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I don't have a lot of formal music training so this may well be a dumb question, but I've been looking at this off and on for a while and I can't figure out why this particular scale doesn't seem to have a widely used name.

I'll over-explain my question a little bit to avoid confusion in case I misuse a term of art:

Musical scale families are often described by an interval "formula" or pattern that specifies the number of semitones between each note in the scale.

For example, a major scale is defined by the interval pattern "whole, whole, half, whole, whole, whole, half" where a whole step represents 2 semitones (e.g., from C to D, skipping over C#/Db) and a half step represents 1 semitone (e.g. from D to D#/Eb, with no notes in between). We can follow that interval pattern (starting at an arbitrary root note) to generate a major scale in any key.

Numerically we might write that as 2,2,1,2,2,2,1. Any sequence of numbers like this that adds up to 12 wraps back to the root note (pitch class) and repeats itself, creating a one-octave scale.

In addition to the more conventional major, minor, pentatonic, blues, etc. scales, there are a number of scales that seem more inspired by abstract mathematical symmetry, like the whole tone scale (all whole steps: 2,2,2,2,2,2) or the four tone symmetric scale (all minor-thirds: 3,3,3,3).

In particular, a repeating whole-step/half-step pattern (i.e. 2,1,2,1,2,1,2,1) is known as a "diminished" scale, and if you offset that pattern by one (i.e., 1,2,1,2,1,2,1,2) that's known as a "Half/Whole Diminished" scale, for obvious reasons.

Similarly, an "augmented" scale is defined by a minor-third-step/half-step pattern (i.e., 3,1,3,1,3,1).

Logically, you could offset that pattern by one also, creating the interval pattern 1,3,1,3,1,3.

Following the naming convention used for the diminished scale you might call this a "Half/Minor-Third Augmented" scale. But virtually nobody seems to do that. In fact I'm not able to find any commonly accepted name for the scale defined by the interval pattern 1,3,1,3,1,3. My question is: why?

I recognize that this is literally a symmetric scale - i.e., that there are only 4 distinct augmented scales (since C~E~Ab, Db~F~A, and so on) - but the same thing (or a very similar thing) is true for the diminished scale (and a bunch of others). So why are the "whole/half" / "half/whole" modes of the diminished scale relatively well known while the "half/minor-third" mode of the augmented scale seems to be so obscure?

Is there another name for the scale defined by the interval pattern 1,3,1,3,1,3? I.e., what would you call the scale in the key of C that contains the notes C, Db, E, F, Ab, A, [C]? I'm not able to find a commonly accepted name for this scale (and it seems like "half/minor-third augmented" is the obvious one, and I can't figure out why that's not used).

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Is there another name for the scale defined by the interval pattern 1,3,1,3,1,3?

This resource provides the following names:

Messiaen truncated mode 3, Hexatonic Set, Prometheus (Liszt), Genus tertium inverse

This page gives the following "common names" for the scale:

Western Modern
   Augmented Inverse
   Six Tone Symmetrical
Named After Composers
   Prometheus
   Ode To Napoleon Hexachord
   Liszt's Prometheus
Messiaen
   Messiaen 3rd Mode Truncated
   T2 Prime Mode
   Messiaen Mode 3 Truncated
Zeitler
   Aerythimic
Dozenal
   FASian

I don't think there's anything particularly "common" about Zeitler's classification and the Dozenal classification.

I'm not sure I've heard the term "Augmented Inverse" (although by definition that's true -- if one inverts the "augmented scale," one gets this scale), which is the preferred name given on that second link. But I have definitely heard this scale referred to as a mode of an "augmented scale." Messiaen's classifications are also somewhat popular.

Academic music theorists would tend to refer to this as a mode of the augmented scale or a mode of the hexatonic scale. While the term "hexatonic scale" (or hexatonic set) can be used for any six-note scale, it is also a term that specifically tends to be associated with an alternating 3-1 or 1-3 interval pattern. Just like the so-called octatonic scale is mostly associated with what the question calls the "diminished" scale of 1-2 or 2-1 interval patterns.

Whether the 1-3-1-3-1-3 version of the 3-1-3-1-3-1 scale is privileged as the hexatonic scale likely depends on the priorities of the classification system. Some academic music theory articles and textbooks will privilege so-called "prime form" of pitch collections, which tend to like to push all notes in the chord or scale collection as far toward the lowest pitch as possible. Using the common music set theory convention of writing 0 for C, 1 for C♯, etc., the two scales would be 03478E and 014589. ("E" here stands for 11.) The prime form would be 014589 (as it is more "compact" to the left), and thus some would consider this 1-3-1-3-1-3 version to be the first "mode" of the hexatonic scale.

In fact I'm not able to find any commonly accepted name for the scale defined by the interval pattern 1,3,1,3,1,3. My question is: why?

Probably because the 1-3-1-3-1-3 version is significantly less common in actual musical practice. The 3-1-3-1-3-1 version allows the creation of both major and minor triads -- as well as augmented -- above the "tonic" note. As the question notes, the scale is symmetrical, so it may not make a lot of sense to think of any note of the scale as "tonic." But when these hexatonic scales are used to classify sections of (tonal or somewhat tonal) music, they often tend to be labeled based on a "bottom" pitch that allows you to build a major and minor triad on it. (With that version of the scale, you also have the possibility of an incomplete dominant or an augmented dominant triad to relate to the "tonic" chord.) In use for jazz theory and improvisation, the note options afforded by the 3-1-3-1-3-1 version are also a bit better/more common for similar reasons. Hence, the popularity of the 3-1-3-1-3-1 version and likely why it has produced more commonly known names.

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There is. In fact, there are two! Due to the pattern, there are two modes of this set of notes.

One is more simply named: Symmetrical Augmented. m3 - s - m3 - s, etc.

The other - s - m3 - s - m3, etc has been given the catchy name Messiaen Inv. III Truncated n2, not really tripping off even a jazzer's tongue, not even a triple tonguing trumpeter's...

Used mainly over 7alt. chords, it's not used too frequently - and not often referred to by name!

Googled to find mdecks.com, which provides lists of scales.

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    Can you provide a citation for the scale names? It would be very helpful given the rarity of their use.
    – Aaron
    Jul 29 at 9:10
  • +1 for the groovy name and the alliteration. I was wondering how I would use that 1-3 inverted scale since it’s so odd but I realized it would work well for descending lines, say use the 3-1 ascending and the 1-3 descending over an augmented chord. Jul 29 at 15:59
  • mdecks has lots of links, but none seems to point to the scale page. Can you provide the direct link?
    – Aaron
    Jul 29 at 16:38
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    @Aaron, here is one, and a interesting article on scales generally: Tymoczko - The consecutive semitone constraint on scalar structure. esm.rochester.edu/integral/wp-content/uploads/2019/06/… Jul 30 at 16:16
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I think there is, but the augmented scale as three semitones/one semitone is maybe the common way it's described.

I know some articles that define a hexatonic scale as a semitone/3 semitones:

At least in these articles from Tymoczko I don't think he really makes a distinction between m2/m3 and m3/m2, it's just alternating the two intervals.

I think you can regard it like the octatonic scales. It's a symmetrical pattern that can be started in one of two positions regardless of whether there are specific, well known scale names for either of the two. Even with the octatonic scale, it's just called "octatonic scale" and then you need to give some interval order, like WH or HW.

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  • +1 for the Tymoczko pulls. He's an important theorist.
    – Aaron
    Jul 30 at 16:36
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This video:

refers to the regular augmented scale as a “minor third/half step scale” at around one minute in. Following that logic there’s no reason not to call your 1-3-1-3-1-3 scale a half (or half step)/minor third scale. In the video it is mentioned at around the 1:48 mark.

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