Book 3

Summary

Book 3 is non-stop proofs about the information we learned about tetrachords and the scales they form in the previous books. According to Henry S. Macran, the author and translator of Aristoxenus' writing to English, Book 3 consists of, "...propositions of details, following one another in logical order like the...Elements of Euclid." While I haven't read Euclid's Elements as I write this, the style that Aristoxenus uses to organize Book 3 is a series of, for lack of a better word, chapters that explain (almost to the point of exhaustion in my opinion) the rules and guidelines that govern harmonic construction in the music of his time.

Because of this, there are going to be less takeaways in the "Cliff Notes" immediately below, and there aren't really themes about aesthetics or philosophy in the same way that the previous books had.

1. Tetrachords can be linked conjunctly (sharing a starting/ending note) or disjunctly (separated by a tone).
2. Nothing explicitly states that tetrachords from different genera can't be made into a larger scale, but there are rules about how to go about it.
3. Tetrachords can only be modified by their inner intervals.
4. Simple intervals are created by successive notes.
5. Species and figure are interchangeable terms, and refer to a set of identival intervals whose sum won't change when rearranged.

Some of the ways these proofs are phrased (or translated, it's difficult to know for sure) makes it not only challenging to understand the significance of the proof, but the operations being performed to illustrate it. I'll mention this when it comes up on a case by case basis.

“Successive Tetrachords are either Conjunct or Disjunct.” (209)

This is referring to whether or not a pair of tetrachords are linked by their starting/ending note, or separated by a tone. Notice how both tetrachords have an identical interval structure whether you start on A for the higher tetrachord in the first example, and B for the higher tetrachord in the second.

"Tetrachords similar in species cannot be separated by a dissimilar tetrachord...dissimilar but successive tetrachords cannot be separated by any tetrachord…" (210)

This quote is interesting because it's the first time I'm aware of that Aristoxenus suggests unlike tetrachords can be chained together. I wish there was more information about this, however the treatise ends before we get a chance to know.

Also, the term "successive" refers to things that are next to each other. So, by that logic, anything that's separated aby something else isn't successive by definition like what's being implied in the second part of the quote.

The first part of the quote says similar tetrachords can't be separated by a dissimilar tetrachord. This is a bit confusing, because this would imply that a scale consisting of 3 or more tetrachords would have to be made up of two similar tetrachords, followed by a dissimilar one.

The second part of the quote says something that results in a similar conclusion. Two tetrachords that are dissimilar can be continuous like in the scenario below, but separating them with a third dissimilar tetrachord isn't allowed because they wouldn't be able to be continuous on account of Aristoxenus' definition from Book 2.

“The interval contained by successive notes is simple. For if the containing notes are successive, no note is wanting; if none is wanting, none will intrude; if none intrudes, none will divide the interval.” (210)

This quote implies that notes that sit next to one another form successive intervals. This gives us a bit of a confirmation about what "successive" means.

The majority of the importance of this proof, to me, comes from the ditone (or intervals greater than a tone for that matter). In an Enharmonic tetrachord, there is a successive ditone from Lichanus to Mese.

The ditone can also not be a simple interval if the notes aren't successive. For example, in a Diatonic tetrachord B-C-D-E, the interval from Parhypate to Mese is a ditone, C-E. However, the D that separates the C and E interrupts the succession. Therefore, the interval isn't simple.

"In variations of genus, it is only the parts of the Fourth that undergo change” (211)

All this quote appears to be saying is that the outer notes in a tetrachord are fixed. This would probably be an imporant reminder in a double or multiple scale context, where the tetrachords are blended together.

“Every genus comprises at most as many simple intervals as are contained in the Fifth” (211)

This is another interesting little theory. If you take a tetrachord and add a disjunct tone on top of it, like you were preparing to add another tetrachord to make an octave scale, the number of unique simple intervals doesn't change.

“A Pycnum cannot be followed by a Pycnum or by part of a Pycnum.” (212)

If you were to take multiple Pycna or part of a Pycnum, one or both of the following would happen. Either, the scale wouldn't span a Fourth, or the scale would have too many successive dieses, which Aristoxenus says can't be sung beyond 2.

“The lower of the notes contained the ditone is the highest note of a Pycnum, and the higher of the notes containing the ditone is the lowest note of a Pycnum.” (212)

In the case of conjunct Enharmonic tetrachords, which are the only tetrachords that can contain a successive ditone like what's being described here, the ditone is bounded by the lowest and highest notes of the Pycnum.

“The notes containing the tone are both the lowest notes of a Pycnum" (213)

This little quote had me scratching my head for the longest time.... This quote involves a Pycnum, and a tone, so Aristoxenus has to be referring to disjunct Enharmonic tetrachords, right? E-E quarter #-F-A-B-B quarter #-C-E. The higher of the two notes of the tone is obvious, because B is the lowest note of the higher Pycnum.

But what about A? A is followed by B, which is a tone, making it not the lowest note in a Pycnum. What I think Aristoxenus wants to prove is that A, when it's part of a conjunct Enharmonic tetrachord scale, is the lowest note in a Pycnum. E-E quarter #-F-A-A quarter #-Bb-D.

“A succession of two Ditones is forbidden.” (213)

A tetrachord isn't possible with successive ditones since they produce an augmented Fifth. Coincidentally, 2 ditones produces a "tritone," which has it's own legacy in Western European music history.

“In Enharmonic and Chromatic scales a succession of two tones is not allowed.” (213)

This is true because two successive tones would mean either the tetrachord is Diatonic, or that the interval between Hypate and Parhypate is larger than 1/2 tone.

“In the Diatonic genus three consecutive tones are permitted; but no more.” (214)

This happens as a result of making a scale out of two disjunct Diatonic tetrachords.

“In the [Diatonic] genus, a succession of two semitones is not allowed.” (214)

Similar to the theory about non-Diatonic tetrachords from earlier, two consecutive 1/2 tones would either make the rightmost interval smaller than a tone (a tone is the smallest it can be by definition), or it would make it too large because the first two intervals would only span 1 tone.

“A ditone may be succeeded either above or below by a Pycnum.” (214)

Related to a theory from earlier. A ditone is always going to be preceded by a pycnum, and it can also be succeeded by a pycnum in the case of a conjunct scale.

“A ditone can be followed by a tone in the ascending scale only.” (214)

Ditones are followed by tones in the case of disjunct tetrachord scales, however when descending a scale, the ditone will always be followed by an Enharmonic diesis (1/4 tone).

“A tone can be followed by a Pycnum in the descending order.” (215)

If you read this quickly, like I did at first, it looks like the case in disjunct Enharmonic/Chromatic scales where there's a tone between the tetrachords.

But actually, what's being described is the opposite. In the descending order, if a tone were to precede a Pycnum, we would have an incomplete tetrachord, since the result isn't a perfect Fourth. Unless the descending order implies that tetrachords can have mirrored structure like this case here, this proof might be a translation error. I don't know for sure.

“In the Diatonic genus, a tone cannot be both preceded and succeeded by a semitone.” (215)

If you try this for yourself, you won't be able to complete the Fourth. For example: B-C-D-Eb.

“A pair of tones, or a group of three tones may be both preceded and succeeded by a semitone.” (215)

This also describes something that can happen only in the Diatonic genus, which is what happens in a disjunct Diatonic scale.

“From the ditone there are two possible progressions upwards, one only downwards.” (215)

Upwards, you can go to either a pycnum, in the case of a conjunct scale, or a tone, in the case of a disjunct scale.

Downwards, you can go only to a pycnum.

“From the Pycnum, on the contrary, there are two possible progressions downwards, and one upwards.” (216)

This is the converse of the previous quote. Upwards you can only go to a ditone.

Downwards, you can go to a ditone, as well, in addition to a tone.

“From the tone there is but one progression in either direction, downwards to the ditone, upwards to the Pycnum.” (216)

This is the third and final proof related to successive movement between the intervals possibly present in the Enharmonic genus: ditone, pycnum, and tone. This case is particularly limited because the tone is only going to occur in disjunct scales, so technically, the tone isn't even part of the tetrachord. Just a bridge between the two.

Upwards, you can only go to a Pycnum.

Downwards, you can only go to a ditone.

Aristoxenus expands more on this quote than the others (probably because it's serving as sort of a summary of the previous 3), but he says that the same logic can be applied to Chromatic and Diatonic scales as well. The only adjustments are that the ditone becomes a smaller interval, and that the pycnum sometimes gets replaced with a larger interval.

“In the Chromatic and Enharmonic scales every note participates in the Pycnum.” (217)

This is another head scratcher... On one hand, the notes that are part of the pycnum must participate in the pycnum. In conjunct scales, Mese also becomes the lowest note of the higher pycnum.

But what about disjunct scales? The top note of the lower tetrachord is no longer part of a pycnum. The only way I can rationalize this is that the top note of the lower tetrachord is already accounted for in the case of the conjunct scale, so it's omitted in the case of the disjunct scale.

“One will readily see that the positions of the notes situated in the Pycnum are three in number.” (217)

This is saying that there are 3 notes in a pycnum. This might seem obvious, but there's a proof later on that supposes a situation that involves a pycnum with multiple same pitches (I don't fully understand why that even needs to be hypothesized, to be honest).

“It is required to prove that from the lowest only of the notes in a Pycnum there are two possible progressions in either direction, while from the others there is but one.” (218)

Here's, yet again, another confusing quote. Like Aristoxenus mentioned earlier, the lowest note of a Pycnum can go down in two ways, by a tone, or by a ditone, depnding on whether the scale is disjunct or conjunct.

From my understanding, this doesn't make sense the way I'm reading it. The lowest note of a pycnum can go up to the next note in the pycnum, but it can go down to a ditone or a tone. The middle note makes sense, since it can only go to one of its neighbors. The last note makes sense, as well, since it can go up to the ditone and down to it's neighbor.

“It is required to prove that from the highest note of a Pycnum there is but one progression in either direction.” (219)

You can check out the diagram above to see this in action.The top note of a pycnum can go one place in either direction. Upwards, it can go to Mese because the pycnum ends at Lichanus. Downwards, it can go to Parhypate because the pycnum continues in that direction.

“It is required to prove, that from the middle note of a Pycnum there is but one progression in either direction.” (219)

You can see this in the above diagram too. Up and down, the because the pycnum is 3 notes long, the middle note will go to one of its neighbors.

“It is required to prove that two notes that occupy dissimilar positions in the Pycnum cannot fall on the same pitch without violating the nature of melody.” (219)

This is the quote I was referring to earlier about notes in the pycnum needing to have their own pitch identities. I'm not sure if I'm missing something here, but the Pycnum by definition is a sum of 2 intervals, and because Aristoxenus doesn't count the unison as an interval, there's no way for two notes to occupy the same pitch in a pycnum.

“It is required to prove that the Diatonic genus is composed of two or of three or of four simple quanta.” (220)

Quanta is replacing the term intervals here. I don't know if there's signficance, but the terms are interchangeable. The Diatonic genus is unique in the sense that it can produce between 2 and 4 unique simple intervals, including the tone added by the disjunct Fifth.

For 2 unique intervals, we have the Intense Diatonic tetrachord. Only 1/2 tone, and 1 tone.

For 3 unique intervals, we can move the Parhypate down a quarter tone, for example. 1/4 tone, 1 and 1/4 tones, and 1 tone.

Lastly, for 4 unique intervals, we need a tetrachord plus a disjunction to give us 4 total intervals. Then, we would need to move both Parhypate and Lichanus to produce intervals other than a tone and a semi tone. For example: a Soft Chromatic diesis, and a Soft Diatonic Lichanus. 1/3 tone, 11/12 tone, 1 and 1/4 tones, and 1 tone. The math gets wonky in this example because we need to find the interval between E third sharp and F three quarter sharp, or 1/3 tone above E, and 1 and 1/4 tones above E.

“It is required to prove that the Chromatic and Enharmonic genera are composed of three or four simple quanta.” (221)

Just like the above theory, we can produce 3 and 4 unique intervals; however, unlike the above theory, we can't produce only 2 unique intervals. Remember, we're adding the tone that completes the Fifth.

For 3 unique intervals, the Pycnum is equal parts. For example, the Soft Chromatic tetrachord: 1/3 tone, 1/3 tone, 1 and 5/6 tones, and 1 tone.

For 4 unique intervals, the Pycnum is unequal. For example, a Soft Chromatic diesis, and a Hemiolic Chromatic Lichanus: 1/3 tone, 5/12 tone, 1 and 3/4 tone, and 1 tone.

“We shall use the terms ‘species’ and ‘figure’ indifferently...when the order of the simple parts of a certain whole is altered, while both the number and magnitude of those parts remain the same...there are three species of the Fourth.” (222)

The last translated paragraph of the book, which defines what "species" means: when a set of intervals can be rearranged and still amount to the same sum. Even though Aristoxenus says there are 3 species of Fourth, don't confuse this with saying there are 3 species of each tetrachord. Tetrachords still follow the same "smallest intervals to the left" rule, along with the other guidelines from the book (check out the calculator for a quick summary, too).

When the pycnum is at the bottom of the Fourth, that's one species, which also happens to be the only melodious tetrachord of the three species described.

Another is when the interval that would typically be spanning the distance from Lichanus to Mese is in the middle of the Fourth.

The last is when the pycnum is above the ditone.