales that Aristoxenus goes into more detail about in book 2.

“...any note in that series will either form with the fourth from it in order the concord of the Fourth, or with the fifth from it in order the concord of the Fifth, while possibly forming both.” (186)

This quote is talking about continuous double scales (and possibly multiple scales, as well). When producing continuous scales consisting of more than 1 tetrachord, each pitch in the lower tetrachord will form a perfect Fourth or perfect Fifth with a pitch higher in the scale. Conversely, the upper tetrachord will form a concord with the lower tetrachord, as well.

“...there are four intervals contained in the interval of the Fifth, two of which are usually equal, viz. those constituting the Pycnum, and two unequal--on the complement of the first concord, the other the excess of the interval of the Fifth over that of the Fourth, the unequal intervals which succeed the equal intervals do so in different order according as we ascend or descend the scale.” (186)

This quote is especially important because there's been so much emphasis on tetrachords here that the fifth has been largely neglected. That being said, Aristoxenus provides a brief and clear explanation (which is very rare in his writing 😂) for how Fifths arise in this tetrachordal system. Basically a Fifth contains all the same properties a tetrachord does, but with the addition of a tone at the top. The interval at the top of the Fifth will always be a tone in both conjunct and disjunct scales.