dolmetsch onlinee-monographs
 



topics on this page :: an article about split key organs


Important: To see and hear our 'live' music examples you will need to install the free Scorch plug-in for PC and MAC systems.



Organs with Split Keys or Added Keys fewer than 17 tones per octave (but more than 12)
Ibo Ortgies, GOArt (Göteborg Organ Art Center)


written: February/March 2001
Latest version: 11th March 2001


List of content

  • Introduction
  • Split Key Design
  • Added Keys or Split Keys in Pythagorean Tuning
  • Meantone Extension
  • Short Octave and Split keys
  • Terminology
  • A Very Short History
  • Use
  • Selected Literature
  • Acknowledgements

    Appendix

  • Pythagorean Comma./.Syntonic Comma = The Schisma

    Note:
    A non-proportional font like "Courier" is necessary to view the diagrams and examples undistorted!


    Introduction
    From the 15th to the 18th century keys were added to keyboard
    instruments as a rather convenient way to exceed the limitations of
    restricted temperaments and tunings in keyboard instruments, since the
    essential features of these temperaments were kept, like for example the
    pure major thirds of meantone temperament.
    Split keys were inserted in those places where there are usually upper
    keys between diatonic notes can be found.
    Usually the split keys between c and c'' were "broken" to provide
    additional keys. 
    Occasionally notes outside this range also were broken, in the treble
    only eb''/d#'' (more frequent) and g#''/ab'' (less frequent) and in the
    bass G#/Ab and Bb/A# occur, but very seldom (s. chapter "A Very Short History").
    Split keys for any f#/gb, in whatsoever octave, are until now not known.
    Split Key Design
    A keyboard may look like this from above:
    ordinary keyboard keyboard where eb 
    without split key is "split"
    back back
    | | | | |
    | | | | |
    | | | | |
    | | | | |
    | | | | |
    | | | | |
    | | | | d# |
    | | | | | | |_____| | ___
    | | | | | | | | |
    | | | | | | | | |
    | | eb | | | | eb | | |
    | |_______| | | |_______| | _|_
    | | | | | |
    | | | | | |
    | | | | | |
    | d | e | | d | e |
    |_______|_______| |_______|_______|
    front front
    In a few preserved keyboards in organ positives by the Manderscheidt
    family (s. in chapter "A Very Short History") the front part of the
    *lower upper key (eb) might be ca. 2 cm long (+/-2-3 mm)
    - the distance being indicated by the vertical line to the right in the drawing.
    Viewed from the front a section will look like
    _____
    / \
    | d# |
    _______ _|_____|
    / \ / : \
    | eb | | : eb |
    ___|_______|___ ____|_______|____
    | :_ _|_ _: | | :_:_|_ _: |
    | | | | | |
    | d | e | | d | e |
    |_______|_______| |________|________|
    And in 3-D:
    a keyboard seen from the right
    / /
    / /
    / /
    / / /
    / d# / / / 
    /____/ / / 
    / | | / / / /
    / / /| | / / / /
    / / / / |____| / / / /
    / / / / /| / / / /
    / / c# / / eb / |/ / / f# /
    / /_____ / /_____ / / / /_____ /
    | | | | / / | | /
    | | / | | / / | | /
    |_____|/ |_____|/ / |_____|/
    / / / /
    / / / /
    c / d / e / f / g
    ___/__________/__________/__________/__________
    | | | |
    | | | |
    | | | |
    ___|__________|__________|__________|____
    Designs for g#/ab etc. would be similar. 
    Added Keys or Split Keys in Pythagorean Tuning
    The pythagorean chain of perfect pure fifths 
    Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E - B
    provides us with the four notes Gb- Db- Ab- Eb, which work as major
    thirds (F#- C#- G#- D#) to some of the main keys of the modal system.
    These major thirds are so close to pure, that they result practically in
    a cut-out of Just Intonation where the diatonic notes (or better, the
    notes from the old hexachord-system - therefore including both B and Bb)
    get major thirds which are only the small amount of 1,9 cents, the so
    called schisma (s. below), lower than pure. 
    The scheme
    ex. 1 Gb- Db- Ab- Eb- ... 
    / \ / \ / \/
    ...Bb- F - C - G - D - A - E - B
    is therefore practically the same as:
    ex. 2 F#- C#- G#- D#- ... 
    / \ / \ / \/
    ...Bb- F - C - G - D - A - E - B
    as which it might have originated once - adding pure major thirds to the
    most used diatonic notes.
    The chain of pure perfect fifths from 
    Cb- Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E
    results like the previous ex. 1 in practice in:
    ex. 3 B - F#- C#- G#-(Eb- Bb- F - C)
    / \ / \ / \ /
    ... Eb- Bb- F - C - G - D - A - E
    Note: the brackets indicating that the repeated notes are just
    continuing the chain of fifths.
    All slashes and lines indicate here and the following
    diagrams *pure intervals!)
    A similar chain of pure perfect fifths 
    Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C - G - D - A 
    will be practically
    ex. 4 E - B - F#- C#-(Ab- Eb- Bb- F) ...
    / \ / \ / \ /
    ... Ab- Eb- Bb- F - C - G - D - A
    The upper rows, for example B - F#- C#- G# in ex. 3. turn out to be
    major thirds, which are only a very small amount, the so-called schisma,
    lower than pure (s. below for an explanation).
    E - B in ex. 1 and A - E in ex. 2. can not be pure fifths:
    The fifth E - B in ex. 3, is in fact not a perfect (impure) fifth, but
    a pythagorean diminuished sixth E - Cb, and is a syntonic comma below
    the pure fifth, because this E is higher than the pure major third to C. 
    In the following ex. 5 the pure E (pure major third to C) indicated by a '
    E in this example is generated by four pure perfect fifths - it is
    audible higher than pure.
    ex. 5 E'
    /
    C - G - D - A - E
    The difference between those two E-s is called the syntonic comma (s. Appendix)
    Another way to show what happens is to combine the chain of fifths and
    the pure major third
    ex. 6 E'
    /
    Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C - G - D - A - E
    Fb is only slightly below the E' which serves as pure major third to C.
    For practical reasons we can say that Fb = E'
    I call those intervalls, actually pythagorean diminuished fourths
    (C-Fb in our previous example), which are practically pure major thirds
    (very close to C-E') "schismatic thirds"
    Especially in an organ the "pulling-effect" ("drawing together") might
    render these "schismatic thirds" pure to the listener. And it is not
    unlikely, that pythagorean tuning might have been derived not from the
    chain of fifths, but from the diatonic notes, which got pure major
    thirds added ...
    Because of this difference another E , pure to C, might have been
    inserted on a keyboard of a pythagorean tuned organ.
    There are hints provided by Christopher Stembridge and L. F. Tagliavini,
    that extra keys were actually already used in organs tuned in
    pythagorean tuning in the middle of the 15th century, but at the same
    time meantone temperament rose and lead to a greater interest in adding
    the "tasti spezzati" or "subsemitonia".
    The pythagorean example given by Stembridge and Tagliavini is from an
    intarsia showing an extra key between E and F. 
    In ex. 3 there is only the E, therefore an E' might have been inserted
    as extra key.
    In ex. 4 there is the E' already there (in the upper row), and E might
    have been added to the lower row to make the pure chord e-g#-b available.
    The split key between E and F would add either the E above C (to ex. 3)
    ex. 7 E - B - F#- C#- G#
    / \ / \ / \ / \ /
    Eb- Bb- F - C - G - D - A - E
    or the pure fifth E to A (ex. 4)
    ex. 8 E - B - F#- C#
    / \ / \ / \ / \
    Ab- Eb- Bb- F - C - G - D - A - E
    It might be that there have been other keys added in other instruments,
    depending on the resp. chain of fifths and the total number of keys. But
    this remains speculative until we'll know of documents...
    Meantone Extension
    In an italian organ around 1500 (4 examples known before 1500), tuned in
    a meantone temperament, the first key to be split might be the a-flat,
    provided with the g# (as back key) - as can be seen in the
    (reconstructed) keyboards of the 2 famous organs in Bologna, Italy, S.
    Petronio, the older one from 1475, getting split keys during the rebuilt
    by G. B. Facchetti in 1528-1531 and the other organ from 1596 built by
    B. Malamini.
    Others started with doubling eb/d#.
    Later it became usual to split consequently: eb/d#, g#/ab (g# being in
    front), bb/a# then c#/db (seldom). There are only two organs known which
    had all these 4 split sharps and in both cases all concentrated in one
    octave span:
    Wolfenbüttel, Germany, Hauptkirche BMV, 1620-1624 G. Fritzsche, and
    Sciacca, Italy, S. Margherita, 1639 G. Sutera and V. Monteleone
    Not one known organ with less than 19 notes per octave (the "enharmonic"
    instruments) had *all five split keys (or more) as subsemitones.
    The pattern with eb/d# and g#/ab (14 notes/pitches per octave) seems to
    have been frequent. It provides the following possibilities,
    symmetrically ordered around the fifth of the 1st mode D-A.
    ex. 5 ~~G#~~D#
    \ / \ / \
    ~~E~~~B~~~F#~~C#~~G#~~D#
    \ / \ / \ / \ / \ / \ / \
    ~~C~~~G~~~D~~~A~~~E~~~B~~
    \ / \ / \ / \ / \ / \ / \
    Ab~~Eb~~Bb~~F~~~C~~~G~~
    \ / \ / \
    Ab~~Eb~~
    dashes and slashes indicate here
    ~~ meantone fifths
    / pure major thirds
    \ minor thirds (as defined by the two previous)
    "wavy" lines (~~) and slashes with resp. "open end" indicate
    continuation in the next row below or above. Which is nothing else than
    Ab~~Eb~~Bb~~F~~~C~~~G~~~D~~~A~~~E~~~B~~~F#~~C#~~G#~~D#
    a chain of meantone-tuned perfect fifths, not indicating the third-relations.
    Adding split keys widens the possible range from the "usual" meantone
    major thirds (8), minor thirds (9) and triads (8 minor and 8 major):
    1 2 3 4 5 6 7 8 (9)
    C~~~G~~~D~~~A~~~E~~~B~~~F#~~C#~~G#
    \ / \ / \ / \ / \ / \ / \ / \ / \ 
    Eb~~Bb~~F~~~C~~~G~~~D~~~A~~~E~~~B
    1 2 3 4 5 6 7 8 
    (with added d# and ab) to: 
    1 2 3 4 5 6 7 8 9 10 (11)
    F~~~C~~~G~~~D~~~A~~~E~~~B~~~F#~~C#~~G#~~D#
    \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ 
    Ab~~Eb~~Bb~~F~~~C~~~G~~~D~~~A~~~E~~~B~~~F#
    1 2 3 4 5 6 7 8 9 10 
    plus db and a#:
    1 2 3 4 5 6 7 8 9 10 11 12 (13)
    Bb- F - C - G - D - A - E - B - F#- C#- G#- D#- A#
    \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \
    Db- Ab- Eb- Bb- F - C - G - D - A - E - B - F#- C#
    1 2 3 4 5 6 7 8 9 10 11 12 
    shown in the following "lattice"...
    ex. 6 ~~G#~~D#~~A#
    \ / \ / \ / \
    ~~E~~~B~~~F#~~C#~~G#~~D#~~
    \ / \ / \ / \ / \ / \ / \
    ~~C~~~G~~~D~~~A~~~E~~~B~~
    \ / \ / \ / \ / \ / \ / \
    ~~Ab~~Eb~~Bb~~F~~~C~~~G~~
    \ / \ / \ / \
    Db~~Ab~~Eb~~
    Short Octave and Split Keys
    d' was also the middle, symmetrical key on the keyboards with short
    octave CDEFGA-c'''.
    The keyboard with a short octave in the bass would look like
    D E Bb
    C F G A B c
    The short octave might have originated from the previous
    FGAB-g'',a''-compass by adding the C to the left of the keyboard and the
    D and E at the indicated places. F-keyboards could have been
    actually based on D (same pattern, but without the C).
    How important the symmetrical aspect might have been regarded, is
    difficult to say. Amazing is in this respect that keyboards until
    ca. 1700 were rather equipped with extra notes in the bass for F# and
    G#, built as split keys (but which are not "subsemitones") as well:
    F# G#
    D E Bb
    C F G A B c
    There are a few keyboard designs, deviating from the above sketched
    patterns, apparently "omitting" certain keys, like doubling
    eb/d# and bb/a#, leaving out the frequent g#/ab), or providing
    g#/ab and c#/db , but not eb/d#
    whichs possible purpose is explained below.
    Terminology
    The "lattice" diagrams above are used as a "informal" way to show the
    symmetry of meantone and the central key d, resp. the fifth D - A is
    also the fundament of the most "popular" pythagorean tuning "Gb"-H ... 
    Interesting that the organ builders kept until the 20th century the
    nomenclature of the pythagorean tuning, saying "d#" even when a "real"
    eb is tuned. This also was the standard in the so-called "new German
    organ-tablature" in the 17th century. This was a notation in which notes
    were expressed by letters and extra signs for rhythms (s. J. G. Walthers
    "Musicalisches Lexicon", table XXI) also stayed with this standard.
    A c-minor-triad would be notated in a keyboard tablature or tablature
    score of the 17th century "c ds g", an Eb-major-triad "d# g b" 
    ("b" indicating here b-flat in the German way - remember B-A-C-H...). 
    The added split keys keys were marked in the Fritzsche- and
    Manderscheidt-tradition with an "#" on the pipes (and in contracts):
    name sounding
    note
    ds eb
    ds# d#
    --------------------
    gs g#
    gs# ab
    This can be found for example in the specification-draft from 1612 by
    Hans Leo Hassler for the future organ in the castle chapel of the
    electoral court in Dresden, completed by Gottfried Fritzsche probably
    end of 1612. A facsimile of this important document is easily
    accessible in Gress 1993 (p. 76-77, transcription p. 102-103).
    A report from Breslau/Wroclaw, that the organ in Stiftskirche St.
    Vincenz, built in the 1660ies by J.C.B. Waldhauser mentions "dis" (eb)
    and "dass" (d#).
    Michael Praetorius ("De organographia", 1619) wrote however (only here?)
    "es" and "ds", "gs" and "as".
    A Very Short History
    It seems to have been Italian organ builders of the mid-15th-C. who were
    the first to apply split keys in organ building. Not surprising Italy
    was the main center of this development for the first 150 years and
    nearly half of the instruments we know of today are to be found in this
    country. Until 1600 we leran about 19 organs. From Italy the idea might
    have spread to Spain (the picture within Spain still remains somewhat unclear).
    In Italy the development seems to faded out after 1660, but shortly
    after 1600 however, Germany, which was dominated by musicians under
    Italian influence, took a central position. Split keys were promoted by
    the leader of the Wolfenbüttel court chapel, Michael Praetorius, and the
    Saxon court organ builder Gottfried Fritzsche. Other promoters were f.
    example Henrich Schütz (Dresden) and Jacob Praetorius (Hamburg), both
    knowing and cooperating with each other and with Fritzsche. Dresden was
    the first organ norther of the alps in which split keys were applied
    (Schlick 1511 reports a failed attempt at an unknown place by two
    unknown builders). The chapel organ was finished by G. Fritzsche in 1612
    after having planned it together with Schütz' predecessor as chapel
    master of the electoral court, Hans Leo Hassler. It is hoped by the way
    that in the near future this organ will be reconstructed in the as well
    reconstructed Dresden castle.
    The Manderscheidt-family originally working in and around Nürnberg
    (Nuremberg) has to be mentioned too, because they produced several
    positives, but also bigger church organs like the recently restored
    choir organ in Fribourg, Switzerland, St. Nicolas, built 1654-1657 by
    Sebald Manderscheidt.
    >From these geographical area the idea spread further to surrounding
    regions and countries:
    - Denmark: only one example known, probably an organ builder
    of Saxon origin (the courts of Denmark and saxony
    were dynastically linked)
    - Sweden: the only country where also local organ builders
    took the practice over.
    - the Netherlands: the northwest-German organ builder family
    van Hagerbeer (1630ies/1640ies)
    - Switzerland: Manderscheidt-family (s. above) ca. 1650
    - England: 2 organs known, by Bernard Smith (Bernhard Schmidt)
    in the 1680ies
    In France this tradition did not prevail, as far as we know today. There
    is no safe evidence for any organ with split keys and the Gamba-player
    Jean-Jaques Rousseau confirmed in 1697 that this tradition was not to be
    found in France, contrary to Italy. Even the only example in Paris, St.
    Nicolas des Champs, 1632-1636, C. Carlier, is not known whether it was
    at all realized or only a contract or draft, not carried out later.
    At the end of the 17th century there are still large organs built with 3
    manuals and pedal in Sweden and Germany, which contained 3 split
    keys/octave, namely eb/d#, g#/ab and bb/a#, sometimes even going down
    below c: Bb/A# occurs in a few cases. Split keys below c were otherwise
    only to be found in some Italian early compasses, but those where
    keyboards extended down to FF - 12-foot-organs thus.
    With the rise of circulating temperaments the practice disappeared in
    organ building soon after 1700. The last organ was built by J. and V. F.
    Bossart in 1716-1721 in Luzern (Switzerland), St. Urban.
    The history of piano's with less than 19 keys/octave is not well known.
    Broadwood is reported to have built a piano in 1766 which had all upper
    keys split.
    About 70 organs with 13-16 keys per octave are known today to have been
    built during the time sketched above. This is however very likely only
    the "tip of the iceberg" and a closer look to literare and archives
    might reveal many more examples.
    The historical informed performance practice movement lead organ
    builders to build organs in meantone tuning again and so the split keys
    came back into modern organ building.
    With the return of historical oriented organ building and the rise of
    historical temperaments and tunings in the practice of instrument
    building and performance practice split keys have become a more and more
    frequent feature again since the 1970ies, after an interlude of nearly
    300 years. Italy, the USA, Sweden and Switzerland might be today the
    countries with the highest concentration of existing organs with split
    keys, while the development in other traditional organ countries like
    the Netherlands or Germany has not yet led to a more frequent re-use again.
    Use
    Split keys could be used in several ways and circumstances: Continuo
    practice, intonation aid, "fancy pieces", etc. 
    Transposition could well be the most important reason for their
    existence. Transposition to a variety of intervals was frequent and
    necessary: The organist had to provide the proper tones in the liturgy,
    to play transcriptions (intabulations) in different applied pitches and
    to accompany ensembles.
    Special designs were developed, which made only sense in transposition,
    for example by apparently "omitting" certain keys, to enable comfortable
    transposition by certain intervals. This might for example reflect the
    relative low or high organ pitch in the respective region. 
    Another organ (Sønderborg Slotskapel, 1626, rebuilt by B. (Zencker [?])
    and restored recently by Mads Kjersgaard) even has two manuals, one of
    them with split keys, that shows some similarity to the concept of the
    so-called ”transposing" instruments by the German/Flemish harpsichord
    builders Ruckers.
    Designs in other organs lead to the conclusion that their split keys
    were used mainly in continuo-playing (Wolfenbüttel, and
    maybe, Sciacca).
    Selected literature:
    Burgemeister, Ludwig. Der Orgelbau in Schlesien. Straßburg, 1925.
    Reprint, edited by, Hermann J. Busch, D. Großmann and R. Walter,
    Frankfurt, 1973.
    Gress, Frank-Harald. "Die Gottfried-Fritzsche-Orgel der Dresdner
    Schloßkapelle: Untersuchungen zur Rekonstruktion ihres Klangbildes." 
    Acta organologica [Germany] 23 (1992): 67-112.
    Lindley, Mark. "Stimmung und Temperatur." 
    In Hören, Messen und Rechnen in der frühen Neuzeit = Geschichte der
    Musiktheorie, Vol.6, edited by Frieder Zaminer, 109-331. Darmstadt:
    Wissenschaftliche Buchgesellschaft, 1987.
    Ortgies, Ibo. "Subsemitoetsen bij historische orgels tussen 1468 en 1721
    [Split keys on historical organs between 1468 and 1721]."
    Het Orgel [Netherlands] 96, no. 6 (2000): 20-26.
    This article appeared slightly earlier in German and Swedish (the Dutch
    version is however the most recent):
    ———. "Subsemitonien in historischen Orgeln. Ein Überblick über die Entwicklung
    zwischen 1468 und 1721." 
    Concerto [Germany] 17, no. 156 (2000): 22-25.
    ———. "Subsemitoner i historiska orglar. En överblick över utvecklingen mellan
    1468 och 1721."
    Tidig Musik [Sweden], no. 2 (2000): 26-31.
    Praetorius, Michael. Syntagmatis Musici Tomus Secundus. De
    Organographia. 
    Wolfenbüttel, 1619. 
    Reprint, ed. by Wilibald Gurlitt, in: Documenta musicologica. 1. Reihe:
    Facsimiles XIV. Kassel, Bärenreiter: 1958, 1980.
    Ratte, Franz Josef. Die Temperatur der Clavierinstrumente.
    Quellenstudien zu den theoretischen Grundlagen und praktischen
    Anwendungen von der Antike bis ins 17. Jahrhundert. 
    Edited by Winfried Schlepphorst. Vol. 16, Veröffentlichungen der
    Orgelwissenschaftlichen Forschungsstelle im Musikwissenschaftlichen
    Seminar der Westfälischen Wilhelms-Universität. Kassel: Bärenreiter, 1991.
    Seydoux, François. "Die abenteuerliche Odyssee eines bedeutsamen
    Instruments oder Das Freiburger Pedalpositiv von Sebald Manderscheidt
    aus dem Jahre 1667." 
    In Musicus Perfectus. Studio in onore di Luigi Ferdinando Tagliavini
    "prattico & specolativo" nella ricorrenza del LXV° compleanno, edited by
    Pio Pellizzari, 49-106. Bologna: Pàtron editore, 1995.
    Seydoux, François. "L'Orgue de Chœer de la Cathédrale de St-Nicolas,
    Fribourg." 
    In Cathédrale St-Nicolas Fribourg. Inauguration de 'lorgue de chœur
    restauré (Sebald Manderscheidt, 1657), edited by Seydoux, François,
    Wolfgang Rehn, Patrice Favre, and Luigi Ferdinando Tagliavini. Fribourg
    [Switzerland], 1998: 14-34.
    Seydoux, François. "L'Orgue de Chœer de la Cathédrale de St-Nicolas,
    Fribourg." 
    La Tribune de l'Orgue 51, no. 1 (1999): 4-12.
    Stembridge, Christopher. "The Cimbalo Cromatico and other Italian
    Keyboard Instruments with Nineteen or More Divisions to the Octave." 
    Performance Practice Review 6 (1993): 33-59.
    Stembridge, Christopher. "Italian organ music to Frescobaldi." 
    In The Cambridge Companion to the Organ, edited by Nicholas J. 
    Thistlethwaite and Geoffrey Webber, 148-163. Cambridge, 1998.
    Stembridge, Christopher, and Denzil Wraight. "Italian Split-keyed
    Instruments with Fewer than Nineteen divisions to the Octave."
    Performance Practice Review 7 (1994): 150-181.
    Tagliavini, Luigi Ferdinando. Considerazioni sugli ambiti delle tastiere
    degi organi italiani. [Thoughts on the keyboard ranges of Italian
    organs]. 
    Edited by Friedemann Hellwig, Studia organologica: Festschrift für John
    Henry van der Meer zu seinem fünfundsechzigsten Geburtstag. Tutzing:
    Schneider, 1987.
    Tagliavini, Luigi Ferdinando. "Notes on Tuning Methods in
    Fifteenth-Century Italy."
    In Charles Brenton Fisk. Organ Builder. Volume One. (Essays in his
    Honor), edited by Fenner Douglass and others, 191-199. Easthampton
    (Mass.), 1986.
    Walther, Johann Gottfried. Musicalisches Lexikon oder Musicalische Bibliothec.
    Leipzig, 1732. Reprint, Kassel, Bärenreiter: 1953.
    Acknowledgements:
    This FAQ was created with the support of members of the Tuning-list and
    others, especially:
    I'd like to thank for valuable hints help:
    Graham Breed, Dale C. Carr, Paul Erlich, Massimilano Guido, John A. de
    Laubenfels, Joseph Pehrson, Margo Schulter.
    Further acknowledgements are listed in my above mentioned article.
    Appendix
    Pythagorean Comma./.Syntonic Comma = The Schisma 
    The schisma is the difference between the 
    pythagorean comma (12 pure perfect fifths - 7 octaves) and the 
    syntonic comma (4 pure perfect fifths - 1 pure major third - 2 octaves)
    The schisma is then 
    (12 pure perfect fifths - 7 octaves) 
    - ( 4 pure perfect fifths - 2 octaves - 1 pure major third)
    Now forget the octaves, because they are just here to keep us in an
    "audible range" and we can use the fourths as complementary intervals of course.
    Resolving the brackets and without the octaves it is
    12 pure perfect fifths 
    - 4 pure perfect fifths 
    + 1 pure major third
    --------------------------------------------------------
    = 8 pure perfect fifths + 1 pure major third
    8 pure perfect fifths + 1 pure major third
    is represented by the chain
    E'
    /
    Fb- Cb- Gb- Db- Ab- Eb- Bb- F - C
    The difference between the Fb and the E' is the schisma (the E' being
    this small amount of 1,955 cents higher than Fb).
    


  •