# An analysis of the 365 double jigs in O’Neill’s, pt. 5

This here is part 5 of my live blogging an analysis of the 365 double jigs in O’Neill’s 1001. Part 1 is here, part 2 is here, part 3 is here, and part 4 is here. Today I will begin to look more closely at the time-interval series of the tunes in the collection.

I first plot all 1,712 8-measure time-interval series from this collection and just look at them to get a sense of what kinds of structures appear. I see some that look like that of jig #89 (“The boys of the town”):
The legend refers to the sections: 1 and 2 are the first and second repeats of the A part, and 3 and 4 are the first and second repeats of the B part. To help with readability I have added some slight offsets in x and y.

The first thing that comes to my mind is this:

I loved that gum when I was kid. The first minute of each piece was glorious! That picture makes my mouth water.

Anyhow, the second thing that comes to my mind is the curious delay between the last two sections (red and green lines). Peeking at the underlying transcription shows how this delay arises:All sections have an anacrusis, but the last measure of the first ending of the B part is a full measure. So the delay we see in the time-interval series comes from a counting mistake. We can correct it simply by removing the B quaver in that last measure. I find about 15 more of these counting mistakes, and so correct them as best I can, reprocess the data, recreate all the features, and plot again.

Let’s have a look at some of the interesting time-interval patterns I see. Here’s the time-interval plot for jig #56 (“The humors of Cappa”):

This shows both parts of the tune share the same intervals in measures 3&4 and 7&8, but do something different in measures 1&2 and 5&6. Here’s the dots confirming that observation:This kind of repetition results in a clear tune structure, and a strong coherence between the parts. If I were to render this as a poem, it would be:

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Here’s the time-interval series for jig #69 (“Philip O’Neill”):

The two parts to this tune echo the same final two measures, and share a bit of the middle section, but otherwise do different things. Here’s the dots to confirm:

Here’s the time-interval series for jig #101 (“The idle road”):

Both parts the last half, but at otherwise different. Look at all that bouncing up and down! Here’s the dots:I imagine a fiddle player in a horse-drawn cart on a bumpy road. It’s curious that O’Neill has notated broken rhythms explicitly. Perhaps the player from whom he transcribed this exaggerated the jig rhythm there. In this classic recording of the tune, Joe Burke (accordion) ignores that and plays the jig quite evenly with the others following suit:

Jig #148 (“The Kinnegad slashers”) is a three part jig with the following time-interval series:

We see a strong relationship between parts 1&2 (A) and 5&6 (C). The B part does something different until its last four measures. The B part also appears more constrained in its use of large intervals, except for the octave leap in its fourth measure. Here’s the dots to confirm:My perusal of these time-interval series inspires a few questions.

What tune features a time-interval series that spends most of the time at zero? Apparently there are two: the A part of jig #69 (“Philip O’Neill”):and the B part of jig #331 (“The foot of the mountain”):Sorting the series according to the time spent on a zero interval results in the following graph:

I think the height of the stair steps comes from using a sampling rate of 6 samples per quaver. There are apparently several tunes that spend no time at zero intervals. One of these is jig #82 (“Doherty’s fancy”):Another question to ask is what tune has a time-interval profile with the most positive mean? In other words, which tune spends most of its time at pitches arrived to by positive intervals? It appears to be jig #96 (“Our own little isle”):

The leap from the D quaver to the g dotted crotchet (an interval of 17 semitones) seems to be contributing a lot to this, even though most of the tune is going downwards.

I find 268 of the jigs in the collection feature a section with a positive mean time-interval profile, and 246 have a section with a negative mean time-interval profile. 110 jigs have a section with a mean time-interval profile exactly equal to zero. One is jig #17 (“The eavesdropper”):Another question to ask is which tune has a time-interval profile with the smallest variance? That prize goes to the A part of jig #84 (“Wellington’s advance”):

There are several semitone intervals in the A part. The jig with the largest variance is #257 (“The Morgan Rattler”):It’s easy to see why that’s the case.

Let’s picture all 1,712 time-interval series in the collection:

Here’s a plot showing this collapsed across the series:

We can see that the time spent at pitches arrived to by steps of ±2 semitones (major second) is greater than the time spent at pitches arrived to by ±1 semitone (which makes sense because most of the intervals in a scale are 2 semitones, and much of the melodic motion in these melodies is stepwise). We also see that the time spent after steps of -3 (minor third) and -4 (major third) semitones is greater than the time spent after steps of +3 and +4. However, more time is spent after an interval of +5 (perfect fourth) than -5 semitones. Spending time at pitches arrived to by intervals greater than a perfect fifth is rare, but if one is to find themselves at a pitch after an octave leap, expect to spend more time resting after leap up than down.

This look at the collection raises an interesting question: What happens when we break the series into smaller pieces, e.g., units of one-measure length? In that case, we would have at most 13,696 time-interval series of dimension 36. How many unique units are there? How do they relate? Are there “prototype” measures? Might we see each series as a concatenation of these units?