We can now name any tertial triad, regardless whether its root, third, and/or fifth belong to the prevailing scale of the music or not. But some chords involve other pitches than the third and fifth above the root, and these may be worth specifying too. Also, it can be important to know which pitch class is the lowest pitch: it isn’t always the root, or even necessarily a pitch class that is otherwise present in the chord at all.
Both situations—chord tones other than the third and fifth and bass notes other than the root—are handled by adding suffixes to our chord symbols. The added chord tones can be complicated to notate and explain, making this page quite a long one; but the bass notes are much simpler to handle, and are covered briefly at the end of the page. Readers familiar with the terminology for chords and intervals may wish to skip the explanations and go directly to the table where the chord suffixes are listed towards the end of the page.
Chord tones other than the third and fifth can arise from continuing the “tertial” process of stacking up pitches each two scale-steps higher than the last. This will produce intervals of a seventh, a ninth, an eleventh, and even a thirteenth above the root. (A fifteenth would be two octaves above the root, so it wouldn’t add a different pitch class to the chord, and the process stops at the thirteenth.) It’s also not unusual to add other pitches—a second, fourth, or sixth above the root—without going through this tertial stacking process. Needless to say, any of these added pitches may or may not belong to the prevailing scale of the music.
Our ultimate goal is to be able to give any chord a name from which we can tell exactly what intervals above the root are contained in that chord, and whether they arise from a tertial stacking process. (We are not trying to specify in what octave each of the constituent pitches occurs: that would be better done by notating the pitches individually on a set of pitch lines.) A secondary goal, subject to the above requirement, is to make each chord name as short as possible: when chords change rapidly, long chord names are hard to fit in to the score. A third goal is to use established symbols and conventions as far as possible, so that the learning curve for users will not be steeper than necessary. When these different goals pull against each other, global notation prioritizes them in the above order. Existing chord notation systems may work well within their own traditions, but their priorities may be different from ours, and global notation has had to mix and adapt ideas from different systems in pursuit of the ability to represent harmony in any music.
Chords involving more than the standard three scale degrees can be indicated by adding numbers to the end of the chord symbol in the manner of guitar chord symbols. The number indicates the interval between the root and the added pitch: for instance, 6 means an added pitch a sixth above the root. But there are different kinds of sixth: a “minor sixth” of 800 cents, a “major sixth” of 900 cents, and an “augmented sixth” of 1000 cents. (Yes, that is the same size as a minor seventh, but there are times when it is better described as a sixth because it is formed of five scale-steps (from a starting point counted as “one”) rather than six; remember that scale-steps are not all the same size.) It’s even possible (though rare in practice) to form a “diminished sixth” of 700 cents (the same size as the usual fifth, though formed of five scale-steps rather than four).
To specify the added pitches precisely (but also concisely), we return to the principle that “major is the default” (see Naming any tertial triad). That is, any number by itself refers to the interval as it would appear above the tonic in the major scale. In the case of 2, 3, 6, and 7, these are called “major” intervals: a major second, major third, etc. The same terms are used for added pitches in a chord, measuring the interval from the chord root, as in “IV with an added major sixth,” which is notated as IV6. With these numbers, the equivalent “minor” interval is 100 cents narrower, so the added pitch would be 100 cents lower. The symbol for a “minor” interval is a lower-case m, so a minor sixth would be m6. A minor triad on scale degree 4 with an added minor sixth would be ivm6.
Those accustomed to guitar chord notation, where the number 7 by itself means a minor seventh, should note that in our system it refers (like the unmodified numbers 2, 3, and 6) to the major version of the interval. If this is too counter-intuitive, the major seventh can be explicitly marked with an upper-case M, as in bIIM7. What the guitarist calls a “seventh chord,” which has a minor seventh and a major third, is written in global notation as Xm7 (using X here to mean a major triad on any root). The guitarist’s “minor seventh chord,” with a minor seventh and minor third, is xm7. This is a case where internal consistency has been prioritized over following existing conventions.
The other degrees in the major scale—the fourth, fifth, and octave—don’t have “major” and “minor” versions like this. For these, the form of the interval that appears in the major scale is called the “perfect” fourth, fifth, or octave. If we want to refer to an interval 100 cents narrower than the “perfect” one (but still formed of the same number of scale-steps), we call it a “diminished” fourth (etc.). If it’s 100 cents wider than the “perfect” interval, it’s an “augmented” one.
In our basic triad descriptions (see Chords: the basics), the terms “diminished” and “augmented” referred to specific “shapes” of triad characterized in part by the size of the interval between root and fifth. The symbols for these triad shapes were ° and + respectively. However, when using the same terms to describe an interval or added pitch rather than a triad shape, it could be confusing to use the same symbols, especially since the symbol must now precede rather than follow the numeral it is attached to. In this context, therefore, global notation substitutes the abbreviations “d” for diminished and “a” for augmented. An added pitch an augmented fourth above the root, for instance, would be written as a4.
“Diminished” and “augmented” can also apply to intervals of a second, third, sixth, or seventh. In this case, the diminished interval is 100 cents narrower than the minor version of the interval, and the augmented interval is 100 cents wider than the major version. Here again, the abbreviations “d” and “a” are used. For example, a bVI major triad with an added augmented sixth (known as a “German sixth chord”) would be bVIa6. A diminished triad (on any root) with an added diminished seventh (known as a “diminished seventh chord”) is x°d7.
When there is more than one numerical suffix, the numbers are written in ascending order. (To save space, they are not separated by slashes as in some existing chord notation formats.)
(Guitar chord notation usually puts such suffixes in superscript, but in global notation this is optional: the meaning is unambiguous without using superscript, which could be difficult to read if the chord symbols are already in a small font.)
With added pitches bearing numbers above 7, the question might arise why (having said that octave transpositions don’t change the identity of the chord) we don’t simply subtract 7 and represent the pitch by its equivalent an octave lower. The answer is that there is a difference between, say, an added ninth and an added second. The second is simply added to the triad, but the ninth arises from tertial stacking, via the seventh. While it’s not always as simple as this, a good rule of thumb is to use the number 9 when there is also an added seventh, and 2 when there isn’t.
The same rule applies to using the numbers 11 and 13 rather than their octave equivalents 4 and 6. Chords with an added eleventh or thirteenth almost always have an added seventh as well, even if one or more other pitches in the tertial “stack” are omitted. (In the next example, some pitch classes appear with octave transpositions or doublings to show the potential for confusion.)
It will be seen that, when a chord has more than one added pitch, the numbers and letters are simply strung together without any spaces or punctuation. This helps to keep the chord names short, and it doesn’t create ambiguity once the principles have been grasped. For instance, the digit 1 only occurs as part of a two-digit number, so “1113” can only mean “an eleventh and a major thirteenth.”
Also in the interest of keeping chord names short, global notation adopts a few conventions by which a single number conveys more information than might otherwise be the case. These conventions have to be memorized for each relevant number, but the effort seems justified by the gain in brevity.
The first of these conventions concerns the number 4. In practice, chords with an added perfect fourth usually omit the third, probably because it would clash too strongly with the fourth or is being withheld as a “consonant” sound for the “dissonant” fourth to resolve to. This practice is so prevalent that we will regard it as the “default” for such chords. That is, the unmodified number 4 in a chord suffix implies that the third is absent. If the third is in fact present along with the fourth, then both numbers can be written in the chord suffix: 34. The same rule applies to 4’s octave equivalent 11, but not to the augmented version of either interval: a4 and a11 do not imply the absence of 3.
Another complication about fourths is that, when the third is absent, it can be neither major nor minor, so we might not know whether to use upper or lower case for the chord’s Roman numeral or letter name. For consistency, we will use upper case for all such chords provided the fifth is “perfect.” If it is diminished or augmented, we will follow our usual format for triads of those types: x° and X+ respectively.
Another type of chord that has no third is the “power chord” of rock music, which consists only of a root and fifth. Since the number 5 is not otherwise needed in our chord notation, we can give it the specific meaning of “fifth but no third.” In power chords, the fifth is usually a perfect fifth, but occasionally it may be a diminished or augmented fifth, so we can use all three suffixes—5, d5, and a5—in this way. In all three cases, the Roman numeral or letter name can be in upper case.
At the other extreme, some chords have no fifth. Often this is the kind of detail that doesn’t need to be specified, as the root and major or minor character of the chord are clear to the hearer even without a fifth; but if you want to make it explicit, you can reverse the above convention so that just as 5 indicates no 3, 3 indicates no 5. To think of it another way, 5 means “5 only” and 3 means “3 only” (among the upper notes of a triad). However, here we have to make an exception since we have already used “34” to mean “both 3 and 4” without implying the absence of 5. Let’s say the “only” is canceled by the “4” (or its octave equivalent “11”).
This rather indirect way of indicating the absence of a third or fifth is based on a preference for describing chords by what they contain rather than what they omit. We can use 3 and 5 to imply each other’s absence because a chord without either a third or a fifth would be rare in the major/minor system, and might be hard to identify as a tertial chord on the supposed root at all. However, such chords do occur, and sometimes even the root of a chord may be present only by implication. In such cases, we may need to indicate directly that a pitch we would “normally” expect to be there is absent. We can do this by using a “minus” sign (-) before the relevant number.
Finally, let’s summarize what we need to know about added chord tones in table form.
|-1||Chord root is implied, not sounded|
|1||(Only occurs as part of a two-digit number)|
|m2||Added minor second|
|2||Added major second|
|3|| Third (major or minor according to case of preceding numeral)|
and (unless chord name includes 4 or 11) no fifth
|-3-5||No third or fifth|
|4||Added perfect fourth and (unless preceded by 3) no third|
|a4||Added augmented fourth (does not imply absence of third)|
|d5||Diminished fifth and no third|
|5||Perfect fifth and no third|
|a5||Augmented fifth and no third|
|m6||Added minor sixth|
|a6||Added augmented sixth|
|d7||Added diminished seventh|
|m7||Added minor seventh|
|7 or M7||Added major seventh|
|Note: remaining suffixes are used only when seventh is also present|
|m9||Added minor ninth|
|9||Added major ninth|
|11||Added perfect eleventh and (unless chord name includes 3) no third|
|a11||Added augmented eleventh (does not imply absence of third)|
|m13||Added minor thirteenth|
|13||Added major thirteenth|
|Note: when more than one suffix is needed, they are added in the above order (ascending numerical order)|
All of this of course assumes that tertial triads are the “normal” or “basic” type of chord, so the chord symbols presented here should only be used for musical styles in which that is in fact the case. Music based on different types of chord will call for different types of chord symbol which can’t be covered on this already lengthy page.
Another kind of information that can be given in chord suffixes concerns what pitch is in the bass. Fortunately, this is simpler than specifying the many and varied intervals that can occur in a chord as described above.
Since the basic identity of a chord is not changed by moving any of its constituent notes up or down an octave, it follows that the lowest note in a chord (the “bass”) is not always the root. Sometimes one of the other pitch classes in the chord appears in the bass; and sometimes the bass note is of a pitch class that doesn’t otherwise appear in the chord or even in the prevailing scale. As the bass makes a big difference to the effect of the chords, we may often want to notate this level of detail too.
By default, the bass note is assumed to be the root of the chord. If it isn’t, the chord name as determined so far (including any suffixes for added pitches) is followed by a slash and then a number to indicate the bass note. The number represents the scale degree of the bass note relative to the tonic of the passage (not the root of the chord). As with chord roots, the major scale is taken as the default, and any pitches outside of that scale are indicated with sharp and flat signs, regardless whether or not they belong to the actual scale used in the music. (Note that sharp and flat signs are used with scale degrees above the tonic, whereas m, M, d, and a refer to intervals above a chord root.)
If the bass notes in your example are not always the chord roots, but you don’t want to specify what they are, you can simply write “bass notes unspecified” below the first chord symbol. If you want to leave the bass note of a particular chord unspecified, you can end the chord name with /x.
We are now ready to notate a melody with chords not limited to the basic triads and a bass line not limited to chord roots. However, as chord symbols with suffixes involve a number of characters, and thus a certain horizontal length, there might be ambiguity as to when exactly a new chord begins. To avoid that, we adopt the convention that the left-hand edge of any chord symbol specifies the moment when the chord begins.
In the next example, the melody line has been written in grey and the bass line as a double black line to clarify where they coincide in pitch (see Counterpoint). Notice how the stepwise movement of the bass is reflected in the sequence of suffixes used for those chords which don’t have the root in the bass. (The octave transposition marking at the beginning of this example indicates that only the sounds represented by double black lines sound an octave lower; see Wide pitch range.)
For an example of music using more varied chord types, let’s look at the beginning of Leonard Bernstein’s song “Somewhere” from West Side Story. The sounds of specified pitch have once more been written with a mixture of grey and double black lines to keep each “voice” distinct when they cross or overlap. A chord name has been written for each half-bar, without trying to reflect small changes that happen within a half-bar. The harmonies show added sevenths, ninths, and elevenths, bass notes that are not always the root or even a “normal” part of the chord, “missing” thirds and fifths, and even a chord without its root. For illustration, the added pitches are marked as they occur above the relevant pitch lines, though this would not normally be done in an actual score. Where the chord root is not in the bass, the root pitch is marked with the number 1.
Melodies relate to chords rather like the way they relate to drones: by combining with them to create various forms of “consonance” and “dissonance.” To represent that in notation, it’s not enough to write the melody and chords separately using different kinds of symbol for each, as in the last two examples. We need to show what pitch classes each chord consists of and whether each melody note “harmonizes” with its accompanying chord or not. To do that, we’ll need to notate melody and chords in a way that makes them more “relatable” to each other, and preferably within the same space. That’s our next challenge.