Book 2

Summary

There's part of me that feels that Book 2 would've been better off as Book 1, since there's a lot more foundational information here that clarifies the concepts of Book 1. However, Aristoxenus wrote so many books that it's hard to say whether or not someone reading this book would've already been familiar with the terms and concepts covered in Book 1. That being said, here are some of what I think are the most important takeaways from Book 2:

1. Sense perception is crucial to the study of music, and it's the main thing separating musicians from mathemeticians.
2. The voice is the most important instrument for understanding the natural laws that inform Aristoxenus' theory of harmony.
3. Just because there are exceptionally small intervals in some tetrachord genera, it doesn't mean that playing them consecutively and out of their scalar context produces a melodious result.
4. Even though the genera have different shades, the ear can still distinguish what genera a tetrachord belongs to.
5. Aristoxenus still has beef with his predecessors and their supposed stripping of music from their theories.
6. The ditone is approximately a major Third, and can be derived by a series of perfect Fourths or Fifths.
7. The Fifth is a Fourth with an added tone.

I also want to add that there's a bit of discussion about Aristoxenus' precessors' organization of the modes near the beginning of Book 2, and while I'd like to talk about it, most of my commentary is about how I'm confused about how they've arrived at the distances between scale degrees. That being said, I think it's better that I focus on the tetrachords and scales they're used to create instead, since the book goes into much more detail about it over a modal context.

“The geometrician makes no use of his faculty of sense-perception. He does not in any degree train his sight to discriminate the straight line, the circle, or any other figure, such training belonging rather to the practice of the carpenter, the turner, or some other such handicraftsman. But for the student of musical science accuracy of sense-perception is a fundamental requirement.” (189)

I'm going to go out on a limb and say that the aesthetic of Aristoxenus' theory of harmony is our senses should guide our theories. Mathemeticians don't readily use their senses to dictate their theories, so why has theory up to this point been so number centric? Aristoxenus likens the musician to the artisan, which I think is an interesting comparison, since artisinal work is often associated with the average person, and education is often associated with the elite. There's almost a sense of trying to distribute music theory to the masses, which seems the most fair considering that music is generally beloved by everyone.

"But as a fact neither clarinets nor any other instrument will apply a foundation for the principles of harmony.” (196)

This quote is easy to read over if you're looking at Aristoxenus' writings through a tetrachordal lens, however, it's a reminder that the voice is ultimately guiding Aristoxenus in regards to his theory of harmony.

“The following fractions of a tone occur in melody…the half, called a semi tone...the third, called the smallest Chromatic diesis…the quarter, called the smallest Enharmonic diesis.” (199)

I wen't into detail about this in comments I made about Book 1, however, this is the actual quote that introduces the three most common melodic divisions of the tone. Something to bear in mind is that he excludes the Hemiolic diesis, which is a 3/8 despite it being one of the more noteworthy shades of the Chromatic genus. Something to bear in mind though is that when combining two Hemiolic dieses results in a 3/4 tone pycnum, and because that's also a sum of quarter tones, it's most likely a melodic division of the tone.

“...many have misunderstood…that melody admits the division of the tone into three or four equal parts…to employ the third part of a tone is a very different thing from dividing a tone into three parts and singing all three.” (199)

This quote reinforces what I said earlier about the tone being divided into smaller fractions, but the biggest takeway is that fractional divisions of the tone don't necessarily mean that playing many of them in succession will produce a melodious result.

“There is admittedly but one interval between the Mese and Paramese, and again between the Mese and Hypate, and in fact between any pair of the permanent notes. Why then should we admit a plurality of intervals between the Mese and the Lichanus?” (200)

Aristoxenus sets up the next excerpt with this question. Before breaking it down, in an octave scale, there are fourth additional scale degree names (in order from lowest to highest): Paramese, Trite, Paranete, and Nete.

This question talks about why the Lichanus is the only variable note in the scale, in comparison to the other notes, which are fixed in order to maintain the perfect Fourth and Fifth needed to complete the octave in this line of thinking. I make the comparison between the Lichanus and the third of a major/minor triad. The third is what constitutes the flavor of the triad, while the root and the fifth are always a fixed distance, so in order to create different genera, the Lichanus must be the variable note because of it's location in the tetrachord.

Something that confuses me, however, is why Aristoxenus puts much more emphasis on the Lichanus being variable, while the Parhypate is conceptually just as variable, albeit the range is smaller.

“We see that the Nete and Mese differ in function from the Paranete and Lichanus, and the Paranete and Lichanus again from the Trite and Parhypate, and these latter again from the Paramese and Hypate; and for this reason each pair has names of its own, though the contained interval in every case is a Fifth.” (200)

All this quote is saying is that two tetrachord scales will always produce a series of 4 Fifths: Hypate to Paramese, Parhypate to Trite, Lichanus to Paranete, and Mese to Nete. Aristoxenus doesn't specify the quality of the Fifths, however, which might be on purpose because some scales could potentially produce a diminished Fifth.

“For the ear detects a motion peculiar to each of the genera, though each genus employs not one but many divisions of the tetrachord.” (201)

Here's another quote that suggests the ear has the ability to distinguish genera from one another. Because of the concept of shading, we know that the slight differences in tuning make the shades sound similar to one another that belong to the same genus, but even if the ear can't distinguish the shades, it can still tell what genus a tetrachord belongs to.

“There are certain divisions of the tetrachord which stand out from the rest as familiar…one is Enharmonic, in which the Pycnum is a semitone, and it’s complement two tones…” (202)

Here is the complete explanation of what an Enharmonic tetrachord consists of. We get a bit of a cryptic explanation in Book 1, however, here we see that it contains a semi tone pycnum (1/4 tone + 1/4 tone) and a 2 tone complement.

“...three are Chromatic, namely, the Soft, the Hemiolic, and the Tonic Chromatic. The division of the Soft Chromatic is… the Pycnum consists of two of the smallest Chromatic dieses, while its complement is expressed in terms of… a semitone taken thrice and a Chromatic diesis taken once. he… Hemiolic Chromatic is that in which the Pycnum is one and a half times the Enharmonic Pycnum, and each Diesis one and a half times and Enharmonic diesis. The… Tonic Chromatic… the Pycnum consists of two semitones, and its complement of a tone and a half." (202)

A long quote, but it fully explains the 3 common shades of Chromatic tetrachord. There was a quote in Book 1 that mentions the genera, and pays the least attention to the Chromatic genus; however, we see that the Chromatic genus has the most common shades of the 6 the Aristoxenus offers here.

The Soft Chromatic is described with some difficult to intuit language. It works out to be 1/3 + 1/3 + 1 5/6 tones, which I talked about in the Book 1 commentary, however his language is a bit wordy here because he's trying to explain the complement in less number oriented terms. It's complement is a semitone taken thrice (1/2 + 1/2 + 1/2) and a Chromatic diesis taken once (1/3). When summing it all together, you get 3/2 + 1/3, which requires you to think in terms of 6ths: 9/6 + 2/6, which equally 11/6, or 1 and 5/6 tones.

The Hemiolic Chromatic is the unusually vague case from Book 1 that's explained much more clearly here. It's pycnum is 1 and 1/2 times the Enharmonic pycnum, which works out to 3/4 tones. This leaves us with a complement of 1 and 3/4 tones.

Lastly, the Tonic Chromatic is a series of 2 semi tones, leaving us with a complement of 1 and 1/2 tones.

“There are two divisions of the Diatonic genus, the Soft and the Sharp Diatonic. The… Soft Diatonic… the interval between the Hypate and Parhypate is a semitone, that between the Parhypate and Lichanus three Enharmonic dieses, that between the Lichanus and Mese five dieses. The… Sharp Diatonic is that in which the interval between the Hypate and Parhypate is a semitone, while each of the remaining intervals is a tone.” (203)

Another longer quote, however it gives us all we need to know about the 2 common Diatonic shades.

The Soft Diatonic is an interesting case because it's the first division that's offered and doesn't have a pycnum. The first interval is 1/2 tone, and the middle interval is "three Enharmonic diesis," which works out to be 3/4 tones, just like the interval from Hypate to Lichanus in the Hemiolic Chromatic. The complement is the same as the sum of the first two intervals, 1 and 1/4 tones, which is why this tetrachord doesn't have a pycnum.

The Sharp (or some resources call it Intense) Diatonic is a 1/2 tone followed by 2 whole tones. This tetrachord is most like the interval relationships we'll find in our everyday Diatonic scales.

“Thus, while we have six Lichani, as there are six divisions of the tetrachord...we have but four Parhypate, that is, two less than the divisions of the tetrachord. For the semitone Parhypate is employed for both diatonic divisions, and for the Tonic Chromatic.” (203-204)

Here, it looks like Aristoxenus is just summarizing what we were just told: that there are 6 common divisions of the tetrachord; however I think there's a bit more to read into here.

For starters, Aristoxenus describes the number of Lichani as "infinite" in Book 1, which would imply an infinite number of divisions of the tetrachord. However, perhaps he's suggesting that despite there being 3 genera, each of the infinite divisions of the tetrachord will always fall into one of these 6 common divisions. I can't say for sure, but just from reading, there's a hint of contradiction.

Second, he talks about the disparity between the number of divisions of the tetrachord, and the number of Parhypate. While the Enharmonic, Soft Chromatic, and Hemiolic Chromatic all have a unique Parhypate, the rest of the tetrachord shades all have a semi tone parhypate. Even though he doesn't say this explicitly, I think this is a hint at why he doesn't call the semi tone a diesis.

“...the Hypate and Parhypate may be equal to that between the Parhypate and Lichanus, or less… but never greater...” (204)

This is an elaboration of what we already know. The lowest interval has to also be the smallest. The middle interval can equal the lowest interval, but it can't be less that it.

"..and also may be ascertained in the Chromatic by taking a Parhypate of the Soft, and a Lichanus of the Tonic Chromatic: for such divisions of the Pycnum sound melodious. ...the opposite order produces an unmelodious result; for instance, to take the semitone Parhypate, and the Lichanus of the Hemiolic Chromatic…...or the Parhypate of the Hemiolic, and the Lichanus of the Soft Chromatic." (204)

Here is an interesting quote that suggests that tetrachords can be formed by mixing and matching elements of unlike genera (how fun 🎉).

The first example is a melodious tetrachord that consists of the Parhypate of the Soft Chromatic (1/3 tone), and the Lichanus of the Tonic Chromatic (1 tone). This works out to be 1/3 tone + 2/3 tone + 1 and 1/2 tones. This is a melodic tetrachord because it's intervals are packed from smallest to largest, left to right.

The next example produces an unmelodic result because the interval from Parhypate to Lichanus is smallest: a semitone Parhypate (1/2 tone) and a Lichanus of the Hemiolic Chromatic (1 and 1/2 tones).

The last example is another unmelodious tetrachord that consists of the Parhypate of the Hemiolic Chromatic (3/8 tone), and the Licahnus of the Soft Chromatic (2/3 tone). This works out to be very unusual fractional math: 3/8 tone + 7/24 tone + 1 and 5/6 tones. The need for 24ths of a tone is unintuitive, but we need it to express the fraction that, when added to the Parhypate, gives us the Lichanus, which is a 2/3 tone above the Hypate. In other words, 3/8 can be expressed as 9/24 and 9/24 + 7/24 = 16/24, which is 2/3s. This is a bit confusing to me since the smallest number we see Aristoxenus use in his math from Book 1 is 1/12 of a tone, so I'm not sure why I haven't encountered 24ths of a tone in any supplemental research I've done on Aristoxenus.

“...the interval between the Parhypate and Lichanus may be equal to, greater than, or less than that between the Lichanus and Mese...equal in the Sharp Diatonic...less in all the other shades...greater when we employ a Lichanus the highest of the Diatonic Lichani, and as Parhypate any one lower than that of the semitone.” (204)

Here, we see Aristoxenus devoting some time to explain the middle interval. Up until this point, we've seen the Parhypate and Lichanus explained as separate entities, which has led us to believe that the middle interval is just a byproduct of following the guidelines that have been set up for us. That being said, we can see that the middle interval can be equal to, less, or greater than the rightmost interval.

The middle interval will be equal to the rightmost interval in one case: the Sharp Diatonic, which conists of 1/2 tone + 1 tone + 1 tone.

The middle interval is greater in shades that involve the greatest Diatonic Lichanus (1 and 1/2 tones) and any Parhypate that's lower than a semi tone. For example, if we have a tetrachord with a Soft Chromatic diesis (1/3 tone) and the greatest Diatonic Lichanus (1 and 1/2 tones), it works out to: 1/3 tone + 1 and 1/6 tones + 1 tone. I think the easiest way to reason this is that, if the Lichanus is 1 and 1/2 tones, the complement has to be 1 tone in order to remain a perfect Fourth. Therefore, the complement will always be smaller than the middle interval in these cases.

In every other shade, we see that the middle interval is less than the right most interval.

“...in investigating continuity the laws of melody must be our guide, nor must we imitate those who shape their account of continuity with a view to the massing of small intervals...the voice’s power of connecting dieses stops short of three.” (204-205)

Another tidbit that Aristoxenus adds in regarding our natural capacity to accurately sing consecutive small intervals. If you read over the quote quickly, you see that the hard and fast rule is "short of three." Why doesn't he just say 2? I don't know 🤷.

I think we also can read into a bit of nuance about the word diesis here. Earlier, I was theorizing why semitones weren't referred to as dieses; however, if this adds additional context to the discussion, semi tones can be sung consecutively without issue, but smaller intervals can't (according to Aristoxenus). Therefore, I'm inclined to believe that semitones aren't a diesis.

"...there is no interval which can be divided ad infinitum in melody, and that the natural laws of melody assign a maximum number of fractions to every interval. We necessarily infer that the notes containing fractions of the said number are consecutive. To this class belong the notes which… have been in use from the earliest times… for instance the Nete, the Paranete, and those that follow them." (205)

Aristoxenus says a lot of the natural order of pitches and intervals in this quote. Not only does he say that fractions of intervals can only be divided so many times before they become impractical to sing, or difficult to distinguish, but he also refers to fractional divisions of the tone as "consecutive."

So, with this quote in mind, why does my calculator divide the tone into cents? First, the math is a little more intuitive with whole numbers, so calculating intervals in terms of cents is easier for me to visualize. Second, all of the limitations that Aristoxenus puts in place for minimum and maximum intervals are in place. All the calculator does is allow for cent divisions of the tone within the range. For example, the Parhypate can be between 1/4 (50 cents) and 1/2 (100 cents) tone, inclusive. So a 13/50 (52 cents) is theoretically possible, but singing it would be basically indistinguishable from a 1/4 tone Parhypate.

I want there to be more to read into this, because it feels complex based on the way that it's written; however, all I think Aristoxenus is saying here is that tetrachords are constructed using consecutive intervals. Sometimes there's nuance, like in the case of adjacent notes that are more than a tone apart (the Enharmonic Lichanus and Mese, for example), but Aristoxenus throws in Nete and Paranete out of nowhere it seems, considering we've been discussing tetrachords from Hypate to Mese up until this point.

I think starting with Nete and Paranete was intentional, however, because this connects with the idea that I mentioned in my comments about Book 1 that tetrachords are sometimes constructed from the top (or what we would consider the top) down.

“Whatever be the genus, from whatever note one starts, if the melody moves in continuous progression either upwards or downwards, the fourth note in order from any note must for with it the concord of the Fourth, or the fifth note in order from it the concord of the Fifth. Any note that answers neither of these tests must be… out of tune...this then [is] a fundamental principle, the violation of which is destructive of harmony. ” (205)

Here, Aristoxenus elaborates about the significance of the Fourth (and the Fifth for the case of disjunct scales, I imagine) as an interval not just as an outline of the tetrachord, but as a means of both determining whether or not it's "in tune", or in other words, continuous.

“It is quite possible that notes of a scale might form the above-mentioned concords with one another, and yet that the scale might be unmelodiously constructed.” (205)

This quote is an extension of the previous. It states that unmelodious scales can produce notes that are seemingly in tune due to having a concordant interval above it. It just so happens that this must be true of melodious scales, despite the fact that unemlodious scales can also produce the same result.

“If any two tetrachords are to belong to the same scale, one or other of the following conditions must be fulfilled…they must be in concord with each other, the notes of one forming some concord or other with the corresponding notes of the other…or they must both be in concord with a third tetrachord, with which they are alike continuous but in opposite directions.” (206)

This quote is a bit of a head scratcher for me due to the second condition, but before I try to tackle that, the first condition is fairly straightforward. Aristoxenus appears to be reiterating what he said in the previous quotes: that multiple tetrachords must have a concordant relationship between each set of corresponding notes. In other words, each note in a tetrachord also has a perfect Fourth or Fifth above or below it.

The second scenario applies to scales containing three or more tetrachords. Saying that the tetrachords must be in concord with a third implies that the relationship of a perfect Fourth or Fifth still applies. The main difference is that because this third tetrachord is what I believe to be the middle tetrachord, due to the fact that the tetrachords are continuous in opposite directions. What confuses me the most is where exactly the middle tetrachord is placed in the scale. My instinct is to assume that it's placed no differently that any other linked tetrachords, either conjunct where it shares a note with the tetrachord below, or disjunct where it's separated by a tone.

This could be worked out in a bunch of different ways, but a unique example would be 3 linked Diatonic tetrachords with no disjunctions, which would start on E and end on G a Tenth above. We know that the multiple scale this scenario forms is continuous and melodious because all of the notes form a perfect Fourth either above or below with the middle tetrachord.

“...the ear is much more assured of the magnitudes of the concords that of the discords.” (206)

A bit of a break from the verbose theory with this quote. It talks about concords being easier to distinguish from one another than discords. Even though Aristoxenus states they're both infinite in Book 1, an infinite number of concords would imply a never ending process of adding octaves to the Fourth and Fifth. Because the Fourth and Fifth are really the only two distinct concords, in comparison to the many discords within the octave, they're easier to pick out.

“...the most accurate method of ascertaining a discord is by the principle of concordance....one should take the Fourth above the given note, then descend a Fifth, then ascend a Fourth again, and finally descend another Fifth...to ascertain the discord in the other direction, the concords must be taken in the other direction.” (206)

Here, Aristoxenus introduces us to the "principle of concordance," which looks quite similar to our circle of fifths. Ascending or descending a more than 1 Fourth or Fifth will eventually produce a series of discords. Aristoxenus contextualizes this by using it to identify the ditone which is literally "2 tones," or a major Third. Even though his procedure stops at the ditone, you could still produce every discord down to the semi tone using this.

For example: Fourth up, Fith down, Fourth up, Fifth down, starting on C works out to: C, F, Bb, Eb, Ab. Therefore, this process will always produce a ditone below the starting note.

Conversely: Fifth up, Fourth down, Fifth up, Fourth down, starting on C works out to: C, G, D, A, E. Therefore, this process will always produce a ditone above the starting note.

“...if a discord be subtracted from a concord by the method of concordance, the remaining discord is thereby ascertained on the same principle.” (207)

Because Aristoxenus' is being vague about what "a discord" is implying here, we don't know if he has a particular interval in mind. The way I interpret this is that if you perform the operation from both directions, the result will give you the complement that adds up to the concord.

For example, C to F is a Fourth. A ditone up from C is E (C, G, D, A, E). The remaining space in the Fourth is a semitone, E to F. A ditone down from F is Db (F, Bb, Eb, Ab, Db). The remaining space in the Fourth is also a semitone, Db to C.

Let's do the same with C to G, which is a Fifth. This time our discord will be Eb, which is a minor Third above C (C, F, Bb, Eb). The remaining space in the Fifth is a ditone, Eb to G. A minor Third down from G is E (G, D, A, E), which leaves a ditone worth of space between E and C.

“The surest method of verifying our original assumption that the Fourth consists of two and a half tones…let us find the discord of two tones above its lower note...and the same discord below its higher note...the complements will be equal, since they are remainders obtained by subtracting equal from equals...let us take the Fourth above the lower note of the higher ditone...and the Fourth below the higher note in the lower ditone...adjacent to each of the extreme notes of the scale…will be two complements in juxtaposition, which must be equal.” (207)

This is an elaboration of the process I went over above. Aristoxenus is using it to prove what we already know: that the Fourth is 2 and 1/2 tones, and that the Fifth is a Fourth with an additional tone.

To prove the Fourth's size, in this case we'll use C to F, we find the ditone above the lower note, which is E. Then we do the same for the ditone below the higher note, which is Db. The complements are equal, like we saw in the previous quote, C to E leaves a semi tone up to F, and F to Db leaves a semi tone down to C. 2 tones and 1/2 tone.

To prove the Fifth's size, we start by taking the Fourth above the bottom of the higher ditone: Db to Gb. Then the Fourth below the top of the lower ditone: E to B. If we make a continuous scale using the pitches, the result is a "Fifth" (or a diminished 6th if you spell it diatonically): B, C, Db, E, F, Gb. 1/2 + 1/2 + 1 and 1/2 + 1/2 + 1/2 = 2 and 1/2 tones (a Fourth) + 1 tone.

“...if the concord formed by the extreme notes…is greater than a Fourth, and less than an octave, it must be a Fifth.” (208)

To preface, this quote is accompanied by more explanation that justifies the point of these procedures, however this quote is really the best summary of what I believe Aristoxenus set out to achieve with the "principle of concords" in this context. This quote is essentially the end of Book 2. After doing all these procedures on my own, I didn't really expect the point to be about determining the size of the concords in question.

Like he said in an earlier quote, concords are easier to distinguish by ear than discords because of their distinct size. Because concords are much more clearly defined in terms of their sizes, if a sum of intervals results in an interval you can identify as a concord, and it's bigger than a Fourth, but smaller than an octave, it must be a Fifth. Honestly, kind of a let down because I feel like he's stating the obvious. Of course a Fifth is in between a Fourth and and Octave, Aristoxenus. Even though I didn't have expectations about how the book was going to end, all of that grappling with Aristoxenus' writing ultimately led to a pretty underwhelming proof. Then again, this is a reminder that Aristoxenus didn't have the centuries of music theory and mathematics that we have today, so proving truths like this one might've been a huge deal back then.