In Chapter 12 we outlined the various considerations surrounding interval progressions—the series of pitch combinations that result when melodic lines sound simultaneously. Our discussion so far has been limited to musical textures consisting of just two voices. Chapter 14 will expand this discussion to include progressions with more than two voices, but let us preview what a four-voice texture might look and sound like here:
Example 13–1. Johann Joseph Fux, Gradus ad Parnassum, Fig. 170 .
This short piece—actually a sample exercise from a composition manual by Johann Joseph Fux—consists of four voices in F major. For the most part, each pair of voices follows the conventions discussed in Chapter 12. Given that all four voices follow the same simple rhythm, we may describe the passage as a series of polyphonic sonorities. The following example reduces these sonorities to their unique pitch classes and arranges each of them on a single staff with the noteheads as close together as possible:
Example 13–2. Reduction of Johann Joseph Fux, Gradus ad Parnassum, Fig. 170 .
With the exception of the final measure, all of the sonorities in Example 13–2 look and sound remarkably similar: three-note stacks of thirds differing only in their position on the staff. Harmonies such as these are known as triads and they appear in virtually every piece of tonal Western art music.
In this chapter we will discuss the construction of triads and the different types one encounters. We also will describe a widely-used system for labeling triads in a key using Roman numerals and the various analytical applications in which these labels come in handy.
A chord or harmony is a musical sonority consisting of two or more pitches. (Most people reserve these terms for sonorities with three or more pitches, though an interval may also be considered a type of chord.) A triad, as the name implies, is a type of chord made up of three unique pitch classes. Not all three-note chords are triads, however. For a chord to be a triad, the pitches contained therein must combine to create specific intervals.
When written as closely as possible on a staff, the two upper notes of a triad must form a third and a fifth—or compound third and fifth—above the lowest note. The three chord members have names that correspond with their position in the chord:
Example 13–3.
When a triad is written in the manner shown above—as a stack of thirds with the three notes occupying consecutive lines or spaces on the staff—the lowest note is called the root. The name is easy to remember since the root provides a stable support for the rest of the chord, just like the roots of a tree. The other notes are named according to the interval they form above the root: the third is a third above the root and the fifth is a fifth above the root. These names stay with their respective pitch-classes, regardless of how the chord is voiced (arranged on the staff):
This chord has the same pitch classes as the one in Example 13–3. Therefore, G is still considered to be the root. Likewise, B and D are still the third and fifth, respectively, even though they are now positioned below the root.
Identify the root in each of the following triads.
What is the root of the triad below?
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1a.mp3
When the three notes of a triad are written on three successive lines or spaces, the lowest note will be the root.
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1b.mp3
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1c.mp3
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1d.mp3
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1e.mp3
Exercise 13–1f:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-1f.mp3
Although there are only three chord members in a triad, this type of harmony frequently appears in textures with more than three voices. (Recall the example from the introduction to this chapter, where a series of triads appears as a result of combining four melodic lines.) When a chord member appears more than once in a voicing, we say that it has been doubled. The following example shows the same triad as above but here voiced in SATB format (see Chapter 5):
Despite the fact that there are four voices, the chord in Example 13–5 is still considered a triad. It presents four pitches, but only three unique pitch classes and in this regard is the same as Example 13–3 and Example 13–4. Notice that both the bass and soprano have G, the root of the chord. We would say, then, that the root of the chord has been doubled.
The excerpt below is written for four voices. This means that any triads will necessarily have at least one chord member doubled. Identify the doubled note in each of the boxed triads.
Fanny Hensel, Gartenlieder (Op. 3), 1. "Hörst du nicht die Bäume rauschen, mm. 1–5 .
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-2.mp3
Which chord member has been doubled in box "a"?
First find the doubled pitch class, then determine if it is the root, third, or fifth of the triad.
third (D#)
Which chord member has been doubled in box "b"?
root (F#)
Which chord member has been doubled in box "c"?
root (C#)
Which chord member has been doubled in box "d"?
Which chord member has been doubled in box "e"?
fifth (F#)
Which chord member has been doubled in box "f"?
B (root)
As with intervals, triads come in different qualities. Triads may be major, minor, diminished, or augmented. To determine the quality of a triad, one must consider the qualities of the intervals contained therein.
The following example shows a major triad and a minor triad built on the same root:
a. major triad
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/example_13-6a.mp3
b. minor triad
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/example_13-6b.mp3
Both of the triads in Example 13–6 are consonant and stable. This is largely due to the fact that both chords feature a perfect fifth between the root and fifth. The difference between major and minor triads lies in the quality of the interval from the root to the third. In a major triad, the interval from the root to the third is a major third; in a minor triad it is a minor third.
In addition to the intervals formed with the root, there is another interval heard between the third and the fifth. Notice that in major and minor triads, the quality of this third is the opposite of the quality of the whole chord. In other words, a major triad has a minor third between the third and the fifth and a minor triad has a major third in the same place. In addition to thinking of a triad as consisting of a third and a fifth above a root, it is also helpful to think of it as two thirds stacked one on top of the other.
Note: A tertian harmony is one that is constructed as a stack of thirds. Triads are tertian harmonies. If one were to add another third on top of a triad, that note would be a seventh above the root. This type of tertian harmony is discussed in Chapter 18.
Triads are named according to their root and quality. The triad in Example 13–6a, for example, is a G-major triad and the triad in Example 13–6b is a G-minor triad. Triads may be built on any note. The following example shows an Eb-major triad:
Notice that an Eb-major triad requires two accidentals to preserve the exact interval qualities shared by all major triads.
A triad with a minor third and a diminished fifth above the root is considered diminished. The following example shows a G-diminished triad:
This triad is much more dissonant than the major and minor triads heard above. It has the same minor third between the root and third as the minor triad, but here the perfect fifth has been replaced with a dissonant tritone: a diminished fifth. (You may also think of it as a stack of two minor thirds.) As a result, this chord is much less stable. We will discuss this chord at length in Chapter 16.
A triad with a major third and an augmented fifth above the root is considered augmented. The following example shows a G-augmented triad:
Like a diminished triad, an augmented triad is dissonant. Like a major triad, it has a major third between the root and third. But here we find another major third stacked on top, making the framing fifth augmented. Of the four triad qualities, augmented triads are the outliers. They have a very peculiar sound and, as we will see momentarily, they differ from the other triad qualities in that they cannot be constructed using only diatonic pitches. As a result, they appear far less frequently than major, minor, and diminished triads. We will look at a few examples of this rare chord in Chapter 34.
Note: Triad names are often abbreviated. An uppercase pitch letter name with an uppercase "M"—or an uppercase pitch letter name by itself—indicates a major triad ("GM" and "G" are short for "G major"). An uppercase pitch letter name with a lowercase "m"—or a lowercase pitch letter name by itself—indicates a minor triad ("Gm" and "g" are short for "G minor"). A raised degree sign (o) attached to pitch letter name indicates a diminished triad ("Go" is short for "G diminished"). Finally, a raised plus sign (+) attached to a pitch letter name indicates an augmented triad ("G+" is short for "G augmented").
Table 13–1 summarizes the intervallic content of the four triad qualities:
Triad quality:
Interval between root and third:
Interval between third and fifth:
Interval between root and fifth:
major M3 m3 P5
minor m3 M3 P5
diminished m3 m3 d5
augmented M3 M3 A5
Identify the quality of each of the following triads.
What is the quality of the triad below?
Rewrite the triad in root position and consider the intervals heard above the bass. Consult Table 13–1 to determine the quality.
In each of the following exercises, you will be asked to spell a triad given the root and quality.
Which three pitch classes are used to spell an E-major triad?
Write the noteheads on a piece of staff paper and add accidentals where necessary.
root: E, third: G#, fifth: B
Which three pitch classes are used to spell an A-diminished triad?
root: A, third: C, fifth: Eb
Which three pitch classes are used to spell a Bb-major triad?
root: Bb, third: D, fifth: F
Which three pitch classes are used to spell a G-augmented triad?
root: G, third: B, fifth: D#
Which three pitch classes are used to spell a C#-minor triad?
root: C#, third: E, fifth: G#
Which three pitch classes are used to spell an Eb-major triad?
root: Eb, third: G, fifth: Bb
It is important that you be able to quickly and accurately identify or construct triads. One useful step in acquiring this skill is memorizing all of the natural triads—that is, all of the triads that can be constructed using the white keys on a piano keyboard.
Consider, for example, a triad built on C using only natural pitches, no sharp or flat pitches. The following example shows such a triad in staff notation and shows the location of the corresponding piano keys:
Example 13–10
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/example_13-10a.mp3
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/example_13-10b.mp3
This chord—built using the pitches C, E, and G—is a C-major triad. With this in mind, it is very easy to identify the following chord:
Example 13–11.
This triad is very similar to the C-major triad shown above. It has the same root and fifth. The only difference is that the third lies a semitone lower, making the interval between the root and third a minor third. By comparing it with the C-major triad shown above, it is clear that the chord in Example 13–11 is a minor triad.
The following table summarizes all of the natural triads:
Root:
Triad:
A C E A minor
B D F B diminished
C E G C major
D F A D minor
E G B E minor
F A C F major
G B D G major
Notice that all of the natural triads are either major or minor with the exception of one: the natural triad built on B is diminished. Notice, too, that there are no augmented triads in Table 13–2. An augmented triad will always require at least one accidental.
In each of the following exercises, you will be asked to identify the chord that results from adding one or more accidentals to a natural triad.
The natural triad built on E (E, G, and B) is minor. What chord would result from raising the third one semitone?
Try playing the chord on a piano and identifying the quality by ear.
E major (E, G#, B)
The natural triad built on G (G, B, and D) is major. What chord would result from lowering the third one semitone?
G minor (G, Bb, D)
The natural triad built on A (A, C, and E) is minor. What chord would result from lowering both the root and the fifth?
Ab major (Ab, C, Eb)
The natural triad built on B (B, D, and F) is diminished. What chord would result from raising the fifth one semitone?
B minor (B, D, F#)
The natural triad built on B (B, D, and F) is diminished. What chord would result from lowering the root one semitone?
Bb major (Bb, D, F)
The natural triad built on D (D, F, and A) is minor. What chord would result from lowering the root one semitone?
Db augmented (Db, F, A)
Of the three chord members, the root of a triad is considered to be the strongest and most essential. In terms of voicing, the bass note—the lowest sounding note—is often heard as supporting the notes that appear above it. When these two things align—that is, when the root of a triad appears in the bass—we tend to hear the chord as being very grounded: the most stable chord member is in the most stable part of the chord. Of course, the third and fifth may appear in the bass as well, in which cases the chord will sound comparatively less stable.
The position of a chord is determined by the chord member sounding in the bass. Since there are three chord members in a triad, there are three possible positions. A triad with the root in the bass is said to be in root position. Triads with the third or fifth in the bass are said to be inverted since the root appears higher up with at least one of the chord's intervals inverted. A first inversion triad has the third in the bass while a second inversion triad has the fifth in the bass.
The following example shows all three positions of a C-major triad.
Notice that we have also included a small stack of Arabic numbers (1, 2, 3, etc.) under each chord. These numbers—collectively refereed to as figured bass—indicate the sizes of the intervals appearing above the bass note and, therefore, the position of the chord. (See Chapter 21 for a more in depth discussion of figured bass.) Each chord position has a unique set of numbers. A root position triad has the root in the bass with the other notes of the triad forming a third and fifth above it. The complete figured-bass signature is thus 5/3. A first inversion triad inverts the interval between the root and third of the chord (C and E in this case) to a sixth and retains the third between the third and the fifth (E and G), hence the figured-bass signature 6/3. A second inversion triad inverts both of the original intervals and therefore contains a fourth and a sixth above the bass, thus the figured-bass signature 6/4. You will frequently encounter triads referred to by their interval content ("six-three triad" instead of "first-inversion triad").
Note: For the sake of clarity, each of the chords in the example above is written using only three notes. Doublings have no bearing on the position of the chord. Similarly, the relative locations of the upper voices do not affect the chord's position. If the E in the root-position triad above was written higher than the G, it would still be a root-position triad.
In this activity, you will be presented with a series of triads in SATB setting. For each exercise, choose the appropriate figured bass signature (5/3, 6/3, or 6/4) to represent the inversion of the triad.
Which figured bass signature would be used to represent this chord?
Remember, the figured bass signature represents the intervals that appear above the bass. Disregarding octave doublings, make sure your answer accounts for all the intervals formed with the bass.
Figured bass originated as a shorthand technique, so the figures used to indicate chord inversions are often abbreviated. Root-position triads are so common that they are generally represented with no figure at all. They are also occasionally indicated with only 5 (the third above the bass is assumed). The following example shows three ways of representing a C-major triad in root position:
First-inversion triads also appear quite frequently, so the 6/3 figure is often abbreviated to just 6, with the third taken for granted. Both of the figures in Example 13–14 can be used to indicate a C-major triad in first inversion:
Second inversion triads are always represented with 6/4.
The following excerpt shows how figured bass can be used to indicate inversions:
Example 13–15. Reduction of Johann Sebastian Bach, "Nun danket alle Gott" (BWV 386), mm. 1–2 .
The first three chords are all A-major triads. The first and second are both in root position even though the bass leaps up an octave. As the bass continues to leap up from A to C# in the first full measure, the figures change from 5/3 to 6/3 indicating the progression from a root position A-major triad to a first inversion A-major triad. A similar situation happens with the two D-major triads on beats three and four of that same measure.
The following table summarizes the various figures for triads and lists the common abbreviations:
Table 13–3.
Figured Bass:
Common Abbreviations:
root position 5/3 5 or no figure
first inversion 6/3 6
second inversion 6/4
Note the absence of 6/4 chords in Example 13–15. Second inversion triads are considered unstable in this style of music and therefore appear far less frequently. We will discuss second-inversion triads at greater length in Chapter 23. You should nonetheless be familiar with all three rows of figured bass symbols in Table 13–3.
As with intervals, it can be helpful to think about triads as they relate to a scale or key. A C-major triad, for example, is built using scale degrees [latex]\hat1[/latex], [latex]\hat3[/latex], and [latex]\hat5[/latex] of the C-major scale:
Notice the similarity in sound between the C-major scale and C-major triad. Every major and minor triad shares this relationship with the corresponding key. But do not let this relationship lead you to confuse scale degrees and chord member names. The terms root, third, and fifth refer to the position of a note within a triad and do not necessarily correspond with scale degree numbers. In fact, we can build triads using any scale degree as the root. The following example shows all of the diatonic triads in C major:
In Chapter 6, we discussed several different ways of labeling scale degrees. The set of scale degree names can also be used to label chords in the context of a key:
These labels are particularly useful when talking about how various chords relate to the key. We may use them, for instance, to point out that an E-minor chord is the mediant of C major. These terms are similarly used for describing key relationships. One could say, for example, that the key of G major is the dominant of C major.
Minor keys have a different ordering of chord qualities:
You may notice that with regard to the qualities of the triads, the pattern in minor is the same as major, but beginning in a different place. (Beginning with the mediant in minor, you'll find the same pattern of qualities as begins on the tonic in major.) This similarity is a result of the relationship between relative keys. We will return to minor-key chord qualities in Chapter 17, when we discuss common alterations made to the natural minor scale.
The following questions will test your knowledge of how certain chords relate to keys.
An E-minor chord is what in the key of C major?
Chords in a key are identified by the scale degree that corresponds with the root.
E minor is the mediant of C major.
An E-minor chord is what in the key of G major?
E minor is the submediant (relative minor) of G major.
An A-diminished triad is the leading tone chord in which major key?
A diminished is the leading tone chord in Bb major.
An E-minor triad is the supertonic in which major key?
What is the dominant chord of E major?
B major is the dominant of E major.
What is the subdominant chord of F major?
Bb major is the subdominant of F major.
Roman numerals are a useful, shorthand way of naming and analyzing chords, and of showing their relationships to a tonic. They are a popular tool in the harmonic analysis of tonal music because they convey two vital pieces of information in a single symbol, indicating both the root and the quality of a chord. The number symbolized by the Roman numeral corresponds to the scale degree serving as the root of the chord. The quality is indicated by the case of the Roman numeral: upper case indicates major triads, lower case minor triads. As discussed above, a raised degree sign (o) attached to a lower-case Roman numeral indicates a diminished triad and a raised plus sign (+) attached to an upper-case Roman numeral indicates an augmented triad.
It is imperative that you familiarize yourself with the qualities of diatonic triads in both major and minor keys. The following example shows the pattern of major, minor, and diminished triads in a major key:
As you can see, the triads built on scale degrees [latex]\hat1[/latex], [latex]\hat4[/latex], and [latex]\hat5[/latex] are major (I, IV, and V), while the triads built on scale degrees [latex]\hat2[/latex], [latex]\hat3[/latex], and [latex]\hat6[/latex] are minor (ii, iii, and vi). Notice that the triad built on scale degree [latex]\hat7[/latex] (viio) is the only diminished triad of the group.
For easy demonstration, Example 13–20 is in C major, but the pattern of triad qualities is identical for all major keys. Roman numerals always refer to the scale degrees of the key at hand. If Example 13–20 were transposed to E major, the Roman numerals would stay the same:
Minor keys, on the other hand, have their own pattern of major, minor, and diminished triads:
Again, the pattern in minor is the same as major, but beginning in a different place. (Beginning with III in minor, you'll find the same pattern of qualities as begins on I in major.)
Note: The Roman numeral system described here is just one of several commonly used methods for identifying chords. In other texts you may encounter analyses that use only uppercase Roman numerals. In such cases, it is assumed that the reader will be familiar enough with the qualities of diatonic chords that they will know that "VI" in a major key indicates a minor triad unless otherwise specified.
Another widely-used system—the so called "Nashville number" system—uses Arabic numerals instead of Roman numerals. In the context of G major, for example, "4" is used to indicate a C-major triad. Many jazz musicians are familiar with a more straightforward system. There, chords are specified by their pitch content alone with no reference to the scale degree on which they are built. A "D," for instance, indicates a D-major triad. Extra letters and symbols are used to indicate other chord qualities: "Em," for example, refers to an E-minor triad. This system is sometimes called the "slash chord" system in reference to how inversions are indicated. If the chord is not in root position, a slash followed by a note letter name is used to specify which chord member is in the bass. "Em/G," for example, refers to an E-minor triad in which G is the lowest sounding note—in other words, an E-minor triad in first inversion.
Switching from one system to another can be confusing at first. But with patience and perseverance you will find that all of these systems can be both useful and intuitive. Each method has advantages and disadvantages for different kinds of music and no system is universally perfect.
As we have seen, Roman numerals are succinct and informative in themselves: they indicate both the root and quality of a given harmony. Beyond this, a Roman numeral may be combined with a figured bass signature to provide an even more thorough summary of a chord. The figured bass, by specifying the intervals heard above the bass note, indicates the position of the triad while the Roman numeral indicates the scale degree of the root and the quality of the chord. The following example shows a first-inversion D-major chord in the context of G major:
Here, the Roman numeral tells us that this is a dominant chord in G major since V corresponds with scale degree [latex]\hat5[/latex]. We know that this is a major triad since the Roman numeral is written with an uppercase letter. Finally, we know that the bass note (F#) is the third of the chord since 6 (short for 6/3) is the figured bass signature for a first-inversion triad.
This combination of analytical tools allows us to summarize the harmonic content of a piece or passage very efficiently. Consider the following analysis:
Example 13–24. Giovanni Pierluigi da Palestrina, Missa Inviolata, III. "Credo in unum Deum," mm. 73–81 .
This passage is in the key of F major. A Roman numeral has been placed under nearly every bass note. The passage begins with an F-major triad: F, A, and C with the root doubled in the alto. This is a tonic chord in the key of F major and so has been labeled "I." The D-minor and A-minor triads that follow, built on scale degrees [latex]\hat6[/latex] and [latex]\hat3[/latex] respectively, have been labeled "vi" and "iii" for the same reasons. The first chord in m. 75 is another minor triad, though here the root is in the soprano voice with the fifth in the alto and the third doubled in the tenor and bass. Since the root (G) is scale degree [latex]\hat2[/latex] in F-major and since the bass note is the third, the chord has been labeled ii6.
Note: Note that some of the notes—we have put them in parentheses—do not belong to the chord with which they sound. They are melodic embellishments that enliven the musical surface. These nonharmonic tones are discussed in greater detail in Chapter 15.
Aside from efficiently cataloging the pitch content of each chord, the Roman numerals allow us to make a number of quick observations about the whole passage. The first set of chords—up to the rest at the end of m. 76—begins with the tonic (I) and ends with the dominant (V). The second set of chords (mm. 77 – 81) does the opposite: it begins with a long V chord and works its way back to I. Most of the chords are in root position, adding to the excerpt's stability and finality. Some chord progressions are common to both sets of chords, including ii – IV and IV – V.
Exercise 13–8:
Add a Roman numeral in F major to each of the chords in the music below The first and last Roman numerals have been provided for you.
Johann Joseph Fux, Gradus ad Parnassum, Fig. 170 .
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-8_answer.mp3
Write out each of the indicated chords.
Write the following chord in Eb major:
The root of the chord is the scale degree indicated by the Roman numeral. The quality of the chord is indicated by the case of the Roman numeral. The position is indicated by the bass figures next to the Roman numeral.
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9a_answer.mp3
Answers will vary, but pitch content and bass pitch class must be the same.
Write the following chord in G major:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9b_answer.mp3
Write the following chord in Bb major:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9c_answer.mp3
Write the following chord in D major:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9d_answer.mp3
Write the following chord in A major:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9e_answer.mp3
Write the following chord in Ab major:
https://milnepublishing.geneseo.edu/app/uploads/sites/61/2019/12/activity_13-9f_answer.mp3
A triad is a type of chord consisting of three unique pitch classes. When the three pitches are written on consecutive lines or spaces on a staff, we refer to the lowest note as the root and the upper notes as the third and fifth, based on the intervals they form above the root. There are many ways to voice a triad: the positions of the three chord members may be rearranged and any of them may be doubled. The names of the chord members, however, stay with their corresponding pitch classes, regardless of how the chord is voiced.
Like intervals, triads come in different qualities. Major and minor triads are consonant. They feature a perfect fifth between the root and chordal fifth and are named after the major or minor third between the root and chordal third. Diminished and augmented triads are dissonant. A diminished triad has a minor third and diminished fifth above the root while an augmented triad has a major third and augmented fifth above the root. Of the four triad qualities, augmented triads are outliers since they cannot be constructed using only diatonic pitches and will therefore always require at least one accidental. Major, minor, and diminished triads, on the other hand, can be constructed using only white (natural) keys on a piano and appear in every major or minor key.
The position of a triad is determined by the bass voice. If the root is in the bass, the chord is said to be in root position. If the third or fifth is in the bass, the chord is said to be in first or second inversion, respectively. We indicate a chord's position with a figured bass signature—a stack of Arabic numerals corresponding with the sizes of the intervals heard above the bass. The figure 5/3 (or 5 or no figure at all) is used to indicate a triad in root position, and 6/3 (or 6) and 6/4 are used to represent triads in, respectively, first and second inversions.
Roman numerals are a convenient means of naming and analyzing chords. They concisely convey important information. The numbers symbolized by Roman numerals indicate the scale degrees on which chords are built. "I" indicates a chord built on [latex]\hat1[/latex], "ii" a chord built on [latex]\hat2[/latex], and so on. Furthermore, the case of the Roman numeral indicates the quality of the chord. Figured bass numerals may be added to Roman numerals to provide an even more thorough account of the content and construction of any given triad.
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