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Chapter Two


Chapter Two

[2.1] Secondary school students are familiar with Pythagoras and the theorem regarding triangles, less so the story of the Smithy. The arguments of interest here pertain to the activities and interactions of Pythagoras, and his contemporaries, with the progression of fifths, or the 3:2 resonance. This journey of seven hundred years evolved towards a new tuning system which would change the musical world and the perception of sound for generations. The very concept of another tuning system to many elementary students is strange for they have only experienced the current tuning system, and the results of centuries of approximate impure tuning (Mathieu, 1997, p.1). Is the fact that it happened, and how it occurred, relevant, to the understanding of the fifths today?

[2.2] Pythagoras was a philosopher who lived in the sixth century BC. He is reputed to have discovered the perfect musical intervals which, as discussed in the previous chapter, were being, or had been, discovered in the ancient kingdoms. The problem with this account is the lack of primary source material for the claim. The first written account appeared in a publication called Harmonikon Enchiridion by Nicomachus of Greece, published in the second century AD, six hundred years after Pythagoras’ lifetime. In this document the author presents the story of the chance discovery made by Pythagoras when he passed a blacksmith shop. The account, found in Chapter Six, states that Pythagoras was deep in thought at the time, regarding the requirement for an accurate measurement of pitch. The sound of the anvils being struck at a combination of harmonious pitches, except one, drew his attention and he became aware that the sounds were the octave, fourth and fifth. The interval between the fourth and fifth is the dissonance sound and the other options were harmonious. Rushing into the blacksmiths workshop it was determined by Pythagoras, through experimentation, that the sound was produced by the varying weight of the hammers as opposed to the item being formed or the density of the iron. The author continues to explain how the discovery was confirmed when Pythagoras hung weights from taut strings. The multiple strings were suspended from a pole which was attached to the walls of his house parallel with the ceiling. The experiments continued with strings and moved onto pitch pipes resulting in an instrument like a lyre (Anderson, 1983).

[2.3] Tang, in a paper entitled Pythagoras at the Smithy, analyses the story with reference to the original language text, where it is commented that in the published narrative Nicomachus makes Pythagoras seem like a mythical character who dashes from one discovery to another with no motivation or explanation. An example can be found in the original text that indicates that there was a change of action from hearing hammers on anvils to an experiment with weights and strings, however there is no indication as to why Pythagoras suddenly changed from hammers and anvils to weights and strings. The lack of clear information provides a confusion of the source of sound. Was the sound produced from plucking the strings or striking the suspended weights? From the sound produced it was determined that the apparatus used a weight of twelve units suspended from a string and compared it to a weight of six units, this produced a 2:1 ratio.  The interval of the fifth was produced by Pythagoras’ apparatus by using the largest weight of twelve with a weight of eight that produced, when divided by four, the ratio of 3:2. Finally the twelve weight was compared with the nine weight, the result showed, when divided by three, that the ratio was 4:3. Nicomachus commented that the nine weight also formed the 3:2 ratio when compared with the six weight and the 4:3 ratio when the eight weight was contrasted with the six weight. (Chi-Chung Tang, 2011). As a primary source the document Nicomachus published provides as much ambiguity as it does an account of the true events. It does not make clear the source of sound, just that the ‘apparatus sounded’. However, in view of the question this paper asks, the text can be accepted to indicate that the core facts of the harmonic ratios embedded in the story of Pythagoras existed in Greece during 570 – 495 BCE. Whether he came across the inspiration at the smithy or elsewhere is lost in time. This account has made a good basis for stories, operas, and tales about the Harmonious Blacksmith, as can be heard in the example by Handel (Handel, 2007) which can be of interest to elementary students.


[2.4] Ptolemy was a Greek music theorist and mathematician who wrote a three-volume work entitled Harmonics. This work is recognised as the most learned discussion regarding music theory of its time. Ptolemy presents an analysis of the theory of the fourth, fifth and octave species. The Pythagoreans are criticised for defining intervals in different ways and for suggesting theoretical relationships that were not consistent with reality (Richter, 2001). Ptolemy, instead, calculated music theory on the results of numerical calculations and ratios, his work arranged intervals and redefined tetrachords. Pythogoras’ experiments with strings, as stated above, were documented a significant time after he died, in a rather haphazard way. In contrast Ptolemy used a monochord for all fifteen notes of the double octave, providing an accurate and re-creatable mathematical account. To complete this task Ptolemy classified intervals as two types, concords and simple scalar or melodic intervals which he explains as being unequal. This topic is beyond the scope of the paper therefore the items included will focus on the concords. Andrew Barker in Mathematical Beauty Made Audible, states that Ptolemy initially divided the ratio of the octave, 2:1, into two epimoric, or superparticular, ratios (Figure 2‑1) of 3:2 and 4:3, dividing it almost in half. A superparticular ratio is a pair of numbers where the greater number exceeds the lesser number by a unit (, n.d.). The Greek methodology was to move inwards by a fourth from either end of the scale thus dividing the scale into two tetrachords with a tone, the ratio of a tone is 9:8, in-between (Barker, 2010).

[2.5] The next influential name in the journey of the fifth is Boethius (475CE – 526CE) who was heavily influenced by Ptolemy’s texts and proficient in translation. Although the source and influences are not clear, it is accepted that he was influenced by the three theorists. Boethius’ work is the division of the monochord in an orderly and mathematical way and dedicates five chapters to this in his treatise De institutione musica (Boethius, 1491). According to Tang, Boethius makes four direct references in this volume, alongside three mythical style references, to Pythagoras by using a pseudonym. The final chapter is one of the most complete sources of theory regarding Greek modes (Bower, 1984). Initially Boethius defines the species as a unique pattern of notes that present a proportion of consonance.  Boethius, unlike Pythagoras and Ptolemy, offers two orders of species, the first showing a descending movement over a four (Figure 2‑2) or five (Figure 2‑3) note progression. The second order shows the ascending movement over four (Figure 2‑4) or five notes (Figure 2‑5). The original table, which can be found on page two hundred and fifty-six in The Modes of Boethius by Calvin Bower, shows additional numbering which I have not included in the notation models shown below as it is irrelevant to the topic of this paper. The pitch is shown on the treble clef for convenience and is not an indication of the lack of pitch registration of the original text. The books written during this period generally did not show notation but rather T for tone and S for semitone, for ease of understanding I have shown these in the common way for this time alongside the T and S from the older ways. The ordering of the First Species is new and shows the semitone, in blue, moving in logical pattern as the four or five notes are commenced from three different starting pitches. The order of the First Species is pleasant aurally and satisfactory in theory. The practice causes an issue on the fourth option of the diapente (fifths) where the inclusion of the lower B, shown on Figure 2‑3 by a * that indicates the adjusted interval, that should be a tone but is now a semitone. The first two lines on this image show the overall interval of E to B on line one, D to A on line two and C to G on line three, all are concords with a 3:2 ratio. The final note of a B has changed the overall interval B to F from a perfect to a diminished fifth or tri-tone, defined as three tones, as opposed to the series of perfect fifths otherwise observed. In ratio form this would be 1024:729 in Pythagorean tuning (Monzo, n.d.). This error does not appear to have been considered important enough to ignore the logic of order that the new semitone movement provides. Boethius sets out an alternative method which is moving towards the modern way of calculating note progressions in equal temperament where the interval of the fourth and fifth are adjusted for convenience over logic.

[2.6] Finally, in this chapter, we approach the shift of tonality from modal to major-minor tuning. During the ninth and tenth centuries three essays were published by Hucbald of Saint-Amand named De harmonica institutione and two anonymous publications named Musica enchiriadis and Scolico enchiriadis. In these publications the modal pitch collections began to look more like tetrachords with a tone and semitone pattern with the final note, a duplication of the first an octave higher, being included. In the eleventh century the characters of each mode and the recognisable patterns start to appear in an anonymous paper entitled Alia musica (Cota, 2018a). Music had moved forward from the Greek concepts of harmonies to include polyphony with melodies overlapping. A composer named Zalino had published a twelve-mode system in Le institutioni attributing names that were number biased as opposed to the initial Greek influenced nomenclature he originally used. This publication was to be used for the new polyphony. The new classifications were shown in table form indicating the fifth, fourth and octave relationships. This is the first published imagery showing a movement towards the visual representation of the relationships of the fifth. As the sixteenth century ended, eight mode song cycles were being composed and a new style of music was developing but the shift between tuning systems was not due to a musical reason or popularity. James Cota in From Guido’s Hand to the Circle of Fifths indicates, on page seven, that the advent of the major-minor tonality was expediated due to the requirement of the printed music score. Figured bass notation was popular with musicians as it required less space, but for it to work efficiently a standard harmonic system was required that was accepted and understood. This requirement led to the concept of a harmonic-tonal system with a choice of major or minor harmonies and the ability to harmonise any pitch (Cota, 2018b). The harmonic experience of Western art music had changed, and by extension the musical experience of future generations

[2.7] The question this paper is tasked with answering is “Is the circle of fifths relevant to elementary music education today?” This chapter addressed the significant arguments focusing on the activities of Greek philosophers and mathematicians leading to composers of the medieval period and the advent of printed music. The introduction ends with the question “Is the fact that it happened, and how it occurred, relevant, to the understanding of the fifths today?”  While it is not directly relevant, the understanding of figured bass will be of benefit when teaching an elementary student about roman numeral analysis and the understanding of common chord nomenclature. For elementary students to understand these foundation principals reinforces the importance of learning intervals, chords, and scales. The fun fact that the printing press changed the sounds of music enables inquisitive minds to ask questions, explore tuning systems that existed before and investigate further early printed music. Whilst not as exotic as ancient history, the journey to equal tempered music and potentially to the instrument that a student is learning is important.

Figure 2‑1. Epimoric Ratio

^ Figure 2‑1. Epimoric Ratio.

Figure 2-2. First Order of Species. Diatessaron (Fourth)

Figure 2-2. First Order of Species. Diatessaron (Fourth).

Figure 2‑3. First Order of Species. Diapente (Fifth)

Figure 2‑3. First Order of Species. Diapente (Fifth).

Figure 2-4. Second Order of Species. Diatessaron (Fourth).

Figure 2-4. Second Order of Species. Diatessaron (Fourth).

Figure 2-5. Second Order of Species. Diapente (Fifth

Figure 2-5. Second Order of Species. Diapente (Fifth).

C. Mathieu, 1997, p.1
C. Anderson, 1983
C. Chi-Chung Tang, 2011
C. Handel, 2007
C. Barker, 2010
C. Boethius, 1491
C. Monzo, n.d.
C. Cota, 2018a
C. Cota, 2018b
C. Bower, 1984
C., n.d.
C. Richter, 2001
Figure 2-1
Figure 2-2
Figure 2-3
Figure 2-4
Figure 2-5
(F2.2 2.3)


Anderson, G.H. (1983). Pythagoras and the Origin of Music Theory. Indiana Theory Review, [online] 6(3), p.36. Available at: [Accessed 19 Aug. 2022].

Barker, A. (2010). Mathematical Beauty Made Audible: Musical Aesthetics in Ptolemy’s Harmonics. Classical Philology, [online] 105(4), pp.412–413. doi:10.1086/657028.

Boethius, A.M.S. (1491). De institutione musica. [online] Internet Archive. University of Pennsylvania Libraries. Available at: [Accessed 23 Aug. 2022].

Bower, C.M. (1984). The Modes of Boethius. The Journal of Musicology, [online] 3(3), p.256. doi:10.2307/763815.

Chi-Chung Tang, A. (2011). Pythagoras at the Smithy. [online] Texas Scholar Works. Available at: [Accessed 19 Aug. 2022].

Cota, J. (2018a). From Guido’s Hand to Circle of Fifths., [online] p.3. Available at: [Accessed 25 Aug. 2022].

Cota, J. (2018b). From Guido’s Hand to Circle of Fifths., [online] pp.7–8. Available at: [Accessed 25 Aug. 2022].

Handel (2007). Suite in E major, HWV 430. [online] Available at: [Accessed 20 Aug. 2022]. Multiple dates, earliest document added 2007.

Mathieu, W.A. (1997). Harmonic experience : tonal harmony from its natural origins to its modern expression. [online] Rochester, Vt.: Inner Traditions International, p.1. Available at: [Accessed 24 Aug. 2022].

Monzo, J. (n.d.). diminished-5th / dim5 / -5 / b5 - diatonic musical interval. [online] Available at: [Accessed 1 Sep. 2022].

Richter, L. (2001). Ptolemy. Oxford Music Online. [online] doi:10.1093/gmo/9781561592630.article.22510. (n.d.). Definition of SUPERPARTICULAR. [online] Available at: [Accessed 1 Sep. 2022].

Anderson (1983)
Barker A 2010
Boethius 1491
Bower CM (1984)
Chi-Chung TANG
Cota J 2018a
Cota J 2018b
Handel 2007
Mathieu 1997
Monzo nd
Richer 2001
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