Interval width changes across the syntonic tuning continuum

If we stack nine tempered major fifths (traditionally called "perfect fifths") above Re, and nine below it, we get the following generated collection:
-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

Plotting these intervals' relationships across the syntonic temperament's tuning continuum produces this chart:

(You might want to open this chart into its own window, so that you can look at it, without scrolling, while reading the text below.)

This chart follows the following JIMS conventions:
- Interval names are traditional, except for
  + 4ths and 5ths: wider is "major," narrower is "minor"
  + (that way, 4ths and 5ths follow the same naming-pattern as all of the other non-octave intervals)

- All intervals follow the standard JIMS color-code:
  + major intervals in blue
  + augmented intervals in cyan (an "extreme blue")
  + minor intervals in red
  + diminished intervals in magenta (an "extreme red")

- All chromatic variations of a given diatonic interval share the same note-line symbol. For example,
  + All the unisons (Ra, Re, Ri) are marked with x's.
  + All of the seconds (Me, Mi, My) are marked with squares.
  + All of the thirds (Fa, Fi) are marked with vertical lines.
  + etc.

The legend, at the right of the chart, displays the generated collection of notes, in the same order (bottom to top) as they appear in the list at the top of this blog post. Each note's name is followed, after a colon (':'), by its interval-from-Re. Observe that the follow a pattern: augmented intervals at the top, then major intervals, then unison (Re), then minor intervals, then diminished intervals at the bottom of the list.

The vertical scale, on the left, indicates the width of a given note from Re.

The horizontal scale, on the bottom, indicates the width of the tempered major fifth (M5), that is, of the generator of the generated collection. The scale includes the valid tuning range of the syntonic temperament, which can be thought of an an extended meantone tuning system.

The widths of the intervals between Re and every other (non-octave) note is controlled by the width of the generator, M5. As the width of the M5 increases, from left to right across the chart, the widths of all of the non-octave intervals change:
- The intervals below Re in the legend, representing minor and diminished intervals, slope downwards as the M5 increases, indicating that they narrow.
- The intervals above Re in the legend, representing major and augmented intervals, slope upwards as M5 increases, indicating that they widen.
- The farther a note is from Re in the legend, the steeper its slope.

Consider, for example, the widths of the unisons. As the generator (M5) increases in width:
- Re (unison) is unchanged at 0, because it is the basis from which all other intervals are measured. Its note-line is shown at the very bottom of the chart area, as a series of black x's.
- Ra (diminished unison, d1), shown with magenta x's, decreases in width. It's note-line drops from 0 cents below Re (i.e., 1200 cents above Re), on the left edge of the chart, to 240 cents below Re (i.e., 960 cents above Re) at the right edge.
- Ri (augmented unison, A1), shown width cyan x's, increases in width, from 0 cents above Re on the left to 240 cents above Re on the right.

All of the unisons start, on the left, at 0, and separate as the width of the generator increases.

Likewise, consider the widths of the seconds-from-Re:
- Me (minor second, m2) drops rapidly from 171 cents to 0.
- Mi (major second, M2) rises slowly from 171 cents to 240.
- My (augmented second, A2) rises sharply from 171 cents to 480.

Just as with the unisons, all of the seconds start together (at 171 cents) and separate as the width of the generator increases. Generally, all of the chromatic variations of a given diatonic degree start at the same point on the left-hand edge of the chart, and diverge as the M5's width increases rightwards across the chart. (Note that 1200 and 0 are the same octave-reduced interval, so that Ra, which intersects the left edge at 1200, intersects it at the same interval as Re and Ri, which intersect it at 0.)

7-edo
The seven left-edge-intersection-points divide the octave into 7 equally-wide intervals, forming a 7-note "equal division of the octave," abbreviated "7-edo."

(The phrase "N-tone equal temperament" and its abbreviation "N-TET," used in Wikipedia and elsewhere, is avoided in JIMS, because it confuses the important distinction between tunings and temperaments...an explanation of which is beyond the scope of this blog post.)

5-edo
Likewise, the right-hand edge of the chart, at M5=720, shows that a completely different combination of notes intersect to divide the octave into five equally-wide intervals: 5-edo. (Again, note that 1200 and 0 are the same octave-reduced interval, so Di, intersecting the right edge at 1200, and Me, intersecting the right edge at 0, are intersecting it at the same interval.)

12-edo
Near the middle of the chart, at M5=700, you can see that seven pairs of note-lines cross. From top to bottom, the crossing pairs are:
1100 - Ra and Di (d1 and M7)
900 - De and Ti (d7 and M6)
800 - Te and Li (m6 and A5)
600 - Le and Si (m5 and M4, traditionally named d5 and A4)
400 - Se and Fi (d4 and M3)
300 - Fa and Mi (m3 and A2)
100 - Me and Ri (m2 and A1)

The notes in the crossing pair are always 12 notes apart in the 19-note stack of M5's (check for yourself, using the chart's legend).

The crossing note-pairs are said to be "enharmonic" (i.e., have the same pitch) in 12-edo. This is the "equal temperament" tuning familiar to most modern musicians -- so familiar, in fact, that many such musicians do not realize that other tunings exist, or that there is such a thing as a tuning (let alone a temperament).

17-edo
Slightly to the right of 12-edo, at M5-706 cents, two other note-lines cross:
352 - Se and Mi (d4 and A2)
847 - De and Li (d7 and A5)

All of the note-lines intersect the vertical line labeled "17-edo" at 17 equally-spaced intervals, so M5=706 is 17-edo tuning.

In 17-edo, the major second is subdivided into three equally-wide intervals by the augmented second and minor second. For example, see how the gap between Re (black x's, at the bottom) and Mi (blue squares, near the 200 cent horizontal line) is evenly divided by Ri (A1, cyan x's) an Me (m2, red squares). Note that at this point along the horizontal axis (M5=706), Me is closer to Re (i.e., lower in pitch) than Ri is.

In 17-edo -- and indeed everywhere rightward of 12-edo -- minor/diminished intervals are lower in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

19-edo
Likewise, the vertical line labeled "19-edo" marks the spot, at M5=695, where the note-lines subdivide the octave into 19 equally-wide intervals: 19-edo tuning.  At this tuning, a major second (for example, Re-Mi) is divided into three equally-wide intervals by and augmented unison (Ri) and a minor second (Me).

In 19-edo -- and indeed everywhere leftward of 12-edo -- minor/diminished intervals are higher in pitch than the augmented/major intervals with which they are enharmonic in 12-edo.

Dynamic Tonality
Despite the changes among the relationships between intervals across the syntonic temperament's tuning continuum, the sound of tonal harmony's basic structure survives, as shown in this video (with over-the-top narration, for which I apologize):


This dynamic flexibility of tuning, combined with the consistent fingering of the Wicki/JIMS keyboard, can be used to create musical effects that are truly new, such as the tuning progression in this piece, C to Shining Sea, by William Sethares. We call the result Dynamic Tonality.

Well-formed scales beyond the chromatic

First, let's review the construction of the chromatic scale.

Stacking 13 tempered perfect fifths (P5's) one atop the other, centered on Re, produces the following 13-note generated collection:
-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

The Le-to-Di and Me-to-Si 12-note subsets of this generated collection are both just transposition of each other, so either can be used to represent a 12-note contiguous subset of the above 13-note generated collection. In the following discussion, the Me-to-Si subset will be used.

The Me-Si generated collection's notes can be adjusted so that they all fall within a single octave. We will arbitrarily define the octave to start on Do. The result is a "well-formed scale," in this case the chromatic scale.

The chromatic scale has the following note sequence and interval sequence:
note sequence:    Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2]
interval sequence:  A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

...where:
A1: augmented unison
m2: minor second

In the syntnonic temperament's valid tuning range -- that is, when the width of the P5 is anywhere between 686 and 720 cents wide -- the m2 is wider than the A1. Hence, in the syntonic temperament, the chromatic scale has the following width sequence:
width sequence: S L L S L S L S L L S L

That's 7 L's and 5 S's.

With that review, we can now go...

Beyond the Chromatic
In the syntonic temperament, then, the well-formed scale with the next-highest cardinality after the chromatic's 12 will have the cardinality:
Cardinality' = 2L + S = (2 * 7) + 5 = (14) + 5 = 19.

Stacking 19 tempered P5's one atop the other, centered on Re, produces the following generated set:
-9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9
De-Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li-My

...with the extra notes (relative to the chromatic scale) appearing the ends and shown in boldface.

Octave-reducing this generated set, and arbitrarily defining the octave to being on Do, gives the following 19-note note sequence and interval sequence:
Do-Di-Ra-Re-Ri-Me-Mi-My-Fa-Fi-Se-So-Si-La-Li-Te-Ti-De-[Do2]
  A1-d2-A1-A1-d2-A1-A1-d2-A1-d2-A1-A1-d2-A1-d2-A1-d2-A1

...where:
A1: augmented unison
d2: diminished second

Clearly, as we sub-divide the octave into more pieces (i.e., into higher-cardinality scales), those pieces must get smaller.
Scale         Cardinality Large Small
Pentatonic          5       m3    M2
Diatonic            7       M2    m2
Chromatic          12       m2    A1
Enharmonic_19      19       A1    d2

At each successively-higher cardinality, the formerly-small interval width becomes the new large width, and a new small width is introduced.

On a 19-note-per-octave Wicki/JIMS note-layout, and played in 19-tone equal temperament (P5=695, at which the A1 and d2 are both 1200/19=63.16 cents wide), this scale looks/sounds like this (source code here):


Now, let's explore the alternative cardinality-successor to the chromatic scale.
Stacking 17 tempered P5's one atop the other, centered on Re, produces the following generated set:
-8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8
Se-Ra-Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si-Ri-Li

...with the extra notes, relative to the chromatic, added to either end, and shown in boldface.

Octave-reducing this generated set, and arbitrarily starting defining the octave to being on Do, gives the following 17-note note sequence and interval sequence:
Do-Ra-Di-Re-Me-Ri-Mi-Fa-Se-Fi-So-Le-Si-La-Te-Li-Ti-[Do2]
  m2-d2-m2-m2-d2-m2-m2-m2-d2-m2-m2-d2-m2-m2-d2-m2-m2

On a 17-note-per-octave Wicki/JIMS note-layout, played in 17-tone equal temperament (P5=706), this scale looks/sounds like this (source code here):


And there you have it: the next-higher-cardinality scales after the chromatic are 17 and 19.

Syntonic and Mavila

In an earlier post, I calculated the cardinalities of successive well-formed scales -- pentatonic (5), diatonic (7), and chromatic (12) -- and animated their interval-patterns on the Wicki/JIMS note-layout.

What we saw -- with some interpretative help from Andy Milne -- was that:

1. each successive well-formed scale came in two versions: one with X large intervals and Y small intervals, and one that was vice versa (Y large and X small); and that
2. the sequence of intervals that defined both versions was the same; the only difference between the two versions was the tuning (that is, the width of the tempered perfect fifth, since that is the generator of the generated set that defines a well-formed scale).

For example, the diatonic "generated set" is Fa-Do-So-Re-La-Mi-Ti, which produces the note-sequence (in Do-mode) Do-Re-Mi-Fa-So-La-Ti-[Do2], which has the inter-note interval sequence M2-M2-m2-M2-M2-M2-m2.

In the syntonic temperament's valid tuning range (P5=(686, 720)), the M2 is wider than the m2, so this sequence can be written as the width sequence L-L-S-L-L-L-S, which is 5 large (L) and 2 small (S) intervals.

However, as P5's width shrinks towards 686, the m2 widens and the M2 shrinks, such that they become equal at around P5=686 cents, producing 7-tone equal temperament tuning.

If one narrows the P5 even further, one leaves the syntonic temperament and enters what Erv Wilson called the Mavila temperament, in which the m2 is wider than the M2. There, this same pattern (note sequence: Do-Re-Mi-Fa-So-La-Ti-[Do2] == interval sequence: M2-M2-m2-M2-M2-M2-m2 ) has the width sequence S-S-L-S-S-S-L, because in the Mavila temperament's valid tuning range, m2 > M2.

Alternatively put, the diatonic note note sequence and (hence) interval sequence are unchanged from syntonic to Mavila; the only thing that's changed is the relationships among the interval-widths, in that syntonic's m2 < M2 becomes Mavila's m2 > M2.

The same meta-pattern applies to the chromatic scale (all from Do):
note sequence:    Do-Di-Re-Me-Mi-Fa-Fi-So-Si-La-Te-Ti-[Do2].
interval sequence: A1-m2-m2-A1-m2-A1-m2-A1-m2-m2-A1-m2

Within the syntonic temperament's valid tuning range, the m2 is wider than the A1 (i.e., m2 > A1), so the above chromatic note/interval sequence produces the following width sequence:
width sequence:     S  L  L  S  L  S  L  S  L  L  S  L

However, if the P5's width is narrowed so that it crosses out of the syntonic temperament's valid range into the Mavila temeprament's valid tuning range, then the width-relationship of the m2 and A1 is reversed, such that m2 < A1 -- producing a chromatic width sequence in Mavila that's the opposite of that in the syntonic:

width sequence:     L  S  S  L  S  L  S  L  S  S  L  S

Apparently, Andy's algorithm for calculating the sequence of cardinalities for successive well-formed scales, and the count of large & small intervals in each, produces a single scale, of which there is a syntonic variant and a Mavila variant. Let's see, in my next post, if that pattern continues, as we explore well-formed scales beyond the chromatic.

Cardinality of well-formed scales

Let’s apply Andy's next-highest-cardinality-MOS-scale-calculating algorithm, starting with the pentatonic scale.

First, here's an animation of the pentatonic scale on a Wicki note-layout (source code here):

As you can see, the pentatonic scale has two large steps (m3’s) and three small steps (M2's).
Name         L   S   Cardinality
Pentatonic   2   3        5

So far, so good.

Remember, a well-formed scale is drawn from a stack of tempered perfect fifths; the stack has the same number of notes in it as the scale does: the cardinality of the scale. The pentatonic scale's stack of P5's (laid on its side) looks like this:
-2 -1  0  1  2
Do-So-Re-La-Mi

Now, let’s apply Andy's algorithm to these values of L and S to get the L and S of the next-higher-cardinality MOS scale, i.e., L’ and S’ respectively.

Cardinality’ = 2L + S = (2 * 2) + 3 = 4 + 3 = 7, which agrees with our expectations for the diatonic scale, which is a god sign. ;-)

X = L + S = 2 + 3 = 5
Y = L = 2

Therefore, the next-higher-cardinality-than-pentatonic MOS scale will have either 5 large steps and 2 small steps, or vice versa.

The diatonic scale has five large steps and two small steps, with cardinality 7, so that seems to be the “right choice.”

However, I am confused. Does this mean that there ALSO exists some “Bizarro-Diatonic” scale of cardinality 7 which has five small steps and two large ones? If so, what is that scale? If not, why not?

This process gives us the following result:

Name              L   S   Cardinality
Diatonic          5   2        7
Bizarro-Diatonic  2   5        7

Hmmmm, it’s a little weird to have the Bizarro-Diatonic scale as a result of this algorithm, but what the heck, let’s press on.

[Edit: Andy Milne was kind enough to point out that the Bizarro-Diatonic scale is more properly named the Mavila scale, following Erv Wilson. I haven't been able to find out much about it on the web. When I understand it better, I'll put up an appropriate animation of its interval pattern.]

The diatonic scale described above looks like this (source code here):

The diatonic scale's stack of tempered P5's is just like the pentatonic's, but it has one additional note at each end (Fa and Ti):
-3 -2 -1  0  1  2  3
Fa-Do-So-Re-La-Mi-Ti

Now, let’s apply the stated algorithm to the diatonic scale’s L and S values, to find the next-higher-cardinality MOS scale (which OUGHT to be the chromatic scale, if all goes well).

Taking the values L' = 2 and S' = 5 from the diatonic scale...

Cardinality’’ = 2L’ + S’ = (2 * 5) + 2 = 10 + 2 = 12, which is the cardinality of the chromatic scale, which is encouraging.

X’ = L’ + S’ = 5 + 2 = 7
Y’ = L = 5

Therefore, the next-higher-than-diatonic MOS scale will have either 7 large steps and 5 small steps, or vice versa.

Hmmmm, that's odd. Can there be two different versions of the chromatic scale?

Yes, it turns out that there can, and the existence of the two different versions answers a question that's been puzzling me for the last couple of weeks.

Here's an animation of the first version of the chromatic scale (source here):

There are 12 intervals in the chromatic scale, so any drawing of them is going to look complicated, and this animation's drawing is no exception. But if you look closely, you can see a lot of structure in its pattern of intervals.

Firstly, the scale has only two interval sizes, as predicted: minor seconds (m2's, in red) and augmented unisons (A1's, in pink).

Second, there are five A1's (smaller intervals) and seven m2's (larger intervals). In 12-tone equal temperament ("12-tet"), the A1 and m2 happen to be equally wide (at 100 cents), but they are still different intervals, so they have different shapes on the Wicki note-layout.

Third, all the interval lines are parallel to each other. Among them, they outline a chromatic staff (well, kinda sorta).

One of the note-buttons, Le, isn't used in the animation above, however -- none of the interval-arrows ever reach it. You might well ask, "why did Jim include Le in the animation, then?"

To answer this leading question, let's look at an animation of the other version of the chromatic scale (source here):

It looks very much the same, as you would expect. The only difference is that, after the scale goes up an m2 from Fi to So, it goes up another m2 from So to Le, rather than turning back towards Si with an A1 as the previous version of the chromatic scale did.

We can call the first version "Chromatic Si," and the second version "Chromatic Le." In 12-tone equal temperament, there is no difference between them, but there would be a difference in (say) 1/4-comma meantone tuning, which had a leading role in Western music for many centuries (and which appears to have been used in the tuning of ancient Chinese bells).

In the stack of tempered perfect fifths that forms the chromatic scale, Le and Si are on opposite ends:
-6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6
Le-Me-Te-Fa-Do-So-Re-La-Mi-Ti-Fi-Di-Si

...which is the same as the diatonic stack, but with extra notes added at the ends (in boldface).

This chromatic stack has 13 notes, but only 12 can be included in the chromatic scale. One must choose whether to include Le or Si; you can't have both, because then you'd have a scale of cardinality 13, not 12.

• If you choose to include Le, you get the Chromatic_Le scale.
• If you choose to include Si, you get the Chromatic_Si scale.

[Edit: I got all excited when I first saw this, because, after about 4am, my counting skills decline precipitously -- so I thought that Chromatic_Le had 5 small and 7 large intervals, and Chromatic_Si, vice versa. They are just transpositions of each other, and nothing to get excited about. One should never trust (or at least, never post) late-night epiphanies.

As further evidence of the decline in my late-night counting abilities, I also declared the down-and-right-pointing intervals (e.g., from Do to Di) to be "diminished seconds," when they were clearly augmented unisons (how "clearly"? In both chromatic animations, the A1 arrows ALWAYS connect notes that begin with the same initial consonant -- that is, chromatically-altered versions of the same note. Hence, the interval MUST be a variation on unison. What an idjit I am!). I have relabeled them accordingly, and updated this blog post's text accordingly.  Thanks to Andy Milne for giving me the heads-up on these errors; see his comment below.

I realy should have changed the color of the A1 arrows to be cyan, while I was relabelling them, to follow my convention that all augmented intervals are colored cyan...but I forgot, and now I'm too tired again. Later, perhaps.]

I've been up all night working on this blog posting (and its animations), so now, at 6am, I'm off to bed. Soon, I'll put up another post that continues walking up the cardinality chain to 17-tone scales, 19-tone scales, and beyond.

Isomorphism & diatonic set theory

There are lots of isomorphic note-layouts -- for example, the Bosanquet, Fokker, Janko, Wesley, Chromatic Button Accordion (B-system and C-system), and Wicki.

JIMS uses the Wicki note-layout for a variety of reasons that are beyond the scope of this post.

The Wicki note-layout is proving to have some interesting mathematical properties. For example, consider any well-formed scale constructed by stacking N tempered perfect fifths and subtracting octaves (an "alpha-reduced beta-chain," where alpha is the octave and beta is the tempered perfect fifth), and N is the "cardinality" of the scale (that is, the number of notes in the scale).

The Wicki note-layout appears to be unique in that such well-formed scales are always tightly packed together on the keyboard, with no "holes" between the notes of the scale.

For example, consider the well-formed scale of cardinality 5 (pentatonic). It's notes [Do Re Mi So La] form a single tight group that (a) has no "holes" in it, and (b) is symmetrical around Re.

The well-formed scale of cardinality 7 (diatonic) is likewise tightly grouped and centered.

So is the well-formed scale of cardinality 12 (chromatic). Notice that both Le and Si are included, which is redundant; they represent the same note in the 12-tone well-formed scale, whether in 12-tone equal temperament tuning or not. I've just included both in the drawing for symmetry. The chromatic scale is the only well-formed scale with even cardinality (well, among those scales with cardinality less than or equal to 19, anyway), which is kinda messing with my head a bit.

And so on, for the well-formed syntonic scale of cardinality 17:

...and 19:

...and 21:

...and so on, ad infinitum.

To put it another way, the Wicki note-layout appears to be unique in that, to increase the cardinality of the syntonic scales playable on a Wicki note-layout, all one needs to do is add more notes to the left & right edges of the note-layout.

The other isomorphic note-layouts do not share this property. Their design intermingles scale notes and non-scale notes. As a result, they do not present the same pattern of notes for well-formed scales of all cardinalities.

By way of comparison, consider the Chromatic Button Accordion's C system note-layout (CBA-C), shown at right.

The CBA-C layout works fine for the chromatic scale, but if you wanted to use it exclusively for the pentatonic or diatonic scales, the note-layout would be full of holes. Alternatively put, neither the pentatonic nor diatonic note-sets map to compact, contiguous button-sets in the CBA-C note-layout.

Likewise, look at the line of "semi-tones" running up-and-rightwardly from C on the CBA-C note-layout. If one wanted to put the Db and C# on separate buttons there's no room. There's only one button-space between C and D; if has to serve for both Db and C#. The CBA-C note-layout does not have a clean "edge" to which the Gb could be added, as the Wicki note-layout does. As a rule of thumb, any note-layout with a contiguous line of "semitone"-controlling buttons has the chromatic scale "baked in," because the "semitone" is only a meaningful concept in chromatic scale (i.e., in the well-formed scale of cardinality 12). In scales of cardinality higher than 12, there is no "semitone." There are augmented unisons and there are minor seconds, but there are no semitones.

Now, look back at the patterns that well-formed scales make on the Wicki note-layout. These patterns all share three characteristics:

(a) They have no "holes" between the notes of a scale of given cardinality.

(b) They are symmetrical around Re.

(c)  All of their notes fall on adjacent rows, with one row being one button/note wider than the other (including the chromatic/12, because I included both Le and Si, which is cheating, just a little).

On the other hand, one can see (using the scale chooser on the interactive keyboard below) that non-well-formed scales, such as the Neapolitan, Melodic, Harmonic Major, Marmonic Minor, and Double Harmonic Minor, do not share all of these characteristics.

This suggests that there is some common element that is shared by (a) the definition of well-formedness and (b) the definition of the Wicki note-layout. I do not yet know what that common element is, but it's pretty obvious that it's in there somewhere.  (I think that it has something to do with the fact that on the Wicki note-layout, the "beta-stack" corresponds directly to one hexagonal line of note-controlling buttons, and the "alpha-stack" corresponds directly to a second, semi-perpendicular line.  But I'm not sure.)

If you can shed any light on this common element, please don't hesitate to let me know.  :-)