Hobbled scale

A hobbled scale is a scale created by either reducing or increasing exactly one step of an existing scale pattern by some step size, resulting in a scale that approximates the original pattern but fits into a different edo. The term hobbled derives from the metaphor of "hobbling" one of the step "legs" to be shorter than the other, analogous to how hobbling an animal restricts its movement by shortening one leg. This concept was explored and refined as an effort of TOTYW2025 in the Xenharmonic Alliance Discord.

Definition / algorithm

Given a scale with a repeating step pattern in some edo and a target edo that we want to fit the scale into, a hobbled version is created by:

1. Calculating the difference between the source edo and target edo
2. Selecting any single occurrence of any step size in the pattern
3. Reducing that step by the calculated difference

This results in a new pattern that fits into the target edo with one step shorter than in the original.

Examples

By taking the standard diatonic pattern from 12edo (in this case we will upscale it to 24edo) or nearby systems, you can create a hobbled scale in an adjacent edo that may not support that scale, like 23edo:

24edo diatonic: 4–4–2–4–4–4–2  

23edo hobbled diatonic: 4–4–2–4–4–4–1

This creates a scale that sounds largely diatonic but with one altered interval, providing a "normal" scale with familiar structure yet xenharmonic character. In this example alone you have 7 different options to choose from, as you could hobble a large step as well, creating a 4–4–2–(3)–4–4–2 scale for instance.

From this you have a rank-3 scale with a large variety of chords. By climbing the "circle of fifths" you alternate between a sharp and a flat fifth (as 23edo is a dual-fifth system). By choosing which note to hobble, you not only nudge the melodic movement (for instance, choosing to have a smaller leading tone by hobbling the last step) but also change harmonic qualities of chords throughout your scale. If you want your tonic chord to have a flat fifth, you would choose to hobble a note that occurs below the fifth, and you would choose a note above if you wanted a sharper fifth.

Furthermore, 23edo is sandwiched between two diatonic-containing edos: 24edo with its basic diatonic, and 22edo with its 709 ¢ hard diatonic. If we instead choose to hobble the 22edo diatonic by increasing an edo step, you create a "superpyth hobbled diatonic" scale:

22edo diatonic: 4–4–1–4–4–4–1  

23edo hobbled diatonic: 5–4–1–4–4–4–1

While there are 7 options to choose when hobbling by increasing step size, if you were to increase one of the small steps, you would get a repeated scale from before (by reducing a 24edo diatonic's small step). Thus in this example, there are 12 options to choose from. (5 from increasing a 22edo 5L 2s large step, 5 from decreasing a 24edo 5L 2s large step, and 2 from increasing/decreasing either scale's small step.)

Given there are 7 modes in the 5L 2s scale, there are now 7 × 12 = 84 combinations of hobbled scales and modes to choose from. Certain combinations of hobbling will could exaggerate or enhance properties of other modes.

Musical applications

Hobbled scales serve several compositional purposes:

Finding familiar scales

You can create surprisingly normal sounding music in very weird tuning systems that are sandwiched between two diatonic tunings like 18edo or 23edo by hobbling familiar scales, which widens the available options for composers wanting to write in a diatonic framework.

You can of course fit any scale into other edos by hobbling. For instance, 22edo's pajara can be fit into 23 (in which case will now have 10 modes instead of 5 due to losing symmetry.)

Modulation pathways

Hobbled scales can facilitate modulation between different mos patterns or edo systems, as the altered intervals may serve as pivot points or transitional harmonies. For example, you could start in 24edo, hobble one small leg into 23 for a brief passage, then hobble it again to balance the legs out into 22edo so that it temporarily is "beaten" into shape.

Dual-fifth systems

It lends particular well for dual-fifth systems, as a hobbled scale will have two different interval types for every scale step, and thus sound less "wrong" when hearing inconsistency in interval quality.

Examples and discoveries

18edo hobbled Lydian

18edo provides an interesting case study for hobbled scales, particularly with lydian patterns. The scale 3–3–3–2–3–3–1 creates an effectively hobbled lydian where one of the short steps has been reduced. This scale exploits the fact that 18edo shares the same wholetone (200 ¢) as 12edo, since both are supersets of 6edo.

The first half of this hobbled Lydian scale is identical to 12edo Lydian, preserving the quintessential dreamy wholetone character. However, the second half becomes distinctly alien with a sharp 733 ¢ fifth, a near-just 12/7 sixth (933 ¢), and a smaller leading tone. This creates a scale that starts familiar but ventures into bright, colorful territory that regular lydian cannot access.

This pattern can also be viewed as a hobbled 17edo or 19edo diatonic, demonstrating how hobbling can bridge between different edos and give a useful lens to understand scales.