# Given a four note combination under the span of a fourth, does this chart correctly show the number of times it will appear in a scale?

I have made this chart that shows how many times will a four note combination under the span of a fourth whether perfect or augmented appear in a scale and some observations, so I wanted to ask you if this chart correctly reflects this and if my observations are correct or utterly wrong.

• All this is very dependent on the definition of tetrachord - and that's where it all falls down.
– Tim
Commented Jan 31, 2022 at 9:44

Yes, regarding the number of times appearing in the scales.

One, you refer to Gregorian modes, relative major/minor, natural minor, etc. All of that could be summed up as simply the diatonic gamut of ABCDEFG, which contains 2 major, 2 minor, 2 phrygian, and 1 lydian tetrachords.

When you mention that the diatonic gamut contains those tetrachords is sort of hints at looking at the tetrachords on each scale step, like this...

C - major tetrachord    CDEF
D - minor tetrachord    DEFG
E - phrygian tetrachord EFGA
F - lydian tetrachord   FGAB (not a P4 tetrachord)
G - major tetrachord    GABC
A - minor tetrachord    ABCD
B - phrygian tetrachord BCDE

While that is true, you will not be able to do the same thing with the non-diatonic scales. Consider, for example, C harmonic minor...

C  - C D Eb G  minor tetrachord
D  - D Eb F G  phrygian tetrachord
Eb - Eb F G Ab major tetrachord
F  - F G Ab B  (not a P4 tetrachord)
G  - G Ab B C  harmonic tetrachord
Ab - Ab B C D  (not a P4 tetrachord)
B  - B C D Eb  (not a P4 tetrachord)

That leads to the second point. When saying a scale contains various tetrachords, it sort of obscures the fact that a lot of the 4 tone sequences of various scale will not be P4 tetrachords.

A more common reference is to show which of the P4 tetrachords (major, minor, phrygian, and harmonic) separated by a P5 comprise various scales. For example, the major scale is two major tetrachords separated by a perfect fifth. Or, the melodic minor scale is a minor tetrachord and a major tetrachord separated by a perfect fifth. Etc. etc.

There is a good theoretical reason to look at scales and tetrachords in that way. Using C as tonic, and using only P4 tetrachords, all of the scales thus derived will have principle tones of C F G C, or a tonic with two tones a perfect fourth and a perfect fifth above the tonic, which gives us the all important tonal scale degrees. From there all the various tones of D E A B become effectively the modal scale degrees. The method provides more than a whole heap of scales, it provides a framework for understanding the tonal structure of the scales.

I'm not saying you shouldn't make your chart, but just pointing out the reason why joining two P4 tetrachords is a common music theory concept.

• I think the first minor tetrachord got a typo, G for F? Commented Jan 31, 2022 at 19:51

From Alfred's Essentials of Music Theory:

A tetrachord is a series of four notes having a patter of whole step, whole step, half step. The four notes of a tetrachord must be in alphabetical order.

In a major scale, there are two places this pattern occurs. Notes 1 to 4, and notes 5 to 8, so it appears twice in a major scale.

I looked at the Wikipedia Tetrachord definition which says:

In music theory, a tetrachord (Greek: τετράχορδoν, Latin: tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cents)—but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system.

While Wikipedia says:

In traditional music theory, a tetrachord always spanned the interval of a perfect fourth

They don't indicate the pattern of a tetrachord, which I've been taught is important. The pattern is whole step, whole step, half step. Also, Wikipedia says:

in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system

So I would say that just because someone calls a four note segment a tetrachord doesn't mean that it is a tetrachord.

So I guess the useful answer is, know your audience. If you're being tested in a music theory class there is one right answer. Learn that answer. When you're in an informal setting with other musicians (who don't know music theory) they might refer to a 4 note sequence as a tetrachord, and you can at least understand what they mean when they say it.

• The definitions (Alfreds and wikipedia) are different. I believe what I've been taught which is the Alfred's definition is the correct definition. But I respect others opinions which is why I shared the Wikipedia definition. In your comment that says I said there are four tetrachords, I think you misunderstood me. There are two tetrachords in a major scale (1-4 and 5-8), they happen to match the Ionion and Mixolydian modes, but that is unrelated to the definition of a tetrachord (IMO).
– PatS
Commented Jan 31, 2022 at 18:45
• Well, a technicality: I'm not sure that "C-C#-E-F" would be considered a segment of any "scale or tone." If you spelled the C# as Db, presumably it would. Commented Jan 31, 2022 at 19:49
• Actually, as I think about it, maybe it could be. The usual definition is, in the context of "12-tone" or serial music, a complete set of all 12 chromatic pitches, but arranged in a specific order—a sort of 12-tone "melody." But the Wikipedia sentence wouldn't make sense with that definition, unless you re-ordered them in order of pitch... in which case it's just a chromatic scale. Maybe it used the term more broadly to cover more exotic "scales." Commented Jan 31, 2022 at 20:01
• Yup. My point was that, for most "garden-variety" meanings of "scale," the enharmonic spellings are such that you don't repeat or skip a letter, thus Db instead of C#. Meanwhile, to the big-picture question of "what 'tetrachord' means," I think you're onto it: it can mean any 4 neighboring notes, but we talk about the major one so often that sometimes we use it as a shorthand, like saying "C chord" without bothering to say "major." Commented Jan 31, 2022 at 20:07
• So the tone row G,Ab,B,C (intervals are half-step, minor 3rd, half-step) are last 4 notes of C harmonic minor scale match this pattern. See diagram at en.wikipedia.org/wiki/Minor_scale. I don't consider this a tetrachord even though Wikipedia's definition allows it.
– PatS
Commented Feb 1, 2022 at 14:14

Yes, the chart is correct. Here it is recreated in text for future readers.

Tetrachord Tone composition Major N. Minor H. Minor M. Minor (asc.) H. Major
Ionic W-W-H 2 2 1 1 1
Doric W-H-W 2 2 1 2 1
Phrygian H-W-W 2 2 1 1 1
Lydian W-W-W 1 1 2
Harmonic H-A-H 1 1
Total 7 7 4 6 4