# Note Derivations In Just Intonation

This post is intended as a resource for exploring the foundations of just intonation.

For musicians working in equal temperament it’s common to think of notes as being “built” from semitones.  What is sol?  It’s seven semitones above do.  You might also say it’s a “perfect fifth” above do, but what’s a perfect fifth?  Seven semitones.

In just intonation you don’t think of notes as being composed of some small indivisible unit.  The semitone is not primary.  Instead, notes are derived from each other, by applying certain “special” intervals in sequence: pure perfect fifths, pure major thirds, and their reciprocals, if you’re working in the “5-limit” variety of just intonation.  A note can be thought of as having intonational “relatives” with which it bears an affinity: a certain inflection of la might arise as the perfect fifth above a certain inflection of re, while another inflection of la might arise as the pure major third above fa.  (These relationships are acoustically significant, because they indicate that each la is an early member of the harmonic series of its parent: our first la is the 3rd harmonic of re while our second la is the 5th harmonic of fa.)  Our two la‘s both sound like a major sixth above do, and they are very close in pitch, but they have subtly different affects, as well as different lineages or derivations.

The genealogy of notes in just intonation is often summarized — with remarkable concision — in lattice diagrams.  Looking at the 5-limit lattice one might say that 27/16 la is three rungs right of do, while the 5/3 la is one rung left and one rung up (assuming an orientation where moving right corresponds to traversing a perfect fifth and moving up corresponds to traversing a major third).

One can spend a lot of time reading materials on just intonation and studying paths in the lattice without ever hearing these paths rendered in sound.  But every path in the lattice is more than a mathematical construct — it’s something you can listen to!  The goal of this post is to illustrate the lattice paths for some of the significant pitches in 5-limit just intonation through short musical examples.  Each example begins on a home note, C, and is designed so that you only hear pure fifths and thirds until you reach the target pitch, which is then sustained over C in the bass, and finally descends to C.  Where there are multiple paths to a note, we traverse fifths first, then thirds.  Notes in the sequence are held long enough so that you have time to judge the quality of each interval traversed.  Each example could be thought of a kind of auditory “proof.” The claim we prove is that a certain pitch can be reached through a sequence of pure, beat-free major thirds and fifths — all the pitches in the 5-limit lattice satisfy this claim, and pitches outside the lattice don’t.

If you do hear beating in the thirds or fifths it’s likely because some of the organ samples that I’ve used in creating these clips have a faint vibrato.  If you’d like to hear an alternate rendering using a different instrument, just let me know.

Also note that the examples assume octave equivalence: there are cases where the target pitch falls within the octave above the bass C, and other cases where the target pitch falls in an upper octave.  So, where I’ve indicated an interval like 3/2, the actual interval you hear against the bass might be 6/2. In addition, there are a few cases where a chain of fifths and thirds is transposed up an octave, midway through, to keep it in a manageable range.  However, octave transposition is never used in such a way that a descending fifth would be heard as an ascending fourth, or a descending third as an ascending sixth.  To keep things simple from the listener’s perspective, I’ve made sure only thirds and fifths are heard until the target pitch arrives.

Here are the scores and audio files for three groups of 5-limit ratios.  I’ve split them into these particular groups simply to make notation easier (I wanted to avoid having multiple tunings of the same letter name in one score).

#### Group 1:

Includes 2/1, 3/2, 9/8, 5/4, 15/8, 45/32, 8/5, 6/5, 4/3, 16/9, 16/15, 5/3.

(View Score)

#### Group 2:

Includes 27/16, 135/128, 27/20, 9/5, 36/25, 25/16, 75/64.

(View Score)

#### Group 3:

Includes 25/24, 10/9, 64/45, 32/27, 128/81, 256/135.

(View score)

For reference, here is a lattice diagram including all the notes in the examples (and more).  The bold numbers are cents, rounded to the nearest integer; below each cent value is the corresponding ratio.  Stepping right corresponds to traversing a perfect fifth (3/2) and stepping up corresponds to traversing a major third (5/4).  To get your bearings when reading cents notation, simply remember that a tempered semitone is 100 cents; other tempered pitches thus fall at multiples of 100.  So if you see 996 cents, for example, you know that it’s 4 cents flat of 10 tempered semitones, which is to say it’s slightly flat of a tempered minor seventh.

# Diaschisma

This post offers an audio example of the diaschisma in musical tuning.

If we start at C and travel four perfect fifths upwards, we arrive at E.  To return to C, we could travel down one major third, or we could travel up two major thirds.  In equal temperament, either path will take us back to an octave equivalent of our initial C, but if we use just intervals — “3/2” fifths and “5/4” major thirds — we’ll land somewhere close, but not exactly on that C.  The first path, C->G->D->A->E->C’, exposes the syntonic comma, which I described in my previous post.  The second path, C->G->D->A->E->G#->B#, exposes the diaschisma.  In the case of the syntonic comma we land a little bit sharp of our starting point, because the final descending major third, when justly tuned, doesn’t span the full distance between our sharp Pythagorean E and the C below.  In the case of the diaschisma we land a little bit flat of our starting point, because even though the sequence of just fifths takes us to a sharp E, those two just major thirds at at the end are too narrow to get us all the way to an octave above our initial C.

My diaschisma example (view score) is set up so that the starting C and the “derived” B# are contrasted and then played simultaneously at the end.  In the equal-tempered version, they match; in the version using just intervals, they clash.  As you listen to the examples, also pay attention to the sound of the thirds before the end; the harsh tempered thirds guide us to a pure octave while the sweet just thirds belie the discord that follows!

Diaschisma — Equal Temperament:

Diaschisma — Pure Intervals:

As I experiment with these examples I keep returning to a synthesized timbre called Bright Pad (used above) as it makes the tuning discrepancies obvious to my ear, but I also made clips using a sampled bassoon:

Diaschisma — Equal Temperament:

Diaschisma — Pure Intervals:

# Syntonic Comma

This post provides an illustration of the syntonic comma in musical tuning.

The audio example (view score) starts at C and ascends in a series of four perfect fifths, touching G, then D, then A, then E.  From this high E, the example descends by a major third, hitting an instance of C two octaves above the starting note.  This high C is restated, and then played together with the original C in the bass.

In equal temperament, the high C that we reach at the end of the progression forms a pure octave with the starting C — you’ll notice that the first clip ends without a trace of dissonance.

However, if we use pure fifths and thirds as opposed to tempered intervals, the progression doesn’t bring us to a pitch that matches our starting point.  At the end of the second clip, you can hear how the starting and ending “versions” of C create a mistuned compound octave that flutters or beats slighlty.  In some timbres, particularly where the sound has a bit of vibrato, the mistuning at the end is imperceptible, but I tried to use a plain sound that doesn’t mask the clash.

Here’s a way of thinking about the progression. A pure perfect fifth is slightly wider than a tempered fifth; obviously, traversing a wider fifth will take us to pitch that’s sharper than the one we would reach via a narrower tempered fifth.  As we traverse four of these pure fifths we end up at an E that’s sharper than a tempered E, which is itself sharper than a justly tuned E.  When we then descend by a just major third from our sharp “Pythagorean” E, we reach a C’ that’s in beautiful agreement with the E; but since that E is so sharp with respect to the original C, descending by a just major third cannot bring us all the way back home.  We land at a C’ that’s 21.51 cents higher than the starting point.

Syntonic Comma — Equal Temperament:

Syntonic Comma — Pure Intervals:

# Diesis

This post is a sequel to my previous post on the Pythagorean Comma.  Here I’d like to illustrate two discrepancies sometimes called the lesser diesis and the greater diesis, though it appears that several other things have been called “diesis” throughout history, so if someone walks up to you and says “diesis” you should probably ask what they mean and not immediately assume they’re talking about the contents of this blog post.

Our example of the lesser diesis (view score) starts on C and traverses three ascending major thirds.  We could spell these thirds as C-E, E-G#, and G#-B#.  The example ends with the B# that we’ve reached, played against the C that we started on.  In equal temperament, B# is the same pitch as C, and when they’re played together we hear an octave.  However, if we ascend using pure 5/4 major thirds, we reach a B# that’s significantly flat of C, and when this B# and our original C are then played together, the dissonance is jarring, especially in contrast to the pure thirds that led us there.

Ascending Major Thirds — Equal Temperament:

Ascending Major Thirds — “pure” 5/4 ratios:

Our example of the greater diesis (view score) starts on A and traverses four ascending minor thirds, which we’ll spell as A-C, C-Eb, Eb-Gb, and Gb-Bbb.  The example ends with the Bbb that we’ve reached, played against the A that we started on.  In equal temperament, Bbb is the same pitch as A, and we hear an octave.  However, if we ascend using pure 6/5 minor thirds, we reach a Bbb that’s quite sharp of A, and the notes clash.

Ascending Minor Thirds — Equal Temperament:

Ascending Minor Thirds — “pure” 6/5 ratios:

In summary, a chain of three pure major thirds falls short of an octave (by 41.06 cents, to be precise), and a chain of four pure minor thirds exceeds an octave by an even greater amount (62.57 cents), hence the qualifiers greater and lesser.

I’ve been thinking about why commas like these are so intriguing.  What comes to mind is an analogy between music and a logical argument.  When I hear notes connected by pure consonances like 5/4 and 6/5, it feels as though I’m being carried from one step of an “argument” to the next, via an impeccable line of reasoning.  At the end of a chain of pristine consonances, I’m inclined to expect that the landing point would “make sense” or fit nicely together with the the previous steps, but the note we reach after three iterations of 5/4 seems very much at odds with the place we started.  In some sense, it’s like listening to an argument that seems flawless and indisputable, only to find that that this perfect argument has whisked you along to a conclusion that seems… absurd?  Or that seems to contradict the premises you started with?  How can it be that such airtight logic leads to such apparent inconsistency?

# Pythagorean Comma

This post contains audio examples of a discrepancy in musical tuning known as the Pythagorean Comma. I’d like to do a full discussion of this topic at some point, but my aim here is just to share some illustrations I made; in this post I’ll assume the reader already understands the underlying theory.

To explore the Pythagorean Comma, I wrote a toy piece (view score) that traverses a series of twelve ascending perfect fifths with some octave skips along the way.  The trick was to keep the example within a constrained pitch range, and to make sure that each interval is sustained long enough that its quality can be judged by ear.  (Since something unexpected happens at the end, I wanted to give the listener a chance to verify that there’s no subterfuge employed along the way!)  The interval palette is limited to ascending fifths and descending octaves; there are no descending fourths.  The pattern works like this: (1) play a note, (2) hold the note and play its fifth, (3) play the two notes again together, (4) play the fifth along with its instance an octave below. Now repeat, choosing the upper or lower instance of the fifth from the previous step as the new starting point — make this choice so that the next fifth will fall in the range C4-C5. For example, we start at C4, then play G4, then C4 and G4 together, then G3 and G4 together — that’s one iteration of the cycle — then we take G3 as our next starting note, play D4 above it, and so on. After doing this twelve times we should land right on C5 if we’re using equal temperament.

First listen to the clip in equal temperament. Pay attention to the end and notice how everything sounds “in tune” as the sequence leads us back to the same note we started with. The clip is rendered using a “Bright Pad” soundfont, and you might ask why I’ve chosen a cheesy 80’s synth for these examples.  No, it’s not my secret love of 80’s music.  Actually, I experimented with lots of high-quality instrument samples but all of the instruments I tried that were capable of long sustain also had artifacts that make it harder to discern whether the tuning is pure.  I tried some beautifully sampled organ stops, for example, but found that some of the notes had a slight wobble or vibrato that could cause intervals to sound like they’re beating.  I’m still looking for other sounds for these examples, but Bright Pad works for now as it has good sustain, no wobble, and lots of overtone content that can help with judging the tuning.  As a side note, if you listen very closely to this clip in equal temperament, you might be able to notice a slight contrast in the texture of the fifths and the octaves: the fifths are just a touch more busy.

Ascending Fifths — Equal Temperament:

Now listen to the same material in Pythagorean tuning and you’ll hear the jarring effect of the comma near the end of the sequence. I’ve tuned this second clip so that each “new” note is a pure perfect fifth above the previous note, meaning that their frequencies always form a 3/2 ratio.  (If you listen closely you might notice there’s less of a contrast in texture between the fifths and octaves here than in the previous clip, but it’s a very subtle difference.)  As we traverse the series of pure fifths, the notes drift increasingly sharp of their equal-tempered counterparts, because a pure fifth is roughly 2 cents wider than a tempered fifth.  However, when the sequence reaches its penultimate step at F, we break the tuning pattern for the sake of comparison: instead of playing a C tuned as a pure fifth above our F, we play an octave-equivalent of our original C — the same note we started with — and now this C sounds flat! That’s the crux of the demo. In order to complete the sequence with pure interval, we’d need a second version of C, one tuned a Pythagorean Comma (roughly 23.46 cents or 1/4 of a semitone) sharp of the C we started with.  The C needs to be sharp because the F we’ve arrived at through the series is itself sharp.  Since we reuse our original C instead, the final F-C interval is too narrow and sounds mistuned in comparison to the pure fifths we’ve been hearing all along.  Here’s the clip:

Ascending Fifths — Pythagorean:

The thing to remember is that both clips start and end on the very same C: in equal-temperament that C still sounds “right” at the end of the sequence, but in Pythagorean tuning it sounds “wrong” because sequence of pure fifths has led us to expect a sharper C.

Another way to observe the gradual pitch drift created by the pure fifths is to play both versions simultaneously, and listen to how the discrepancy between versions grows with each ascending fifth: you will hear this as increasing wobble or beating in the sound.  To save you the trouble of trying to start the two previous clips at the same time, I’ve prepared one clip that superimposes them.  I made one slight modification at the end of the pure fifths version, so that it concludes with the sharp C that the sequence demands, instead of returning to original C; that original C will be heard in the equal-tempered version playing simultaneously.

Ascending Fifths — Pythagorean and Equal Temperament Superimposed:

Here’s another example (view score) that works much like above, except that it follows a series of descending fifths. This time, our penultimate note is a G that’s significantly flat of its 12ET counterpart. Right before the sequence ends, we break the tuning pattern as above. Instead of descending from our flat G to the flat C that would make pure fifth with it, we descend to our original C, and now this C sounds sharp!  Again we’re left with a fifth that’s too narrow and sounds wobbly.

Descending Fifths — Equal Temperament:

Descending Fifths — Pythagorean:

In summary, if you pick a starting note and depart on a series of ascending pure fifths, your starting note will begin to sound flat; if you use descending pure fifths instead, your starting note will begin to sound sharp.  Remarkable!

Appendix