Thirteenth chords are built on base of two octave scale similar to seventh chords, which are built on base of one octave scale. There are seven degrees within an octave and fourteen - within two octaves correspondingly. If we continue to build chords in thirds similar to seventh chord (using odd degrees 1, 3, 5, 7, 9, 11, 13), we will get: seventh chords (1, 3, 5, 7), ninth chords "chords of the ninth degree" (1, 3, 5, 7, 9), eleventh chords "chords of the eleventh degree" (1, 3, 5, 7, 9, 11), thirteenth chords "chords of the thirteenth degree" (1, 3, 5, 7, 9, 11, 13). Pay attention, the thirteenth chord include all seven mode degrees, arranged in thirds. When we build a seventh chord in one octave, four of seven notes within an octave are used. The second, the fourth and the sixth degrees, skipped while the chord construction, are correspondingly the ninth, the eleventh and the thirteenth degrees in the next octave. Thirteenth chords consist of basic seventh chord and extensions. The ninth, the eleventh and the thirteenth degrees altogether are actually extensions. If you know the structure of a key, it is easy to remember the structure of diatonic thirteenth chords, because the next triad of the mode is always extensions of the basic seventh chord. For example, the thirteenth chord, built on the first degree of the G Major key, consists of the basic Major seventh chord Gmaj7 and extensions, which is A Minor triad. The thirteenth chord, built on the second degree of the G Major key, consists of the basic Minor seventh chord Am7 and extensions, which is B Minor triad, the next degree in the tonality. This is the structure of all diatonic thirteenth chords. Look carefully at shapes of all thirteenth chords and make sure, that it is true. A chord of the first degree of the major key as well as the chord of the sixth degree of the same tonality (or the main minor thirteenth chord) are "standard", so as they don't have altered degrees. So, if you know shapes of these two chords, you can get all other shapes altering mentioned degrees.
It is very important to remember, that all keys are of the same structure. So, having mastered an exercise in, for example, E Major key, we can use learned shapes in any other key. We can build a triad on each degree of every diatonic mode. As it was already mentioned in previous parts, we need three different fingerings (major, minor and diminished triad) to play all sequence.
A triad consists of three notes and has three inverted positions. To understand the essence of the inversion, let's examine it on the piano. C Major Triad consists of notes: C - the first degree, E - the third degree, G - the fifth degree. Let's play these notes simultaneously; such position of notes within a triad is called "the root position" or just a triad. Now we shift C an octave higher, so E is the lowest. This shape is called the sixth chord (because of the interval of a minor sixth between the last sounds) or the first inversion. If we shift E by an octave higher, we will get the last inversion of a given triad. G is the lowest note now. This inversion is called the six-four chord (because of a fourth between the first and the second sounds and the interval of a major six between the first and the third sounds) or the second inversion. The next transfer of a G note an octave above will lead us to the initial position C, E, G, that is to the root position but one octave higher.
Now let's view triad's shapes on the bass neck. Every bass string (G, D, A, E) is tuned in fourths, so shapes of triads from the E string and from the A string are the same. Pictures below show the arrangement of a triad and its inversions on the bass neck (by example of C major and relative A minor keys). So we have: the root position of a C triad on the E string (eighth fret), then the sixth E chord on the E string (twelfth fret), a six-four G chord on the E string (fifteenth fret). And again the root position of a C triad appears on the E string, it is one octave higher than the initial one (twentieth fret). Triads are separated by green lines for clearness.
Diatonic progression of a (major) triad in its root position