While Renaissance and Manneristic theorists proposed various refinements or simplifications for the system of hexachords and solmization, Ramos (1482) not only criticized the system of Guido but proposed a radical alternative based on the complete octave rather than the hexachord.
In 1610 and 1612, a young scholar named Johannes Lippius produced two editions of his Musical Synopsis, a compact introduction to music written both from a theological and practical view. As part of his treatise, Lippius enthusiastically reports on a "Belgic" innovation avoiding the complications of hexachord mutation: a system with seven syllables spanning the octave.
In approaching the rudiments of singing, Ramos takes note of various medieval systems using letters or syllables to name the notes, asserting that although Guido chose his hexachord syllables "by chance..., his followers afterwards adhered so much to his syllables that they thought them entirely necessary to music, an idea which must be scorned." 
Not only does Ramos assert that Guido's system is not the only one possible; he announces that he has invented a better one to assist the new student :
"We, therefore, who have labored a long time in nightly lucubrations and vigils to seek the truth of this art, arrange new syllables for individual strings and show the tones of the entire series, so that on the lowest pitch psal is sung, on the next li, on the third tur, on the fourth per, on the fifth vo, on the sixth ces, on the seventh is, and on the eighth tas. And thus the connection of the syllables will be psallitur per voces istas, for the entire series consists of eight syllables. We arrange them from low c to high c, since they teach how to sing with perfection. Therefore they begin on letter c because the musical series begins on that letter..."
This new system thus uses eight syllables to span the octave c-c, with an interesting feature which might give it some kinship with the traditional hexachord system. While Ramos' first six steps are identical to those of a hexachord (T-T-S-T-T), the seventh step is of the octave is fluid, either B or Bb, with a semitone located below (A-Bb) or above (B-C). The syllables Psal-li-tur per vo-ces is-tas might be translated: "It is sung through these notes."
Thus Ramos' syllables encompass the same notes as the standard hexachord system of musica recta: the seven diatonic notes plus Bb, with B/Bb a flexible degree. A mnemonic feature of these syllables is that semitones occur between syllables ending with the same letters:
or ______ __________________ | | | | | psal - li - tur per vo - ces is tas C D E F G A Bb/B C T T S T T S/T T/S
Thus the first semitone (E-F) occurs between the syllables tur per, both ending in r. Depending on whether the fluid seventh syllable is is sung as Bb or B, the second semitone occurs at either ces is (A-Bb) or is-tas (B-C), in either case between syllables ending in s.
For Ramos, the form with B rather than Bb is evidently primary, since he urges singers to begin their practice with this step, noted with the square-B sign which I show in the German fashion as "H" :
"[W]e wish to confine students to certain rules, for we enjoin them not to begin from just any letter, but from letter c, to another c; also they shall not touch the first b but the second H, which is a whole tone from a in ascent and descent."
Ramos himself notes that, strictly speaking, seven syllables rather than eight might suffice "since there are only seven different pitches"; that is, on reaching the octave one could simply revert to the first syllable of the series (as happens with the letter names of the notes, e.g. C3-C4). However, having distinct first and eighth syllables - Psal and tas - serves to show that these notes "are different in highness and lowness." 
Therefore, Ramos describes a kind of "mutation" when continuing an ascent from one octave series to the next: One sings the disyllable tas_Psal on the connecting note C, marking the end of one octave and the beginning of the next, for example :
| Psal- li- tur per vo- ces is- tas_Psal- li- tur per C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 ...
Likewise, in descending to a lower octave series, one makes the mutation Psal_tas :
F4 E4 D4 C4 B3 A3 G3 F3 E3 D3 C3 B2 A2 G2 ... per tur li Psal_tas is ces vo per tur li Psal_tas is ces vo
Thus "those who will have wished to sing with our syllables will be obliged to make only a single mutation in one octave," and Ramos adds that even this ascending tas Psal or descending Psal tas may be regarded as optional. "Students may not be compelled to do this at all times, for sometimes we allow then to say one [syllable] for the other." 
However, Ramos does expect his students "to observe the species of whole tones and semitones, so that they may not use one for the other, as happens with those using [solfa], as they say, with the Guidonian syllables." 
This last precept raises a problem which Ramos thus introduces but does not fully address: the treatment in the new system of musica ficta, inflected notes beyond those included in either the basic Guidonian hexachord system of musica recta, or in Ramos' basic octave series.
Ramos himself (Section 3.1) has noted how singers are continually either making explicit mutations to ficta hexachords outside the basic system, or else supplying "understood semitones" without the benefit of an explicit mutation. While emphasizing that semitones can occur in practice at other places than mi-fa, he does not give examples of how his own system would handle such inflections.
He does, however, supply an equivalent of the Guidonian hand (Section 1.5) for his system. His gamut begins on C2 - a fifth below the Guidonian gammaut at G2 - but the notes D2-F2 are not shown in his printed diagram, being visualized as located on the outer or back joints of the thumb (and thus not seen in a two-dimensional drawing?). The notes from G2 to C4 are shown in an arrangement differing from that of a typical Guidonian hand: Rather than ascending the gamut in a kind of spiral motion, the notes proceed up one finger and then down the next.
Where the ascending gamut moves down a finger toward the base of the hand, Ramos signals this orientation by printing the note names and syllables upside-down; if one rotates the page 180 degrees when reading these fingers, the notes will ascend in a visually upward direction.
The following diagram cannot show this typographical refinement, but does show the notes of Ramos' hand, with "H" again used for the square-B sign, and "b" for the round-B sign, the two alternate notes (B/Bb) for the syllable is. Jumping from C2-Psal on the palm of the hand, the ascending notes move down the thumb, up the second finger, down the middle finger, up the fourth finger, and down the little finger :
________ ________ / \ ________ / \ | | / \ ________ ____ | f per | | g vo | | f per | / g vo \ / | | | | | | | | | / | | | | | | | | | | \ | e tur | | a ces | | e tur | | a ces | | \ | | | | | | | | \ Gamma vo \ | | | | | | | | \ \ | d li | | b is H | | d li | | b is H | \ a ces \ | | | | | | | | \ \ | | | | | | | | \ b is H \|c tas psal|__| c tas |__| [c] psal |__| c tas | \ \ | | | | | [C] psal |
The note at the bottom of the middle finger concluding the octave C3-C4 (c-tas, C4) is immediately followed at the bottom of the fourth finger by the same note with a syllable mutation (c-psal, C4) marking the beginning of a new octave (C4-C5). In this scheme, each finger of the hand except the thumb has a C at its lowest joint, lending emphasis to the successive octave series.
An interesting parallel between this hand of Ramos and the Guidonian hand is that both hands have G2 - Gammaut, or in Ramos' scheme Gamma-vo - at the tip of the thumb, proceeding down the thumb for G2-A2-B2 before continuing with their divergent systems. A distinction in these notes on the thumb, however, is that the Guidonian Bmi (B2) is always a square-B (B-natural), belonging in the regular gamut only to the first hard hexachord G2-E3, where it forms the third degree mi; the note Bb2 in the untransposed gamut would be musica ficta. In Ramos' system, however, B2/Bb2 (B/Bb-is) is a fluid degree, just as in higher octaves, where the Guidonian system has corresponding fluid degrees BfaBmi.
While Ramos shows a beginner how to sing basic intervals and negotiate octaves using his system, he does not delve into such practical details as musica ficta.  Thus although this octave-based system might be readily adapted to plainsong, typically calling for only the musica recta notes of the basic system, its application to late medieval or Renaissance polyphony with the usual musica ficta inflections (written or unwritten) would be more problematic.
Ramos does give a possible clue when he urges that students "observe the species of whole tones and semitones, so that they may not use one for the other."  Might this mean that singers should always sing their semitones at the syllables tur per or ces is or is-tas? If so, singers using Ramos' system would face problems of musica ficta mutations similar to those of the hexachord system. One difference might be that while in conventional solmization a semitone always occurs at mi-fa, a singer using the Ramos syllables could have up to three sets of syllable pairs from which to choose.
Predictably, Ramos' critical remarks about Guido and his followers, as well as his theoretical novelties, sparked considerable controversy; but hexachordal solmization remained the norm throughout the 16th century. Reconstructing the possible application of Ramos' octave-based system to various kind of early Renaissance music could be a very interesting area to explore.
Johannes Lippius is famed for what he himself presents as the central concept in his Synopsis both for the theological meaning of music and for its practical craft: the trias harmonica or "harmonic triad." 
However, in the course of presenting his original approach to music, he reports on a "Belgic notational innovation, which is very compendious and apt": the use in singing of seven syllables to cover the complete range of an octave, a development enthusiastically championed by some other German theorists of the epoch.  (Here I use the English term "Belgic," after the Renaissance Latin Belgicum of Lippius, to include the Low Countries generally, later divided into Belgium and the Netherlands; "Franco-Flemish" is another possible translation.)
These syllables, bo-ce-di-ga-lo-ma-ni, returning then to bo, form an octave always of the pattern T-T-S-T-T-T-S, with the first six notes identical to those of a hexachord (T-T-S-T-T), e.g.
T T S T T T S C3 D3 E3 F3 G3 A3 B3 C4 ... bo ce di ga lo ma ni bo ...
Lippius recognizes two forms of this octave series, one in cantus durus or "hard song" with B-natural, where "bo occurs on C," and one in cantus mollis or "soft song" with Bb, where "it occurs on F" .
Thus we have:
etc. G4 lo G4 ce F4 ga F4 bo E4 di E4 ni D4 ce D4 ma C4 bo C4 lo B3 ni Bb3 ga A3 ma A3 di G3 lo G3 ce F3 ga F3 bo E3 di D3 ce C3 bo durus mollis (Bb)
Like the regular hexachord system and Ramos' system, the basic system of bocedization presented by Lippius includes the seven diatonic notes plus Bb. It shares some additional features in common with each of the other systems.
Like the hexachord system, it offers an interlocking scheme of seven-note "heptachords" with either B-natural (C-C, hard) or Bb (F-F, soft). It would be possible, emulating hexachord nomenclature, to name each note for the syllables it may carry, e.g. A3 as Amadi, D4 as Dmace, etc. In such a naming system, the notes B and Bb would each uniquely belong to a single heptachord: Bni, Bbga (or, in the fashion of hexachord names, BniBga.) 
Like the system of Ramos, bocedization is based (in its cantus durus form) on the octave C-C, and more specifically is identical to the version of this octave which Ramos recommends for practice by beginners, with B-natural rather than Bb. Lippius includes a second heptachord in the same pattern T-T-S-T-T-T-S, the octave F-F with Bb.
In terms of modal theory, the heptachords of Lippius realize the Ionian mode recognized by Glareanus (1547) in its two common positions: its natural form at C-C, and its transposed form at F-F with consistent use of Bb. Both Zarlino and Lippius regard Ionian as properly having pride of place in the system of 12 modes, and this octave-species also serves as the basis for major keys in the tonal system of the late 17th-19th centuries.
Lippius, like Ramos, argues that an octave-based system can eliminate the complexities of hexachord mutation: "[W]hen something can be done more easily in fewer steps, it will be poorly done in more."  Indeed, Lippius proposes to discard not only the hexachord system but "the usual musical alphabets," with the first letters of the bocedization syllables replacing them (b c d g l m n). The result, as in hexachord solmization, would be a notation for relative interval relations and patterns rather than absolute note positions ; compare, for example, the use of hexachord syllables in the early 14th century by Jacobus of Liege to describe two-voice progressions without a need to specify letter note names (Section 2.2).
However, also like Ramos, Lippius presents an octave-based system of syllables for the basic musica recta notes without addressing the problems of musica ficta inflections.
The variety of approaches to syllables for such inflections in later European octave-based systems as well as such systems in other world musics suggests that complications in handling accidentals may not be unique to the hexachord system, a system remarkable for its expansion and adaptation over the centuries.
The original hexachord system of solmization (ut-re-mi-fa-sol-la), in its many forms as they developed from the era of Guido (early 11th century) through the later Manneristic Era (early 17th century) remain very useful and appropriate for the music of these many centuries. Of course, solmization alone cannot resolve questions of accidental inflections, since these are motivated by vertical as well as melodic factors, with solmization a means of realizing the desired effect. However, the hexachord system may give important clues to how a performer might have approached such decisions, at least on the first reading of a piece, with further adjustments possible in case of complications.
During the 17th century, two trends became linked by history, although not by any inevitable logic: the shift from a hexachord-based to an octave-based orientation for solmization schemes; and the shift by the last quarter of the century from a system of fluid and often intermingled modes (e.g. Vicentino, the Monteverdi brothers) to a scheme of major and minor keys.
While octave-based schemes were proposed and in some cases won local acceptance for the modal music of the Renaissance and Manneristic eras (e.g. Ramos, bocedization), by the late 17th century the seven notes of a scheme such as the Italian do-re-mi-fa-sol-la-si could represent the seven notes of a key; mutation (shifting from one heptachord to another) could often be associated with a change of key, or modulation as 18th-century theorists would soon define it.
Even today, the popular do-re-mi-fa-sol-la-si-do scheme - with ti rather than si favored in some countries - is often presented mainly in terms of major and minor keys rather than modes, with major keys having their tonic on do and minor keys on la.
A more ambitious expansion of the hexachord system, amenable to either modal or tonal music, provides additional syllables for accidentals. Thus one can engage in the musica ficta inflections of early music, or the modulations of tonal music, without any need for mutations. Such systems, proposed since the early 17th century, change the nature of solmization: it now has reference not only to interval relationships, but to absolute notes, like those on a keyboard (realized at whatever pitch the singer or ensemble adopts).
Thus in 1623, a theorist named Daniel Hitzler (1576-1635) published a scheme for a 13-note chromatic scale basing the syllable names for the diatonic notes rather minimalistically on the letter names of the notes, and including distinct syllables for both Eb and D#, notes separated by a diesis of about 1/5-tone either on a contemporary keyboard tuned in meantone temperament, or in an ideal vocal tuning approximating pure fifths and thirds.
ci di me fi gi be C# D# Eb F# G# Bb ------------------------------------------------------------- C D E F G A B C ce de mi fe ge la bi ce
This gamut for bebization, as Hitzler's method is called , is equivalent to that of a keyboard instrument equipped (in a fashion not uncommon in the 16th and 17th centuries) with a split key whose front portion produces Eb and its back portion the less frequent D#. The inclusion of D#, as well as the five most usual accidentals of 14th-16th century music (Eb, Bb, F#, C#, G#), may reflect the increasing popularity of this note in the epoch around 1600.
As one critic remarks, this system has "a monotony of vowel sounds"  by comparison to either Guido's syllables or schemes such as bo-ce-di-ga-lo-ma-ni (which borrows most of Guido's vowels). In 1659, however, an author named O. Gibel (or Gibelius) proposed a similar scheme based for the most part on the traditional solmization syllables, with do substituting for ut. Gibel's 14-note gamut  resembles a meantone keyboard of a kind dating back at least to the Lucca organ of the 1480's , with split keys for G#/Ab and Eb/D#:
di ri ma fi si lo na C# D# Eb F# G# Ab Bb ------------------------------------------------------------- C D E F G A B C do re mi fa sol la ni do
Here the syllable ni for B-natural is the same as in Belgic bocedization; both this note and the sharps have the same vowel as traditional mi, being steps below a semitone; and the flats (including Bb-na of traditional musica recta) generally share the same vowel as traditional fa, being steps above a semitone. The one exception to this pattern is Ab-lo, since the usual vowel for a flat is already preempted by the uninflected A-la.
While systems ranging from the usual do-re-mi to such a chromatic keyboard-like scheme represent expansions of the hexachord system, another kind of development involves its contraction to a four-note or tetrachord system using only the syllables mi-fa-sol-la. Many examples of solmization in Morley's treatise of 1597 use only these four syllables, although he presents the full Guidonian scheme. He does restrict the use of ut to the lowest note of a song, and in a series of exercises at the conclusion of his introduction to singing "plainsongs" (simple unmeasured melodies, not necessarily chants or the like) consistently uses only mi-fa-sol-la. 
This four-note or tetrachord system became established in some traditions of religious music in England's American colonies as "fasola," and provided the basis for what is called "shape-note notation," in which each of the four syllables corresponds to a distinctive note head shape: triangular for fa; round for so (a shortened name for sol, also common in the seven-note do-re-mi system); square for la; and diamond-shaped for mi.
In fasola, an octave-species or mode is characteristically built from tetrachords with two points of mutation: a la-fa semitone and a la-mi whole-tone (immediately followed in the new tetrachord by mi-fa semitone):
D Dorian E Phrygian T S T T T S T S T T T S T T d3 e3 f3 g3 a3 b3 c3 d4 e3 f3 g3 a3 b3 c4 d4 e4 sol la fa sol la mi fa sol la fa sol la mi fa sol la F Lydian G Mixolydian T T T S T T S T T S T T S T f3 g3 a3 b3 c4 d4 e4 f4 g3 a3 b3 c4 d4 e4 f4 g4 fa sol la mi fa sol la fa sol la mi fa sol la fa sol A Aeolian C Ionian T S T T S T T T T S T T T S a3 b3 c4 d4 e4 f4 g4 a4 c4 d4 e4 f4 g4 a4 b4 c4 la mi fa sol la fa sol la fa sol la fa sol la mi fa
In fasola theory, strongly influenced by conventional 18th-century theory, the mi step is identified as the step below the tonic fa of a major key: thus with no signature (C major, the same basic octave pattern as C Ionian), mi is found at B. With a Bb signature (F major), it is found at E, and so forth. 
However, in shape-note hymns and other songs in this tradition as notated in various volumes from the second third of the 19th century on, and as still performed by those following the tradition, the written key signature does not always reflect the actual whole-tones and semitones sung. Specifically, songs notated with minor key signatures are actually performed in what might be termed the Dorian mode, with the notated flat for the sixth degree of the mode being disregarded. Thus a song centered on G with a signature of Bb and Eb would in practice be sung with E-natural substituted for Eb. Also, any indicated sharps for the seventh degree of the scale are generally disregarded in performance. 
In these pieces, not only the key signature but the note shapes may suggest a semitone where a whole-tone is sung in practice. For example, in a piece on G with a signature of Eb and Bb, the note shapes may indicate a descending semitone E(b)fa-Dla, although this step is actually sung E-D - following the general patterns of the system, a notation for this step would be Emi-Dla. 
One musicologist, Charles Seeger, refers to this practice as musica ficta - evidently in the modern colloquial sense of "unwritten accidentals or inflections." 
While an obvious advantage of an octave-based system over a hexachord (or tetrachord) system is the avoidance of many mutations, at least one modern musicologist has suggested that such mutations can at times very usefully reflect the structure of the music.
Thus Pal Jardanyi, writing on "The Determining of Scales and Solmization in Hungarian Musical Folklore" , argues that using a single solmization heptachord for Hungarian melodies apparently falling within a single diatonic mode (e.g. Dorian, Phrygian) may fail to reflect the actual musical structure of two pentatonic scales or "strata" a fifth apart.
For example, Jardanyi cites a melody which could conventionally be analyzed as in "G Dorian" (final of G, with Bb signature), but which might better be described as combining two such pentatonic strata: upper C5-D5-F5-G5-A5, lower G4-Bb4-C5-D5-F5, with G4 in the lower stratum corresponding to D5 in the upper. Using a do-re-mi system, Jardanyi suggests a mutation: D5 should be sung as la in the first portion of the sung based on the upper pentatonic stratum, but G4 as la in the second portion based on the lower stratum.
Interestingly, Jardanyi cites another Hungarian song fitting a pattern of two "pentachord" or "tetrachord" scales where Guidonian hexachord solmization (although not mentioned by this author) might reflect such an analysis more clearly than an octave-based system. Here is the melody (G final, Bb signature), based on the printed version in 2/4 meter, with hexachord syllables (first part in natural hexachord C5-A5, second part in soft hexachord F4-D5); plus syllables for a heptachord system (soft heptachord F4-E4/Eb4). As in the printed version, double bars appear at a number of places; the music bears the indication "Tempo giusto" :
1 & 2 & | 1 & 2 || 1 & 2 & | 1 & 2 || G5 G5 D5 E5 F5 E5 D5 G5 G5 D5 E5 F5 E5 D5 hexachord: sol sol re mi fa mi re sol sol re mi fa mi re heptachord: re re la ti ut ti la re re la ti ut ti la 1 & 2 | 1 & 2 || 1 & 2 & | 1 & 2 || Bb4 C5 D5 Bb4 C5 D5 D5 C5 Bb4 A4 G4 G4 G4 hexachord: fa sol la fa sol la la sol fa mi re re re heptachord: fa sol la fa sol la la sol fa mi re re re
Although we must be cautious in applying medieval Western European theory to Hungarian music, Jardanyi's remark that this melody has "two layers" may reflect an aspect of its construction also conveyed by the hexachord solmization. The twice repeated phrase in the first section concludes on D5-re, while the final phrase concludes on G4-re, thus suggesting an affinity between these tones as points of repose.
In medieval Western European terms, these two tones are respectively the confinal and final of the Dorian mode transposed to G, D5 being at the base of the fourth or tetrachord D5-G5, and G4 at the base of the fifth or pentachord G4-D5. Singing both the confinal D5 in the first section and the final G4 in the second section with the syllable "re" may clarify this symmetry, and underscore the role of D5 as the confinal or "co-center" of the melody.
In contrast, an octave-based solmization with G always sung as re, and D as la, may (not surprisingly) emphasize the concept of a mode as an octave species unified around its final. From this viewpoint, we are in "the mode of re," a center of organization and repose for the melody as a whole. Singing the opening note in the heptachord version as G5-re focuses on its affinity to G4-re as "the octave of the final"; singing this same note as G5-sol in the hexachord version may focus on its local role as "the fourth above the point of repose for this phrase, D5-re."
Turning to the early music of Western Europe, we may thus find that while either a hexachord or an octave-based system can lend itself to modal music, these two approaches may tend to emphasize different aspects of the modal system. An octave-based system suggests the concept of "mode as unified octave species," while a hexachord system may place more emphasis on the concept of a mode as a coalition of tetrachords, pentachords, or "melodic orbits." 
Thus Allaire  urges us to understand the modes "as interlocking hexachords, rather than as scales in the modern sense." Whatever Allaire's qualifying phrase "in the modern sense" may mean, medieval perspectives focus on the modes both as octave species and as unions of tetrachords or hexachords. However, as Allaire suggests, the hexachord system may have the unique virtue of highlighting subtle relationships of the latter type, while the octave-species aspects of modality may be obvious enough under any system of solmization.
As an introduction to the illustrations which follow, I might suggest that if one does want to experiment with octave-based solmization alternatives for medieval-Manneristic music, the best approach might be a gamut of heptachords with a fluid seventh degree, e.g. Bb-sa/B-ti for the natural hexachord C-A (or Bb-sa/B-si, if one prefers a widespread European tradition of B-si going back to around 1600). 
Thus we would have a basic musica recta gamut with a natural heptachord of C-Bb/B; a soft heptachord of F-Eb/E; and a hard heptachord of G-F/F#. The notes of the natural heptachord correspond to those of the standard untransposed gamut or Guidonian hand (diatonic notes plus Bb); and the notes of the soft heptachord to those of the gamut system transposed by a Bb signature (C-A, Bb, and Eb as a "naturalized" recta tone - but with B-natural relegated to musica ficta).
Adding a fluid seventh degree to the hard hexachord (thus G-F/F#) interestingly "naturalizes" F# as an element of the regular gamut. While this note does not typically occur in plainsong, it does occur in certain polyphonic conducti around 1200, for example, where fluid alternations or juxtapositions of F/F# are common, and likewise in some 13th-century motets. 
Such a system, like the classic hexachord system, is readily expanded to include an unlimited number of ficta heptachords, just as vital in the new system as in the old when accidentals are desired beyond the range of musica recta (now Eb-F#).
For our purpose here, however, we are concerned only with the basic structure of a mode as reflected by the classic hexachord system and our octave-based heptachord alternative. Each system reveals a certain aspect or dimension of "mode" recognized by medieval theorists, but from different perspectives.
Let us consider, for example, the typical solmization using either system of the Dorian mode (D3-D4):
D3 E3 F3 G3 A3 B3 C4 D4 hexachord: re mi fa sol re mi fa sol heptachord: re mi fa sol la ti ut re
Our hexachord solmization reveals a striking symmetry within this mode on a tetrachordal level: we have two identical sequences re-mi-fa-sol, starting in the natural hexachord (C3-A3) and then mutating to the hard hexachord (G3-E4).
The heptachord solmization, in contrast, places emphasize on two other dimensions of symmetry, both noted by Benito Rivera in his study of Johannes Lippius . The most obvious is the kinship of the mode's final D3 and the octave D4 above it, both sharing the syllable re. Guido himself noted this octave affinity, comparing it to the cycle of seven days in a week, with names of notes repeating as the names of days.  A heptachord system of solmization makes this cyclicality an obvious feature in a way that a hexachord system may not, at least in itself.
While the hexachord system focuses on the dimension of tetrachord symmetry in this mode, a heptachord system reveals a different aspect of symmetry relating to a theme in medieval and Renaissance theory: the modal octave as the union of a fifth and a fourth - here starting from the final D3, with the fifth D3-A3 below the fourth A3-D4.
In our hexachord version, we have D3-A3 as re-re, and A3-D4 as re-sol. With our heptachord system, however, we have D3-A3-D4 as re-la-re - the fifth re-la symmetrically "mirrored" by the fourth la-re. Thus A3-la forms a modal axis of symmetry, dividing the octave into lower fifth and upper fourth.
This fifth/fourth relationship may become clearer if we consider the two forms of the Dorian mode. In the authentic version we have just considered (Mode 1), the modal octave starts with the final as its lowest note, and is divided into a fifth above and a fourth below. In the plagal version (Mode 2), the octave starts a fourth below the final, and adds to this fourth a fifth above. Here the heptachord system underscores the fifth/fourth or fourth/fifth symmetries within each version of the Dorian mode and between the two versions:
Mode 1 (authentic Dorian) Mode 2 (plagal Dorian) 5 4 4 5 |---------------|-----------| |-----------|---------------| D3 E3 F3 G3 A3 B3 C4 D5 A2 B2 C3 D3 E3 F3 G3 A3 Hex: re mi fa sol re mi fa sol re mi fa re mi fa sol la Hept: RE mi fa sol La ti ut RE La ti ut RE mi fa sol La
In our hexachord solmization, the D3-A3-D4 structure of authentic Dorian is sung as re-re-sol; the A2-D3-A3 structure of plagal Dorian (or Hypodorian) is sung as re-re-la.
In our heptachord version, we have D3-A3-D4 as re-la-re and A2-D3-A3 as la-re-la - in both modes, a fifth re-la plus a fourth la-re. Only the order of these two principal intervals of the mode changes as we move from the authentic to the plagal version.
Thus an octave-based system can nicely reflect the medieval concept of a mode as octave species consisting of fifth-plus-fourth (authentic) or fourth-plus-fifth (plagal), a concept emphasized also by a Renaissance (or early Mannerist) theorist such as Vicentino. 
The hexachord system, however, can help to bring out other important symmetries and relationships, for example on a tetrachordal level, which may be easier to overlook. It should be seen (and heard) not merely as an incomplete octave system, but as a musically valuable perspective in its own right as well as a central element of medieval-Manneristic musical culture and tradition.
To Table of Contents.Margo Schulter