Tag: tuning

Medieval Solfeggio within the Heptagonal Church of Rieux Minervois

This paper responds to Reichart and Ramalingam’s study of three heptagonal churches[1], particularly the 12th century church at Rieux Minervois in the Languedoc region of France (figure 1a).

image: The Church in situ

Reichart and Ramalingam discuss the close medieval association of the prime number seven[2] with the Virgin Mary, to whom this church was dedicated. The outer wall of the original building still has fourteen vertical ribs on the inside, each marking vertices of a tetraheptagon, and an inner ring of three round and four vertex-like pillars (figure 1b) forming a heptagon that supports an internal domed ceiling within an outer heptagonal tower. The outer walls, dividable by seven, could have represented an octave and in the 12th century world of hexachordal solmization (ut-re-mi-fa-sol-la [sans si & do])[3]. The singing of plainchant in churches provided a melodic context undominated by but still tied to the octave’s note classes. Needing only do-re-mi-fa-sol-la, for the three hexachordal dos of G, C and F, the note letters of the octave were prefixed in the solmization to form unique mnemonic words such as “Elami”.It is therefore possible that a heptagonal church with vertices for the octave of note letters would have been of practical use to singers or their teachers.

The official plan of Rieux Minervois

12th Century Musical Theory

In the 10th Century, the Muslim Al-Kindi was first to add two tones to the Greek diatonic tetrachord of two tones and single semitone (T-T-S) and extend four notes to the six notes of our ascending major scale, to make TTSTT. This system appeared in the Christian world (c. 1033) in the work of Guido of Arezzo, a Benedictine monk who presumably had access to Arabic translations of al-Kindi and others [Farmer. 1930]. Guido’s aim was to make Christian plainsong learnable in a much shorter period, employing a dual note and solfege notation around seven overlapping hexachords called solmization. Plainsongs extending over one, two or even three different hexachords could then be notated.

Hexachords conceptually overlapped (figure 2, left); another starting when the previous hexachord reached fa, the fifth, or when the melody again reached a given hexachord’s do of G, C or F. The Solmizations, prefixed with their note letters using Boethius’ Gamut system (Starting with our G (for Gamma ut G to e”) then A, etc., as we do today). For example, e’ would be uniquely called “Elami” since it was the note E, and the solfege la, for hexachord of G, whilst also mi, in the hexachord of C.

In contrast, the modern solfége of key signatures, without note letters, refers us to the major diatonic scale when equal tempered keyboards enable modulation of key signatures. By retaining the note letters, the hexachordal world could still reference the octave as a locational framework whilst also loosening the grip of do as tonic, as with modulation. The white notes of our keyboard were the basis of solmization with one exception: the minor hexachord starting at F had to impose the major diatonic T-t-s upon the T-t-T sequence of the diatonic scale by the solitary chromatic Bb.

Figure 2 The relations between Hexachords and the Octave of note names.
[on left, Willi Apel, 1969]

The solmization code created a namespace of unique composite words[1]. By combining note letters and hexachord positions, notes became unique words like Elami. Each note became linked to Beothius’ Gamut from G to e”, the solomised names explicitly identifying their context in the octave as well as the hexachords they belonged to (Figure 3).

Figure 3 The Solmization namespace combining Boethius’ note letters and Guido’s Solfeggio [Willi Apel, 1969]

When melodies exceeded the hexameter within which they were currently set; “In order to accommodate melodic progressions exceeding the compass of one hexachord, two (or more) hexachords were interlocked by a process of transition, called mutation”, since “in medieval theory the compass of tones was obtained not by joined octaves but by overlapping hexachords” and “tones of higher or lower octaves were not considered ‘identical’ within a Boethian scale of G to e””. [Willi Apel. 1969]

The Church as Octave within Solmization

If do of the “natural” hexachord (C) is placed on the (exactly) northern outer vertex of the fourteen vertices, then the three round pillars land, using Just intonation, in the midst of the Pythagorean tones of the major diatonic whilst the four vertex-like pillars coincide with the uniquely Just tones and semitones[1] (figure 4). The southern door marks the tritone between fa (F the minor hexachord) and sol (G, the hard hexachord). The walls of the church could therefore have usefully symbolised the intervals and note classes of the major scale[2] during the perambulation of the hexachordal plainchant, verbalized using Solmization. That is, if the church symbolized the successive octaves of the tonal world notated using hexachords, the building might have been a regional school for training singers, outside the customary cathedral and monastic schools of the 12th century. Guido’s method (staff notation and solfeggio and solmization) rapidly became famous and was widely adopted throughout north Italy and elsewhere[3]. When built, 12th century Languedoc and northern Italy was strongly populated by Cathars, so triggering the crusade from Rome and hence the subsequent confiscation of the church from its feudal owner.

[1] The practical scale of the day would have been the major diatonic since its three major thirds (between do and mi, fa and la and between sol and si) are achieved using the fifths and fourths of Pythagorean tuning in combination with the major thirds. This automatically generates the different tones and semitone found in Just intonation: T = 9/8 and t =10/9 form, in combination, the major third of T × t =5/4, short of the perfect fourth by the new Just semitone, s = 16/15.

[2] the natural scale for Just intonation when tuned using fifths, fourths and major thirds

In numerical tuning theory, the Virgin Mother would be the perfect symbol for an heptagonal church since the world of music springs from an octaval womb (whose number symbol is 2); only the male numbers (3 and 5) can reach into and divide the octave to create octaves of Pythagorean and Just intervals, then symbolic of Christ’s birth. The seven intervals and the notes of the diatonic scale provide a framework within which the magic of hexachordal singing expresses melody with a suppressed Ego or tonic. Hexachordal music strays across many tonic contexts. Numerical harmonists may have viewed tonics as titular deities of the limiting numbers required to theoretically generate Just Intonation[1], like the demiurges creating worlds but becoming an enemy of melodic freedom within them, by seeking to reference everything to their tonic. Arguably the natural tension, between static tonics of the octave and developmental movements like those found in hexachordal music, manifested the Classical traditions of sonata, concerto and symphony.

Drawing the intervals within the Church

If the two types of tone are each given a span of two or three sides of the tetradecagon, and the semitone a span of only one side, the total would be 3 + 2 + 1 + 3 + 2 + 3 + 1 equalling 15 sides rather than 14. But if one respects the natural symmetry of the tone circle about Re, as the (modern) Dorian scale, then one can make the initial tone of 9/8 symmetrical with the following tone of 10/9. In practice, nothing is lost since the church is only loosely a tone circle, with no imposition of logarithms except for those native to the ear, that hears intervals of the same size as the same size irrespective of pitch. Modes other than major could then have similarly been expressed by choosing other starting notes and vertices explicitly given within the fixed solmization words as the note letter prefix[2].

Figure 3 The encoding of intervals within the church

In the arrangement proposed, the disposition of round pillars coincides with the disposition of Pythagorean tones (of 9/8) on the outer wall, whilst the vertex-like pillars face the Just tones of t = 10/9 and s = 16/15. Pythagoras saw these now-eponymous tones of 9/8 as divinely perfect and hence a circular form is appropriate: The pure tones 9/8 are born (in numerical tuning theory) only by the divine male prime number 3 and the female octaval number 2 seen in 9/8[1]. In addition, Just tone 10/9 and semitone 16/15 require the humanly-male prime number 5 to birth them within the womb of the octave’s tone circle. The northern round pillar would also identify the necessarily shortened whole tone as Pythagorean, despite its being shortened, thanks to the association of pillar shapes with either type of whole tone.

As stated above, one can imagine that in a church, designed to represent an octave in the round, one could conduct the choir in Solfeggio.

My book on the role of musical theory in terms of both the number involved and ancient cosmological thinking is called The Harmonic Origin of the World. It came about through a virtual apprenticeship with Ernest G. McClain whose books The Myth of Invariance and The Pythagorean Plato revolutionized the subject (both books can be read in pdf at his posthumous website.)

References

Apel, Willi. Harvard Dictionary of Music. 1969.

Farmer, Henry George. Historical facts for the Arabian Musical Influence. 1930.

[1] Three Heptagonal Sacred Spaces by Sarah Reinhart and Vivian Ramalingam, pages 33-50 in Music and Deep Memory: Speculations in ancient mathematics, tuning, and tradition. in Memoriam Ernest G. McClain. ICONEA Publications 2018.

[2] which cannot join with any other number below ten or even twelve.

[3] then known as Ut–re–mi-fa-so-la-Sa-Io after the mnemonic “Ut queant laxis, resonare fibris, Mira gestorum, famuli tuorum, Solve pollute, labii reatum, Sancte Iohannes”: So that your servants may, with loosened voices, resound the wonders of your deeds, clean the guilt from our stained lips, O Saint John.

[1] Ernest G. McClain, The Pythagorean Plato 1978.

[4] a namespace arises when each name is unique whilst shared elements common to the other words, such as note letters and the solfege within hexachords.

[2] This transpositional modality is reminiscent of our later key signatures to which solfeggio is now applied.

[3] https://en.wikipedia.org/wiki/Guido_of_Arezzo.

[1] Ernest G McClain The Myth of Invariance, 1976

Starcut Diagram: geometry to define tuning

This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClain had been working on tuning via Gothic master Honnecourt’s Diagram of a Man (fig. 2), which is effectively a double square version of the starcut diagram.

The square is the simplest of two dimensional structures to draw, giving access to many fundamental values; for example the unit square has the diagonal length equal to the square root of two which, compared to the unit side length, forms the perfect tritone of 1.414 in our decimal fractional notation (figure 1 left). If the diagonal is brought down to overlay a side then one has the beginning of an ancient series of root derivations usually viewed within the context of a double square, a context often found in Egyptian sacred art where “the stretching of the rope” was used to layout temples and square grids were used to express complex relationships, a technique Schwaller de Lubitz termed Canevas (1998). Harmonically the double square expresses octave doubling (figure 1 right).

Figure 1 left: The doubling of the square side equal 360 units and right: The double square as naturally expressing the ordinal square roots of early integers.

Musical strings have whole number lengths, in ratio to one another, to form intervals between strings and this gives geometry a closer affinity to tuning theory than the use of arithmetic to calculate the ratios within a given octave range. The musicology inferred for the ancient world by Ernest G. McClain in his Myth of Invariance (1976) was calculational rather than geometrical, but in later work McClain (Bibal 2012-13) was very interested in whatever could work (such as folding paper) but was especially interested in the rare surviving notebook of 13th century artist Villard de Honnecourt, whose sketches employed rectilinear frameworks within which cathedrals, their detailing, human and other figures could be drawn.

“I believe we have overlooked Honnecourt as a prime example of what Neugebauer meant in claiming Mesopotamian geometry to approach Renaissance levels illustrated in Descartes. If Honnecourt is 13th c. then he seems more likely to be preserving the ancient picture, not anticipating the new one.”

This draws one into significant earlier traditions of sacred art in Egypt (Canevas) and in Indian temple and statue design, and to Renaissance paintings (see end quote) in which composition was based upon geometrical ideas such as symmetry, divisions into squares and alignments to diagonals. Figure 2 shows one of Honnecourt’s highly stylised sketches of a man, using a technique still in use by a 20th century heraldic artist.

Ernest McClain, Bibal Group: 18/03/2012

Figure 2 The Honnecourt Man employing a geometrical canon.

The six units, to the shoulders of the man, can be divided to form a double square, the lower square for the legs and the upper one for the torso. The upper square is then a region of octave doubling. McClain had apparently seen a rare and more explicit version of this arrangement and, from memory, attempted a reconstruction from first principles (figure 3), which he shared with his Bibal colleagues, including myself.

Figure 3 McClain’s final picture of the Honnecourt Man, its implied Monochord of intervals and their reciprocals.

To achieve a tuning framework, the central crossing point had been moved downwards by half a unit, in a double square of side length three. On the right this is ½ of a string length when the rectangle is taken to define the body of a monochord. McClain was a master of the monochord since his days studying Pythagorean tuning. Perhaps his greatest insight was the fact that the diagonal lines, in crossing, were inadvertently performing calculations and providing the ratios between string lengths forming musical intervals.

Since the active region for octave studies is the region of doubling, the top square is of primary interest. At the time I was also interested in multiple squares and the Egyptian Canevas (de Lubitz. 1998. Chapter 8) since these have special properties and were evidently known as early as the fifth millennium BC (see Heath 2014, chapter two) by the megalith builders of Carnac. In my own redrawing of McClain’s diagram (figure 4) multiple squares are to be seen within the top square. This revealed that projective geometry was to be found as these radiant lines, of the sort seen in the perspective of three dimensions when drawn in two dimensions.

Figure 4 Redrawing McClain to show multiple squares, and how a numerical octave limit of 360 is seen creating lengths similar to those found in his harmonic mountains.

Returning to this matter, a recently developed technique of populating a single square provides a mechanism for studying what happens within such a square when “starcut”.

Figure 5 left: Malcolm Stewart’s 2nd edition book cover introducing right: the Starcut Diagram, applicable to the top square of Honnecourt’s octave model .

Malcolm Stewart’s diagram is a powerful way of using a single square to achieve many geometrical results and, in our case, it is a minimalist version that could have more lines emanating from the corners and more intermediate points dividing the squares sides, to which the radiant lines can then travel. Adding more divisions along the sides of the starcut is like multiplying the limiting number of a musical matrix, for example twice as many raises by an octave.

A computer program was developed within the Processing framework to increase the divisions of the sides and draw the resulting radiants. A limit of 720 was used since this defines Just intonation of scales and 720 has been identified in many ancient texts as having been a significant limiting number in antiquity. Since McClain was finding elements of octave tuning within a two-square geometry, my aim was to see if the crossing points between radiants of a single square (starcut) defined tones in the just scales possible to 360:720. This appears to be the case (figure 6) though most of the required tone numbers appear along the central vertical division and it is only at the locations nearest to D that eb to f and C to c# that only appear “off axis”. The pattern of the tones then forms an interesting invariant pattern.

Figure 6 Computer generated radiants for a starcut diagram with sides divided into six.

Figure 7 http://HarmonicExplorer.org showing the tone circle and harmonic mountain (matrix) for limit 720, the “calendar constant” of 360 days and nights.

Each of the radiant crossing points represents the diagonal of an M by N rectangle and so the rational “calculation” of a given tone, through the crossing of radiants, is a result of the differences from D (equal to either 360 or 720) to the tone number concerned (figure 8).

Figure 8 How the tone numbers are calculated via geometrical coincidence of cartesian radiants which are rational in their shorter side length at the value of a Just tone number

It is therefore no miracle that the tone numbers for Just intonation can be found at some crossing points and, once these are located on this diagram, those locations could have been remembered as a system for working out Just tone numbers.

Bibliography

Heath, Richard.

  • 2014. Sacred Number and the Lords of Time. Rochester, VT: Inner Traditions.
  • 2018. Harmonic Origins of the World: Sacred Number at the Source of Creation. Inner Traditions.
  • 2021. Sacred Geometry: Language of the Angels. Inner Traditions.

Lubitz, R.A. Schwaller de.

  • 1998. The Temple of Man: Apet of the South at Luxor. Vermont: Inner Traditions.

McClain, Ernest G. 

  • 1976. The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas Hays.

Stewart, Malcolm.

  • 2022. Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology. Inner Traditions.

Introduction to my book Harmonic Origins of the World

Over the last seven thousand years, hunter-gathering humans have been transformed into the “modern” norms of citizens (city dwellers) through a series of metamorphoses during which the intellect developed ever-larger descriptions of the world. Past civilizations and even some tribal groups have left wonders in their wake, a result of uncanny skills – mental and physical – which, being hard to repeat today, cannot be considered primitive. Buildings such as Stonehenge and the Great Pyramid of Giza are felt anomalous, because of the mathematics implied by their construction. Our notational mathematics only arose much later and so, a different maths must have preceded ours.

We have also inherited texts from ancient times. Spoken language evolved before there was any writing with which to create texts. Writing developed in three main ways: (1) Pictographic writing evolved into hieroglyphs, like those of Egyptian texts, carved on stone or inked onto papyrus, (2) the Sumerians used cross-hatched lines on clay tablets, to make symbols representing the syllables within speech. Cuneiform allowed the many languages of the ancient Near East to be recorded, since all spoken language is made of syllables, (3) the Phoenicians developed the alphabet, which was perfected in Iron Age Greece through identifying more phonemes, including the vowels. The Greek language enabled individual writers to think new thoughts through writing down their ideas; a new habit that competed with information passed down through the oral tradition. Ironically though, writing down oral stories allowed their survival, as the oral tradition became more-or-less extinct. And surviving oral texts give otherwise missing insights into the intellectual life behind prehistoric monuments.

Continue reading “Introduction to my book Harmonic Origins of the World”

Agni, the Indian God of Fire

Those new to Ernest McClain and his The Myth of Invariance, should know this book was a seminal work for anyone in my generation, that opened up a Pythagorean vision; of how number operates in the domain of harmony. This world of harmony can be numerically defined in a quite extraordinary and specific way and we, as human beings, can receive it through our mind whilst also through the senses. This relates to the unusual fact that, whilst all notes can be doubled in frequency through the number two, with a perfect consonance, a new population of notes is then opened up, within an octave, of intervals that are also harmonious, through the use the two next prime numbers: three and five. Thus music, so effective upon the human heart, can build a world of meaning, sometimes referenced in myths as sacred numbers, written through understanding harmony as fundamentally generated through numeric transitions within music.

In 2008 I prepared a summary of Ernest McClain’s statements about Agni because, in the midst of the perfect symmetry of musical harmony lies something new, born to the world opposite its beginnings and endings. I originally made the pdf below for my friend Anthony Blake, part of our attempt to study the origin of creativity within the existing world. It appears that something important comes into being at the centre of this issue of octaval harmony, just as we ourselves come into existence in the middle of the universe, as conscious beings, conscious then of our incompletion.

It occured to me to include this in an email to Ernest and, all in, he said in reply “I can’t imagine anyone improving on your few pages” and “Put it out now on your own website stamped with my approval”. Please enjoy this transmission from the centre of the octave:

What Ernest McClain says about Agni in The Myth of Invariance:

Visit Ernest McClain website: Musical Adventures in Ancient Mythology. In the section of online documents, his books are available for your to study as links to pdf downloads.