B31 The stem tones
From the wealth of possible variants, a sequence of tones has emerged in the West, which is used as a binding basis for understanding music theory. This sequence of tones, which has the appearance of a Latin numbering, is composed of physical observations as well as aesthetic and theological considerations.
The root tones marked with * are characterized by the special feature that they can be found directly on the monochord without arithmetic operations, because it is the fundamental frequency itself or integer multiples of it - i.e. natural tones - while the rest can only be determined by addition or subtraction. This happened out of the need to achieve step sequences that were as uniform as possible – comparable to the preference for stairs of the same height. The perfect third* and the perfect sixth* - which Archytas of Taranto (*435 - †410 BC) already used - were initially ignored for the same reason and were only appreciated in the second half of the 15th century.
The 8 levels are theologically based and were not significant for the Greeks. However, the congruence with the 8 heavenly spheres, the 8 days of the week (according to the biblical count) and the 8 beatitudes belonged to the Catholic, i.e. all-embracing world view. This explains the blatant discrepancy between the designation and the interval size measured in each case.
Later additions such as chromaticism were made in such a way that the stem tones as such remained untouched - and could be used as a starting point for all further considerations. Therefore, additional designations such as perfect, major, minor, augmented and diminished were required. The term chromatic already indicates that the original tonal appearance should only be changed slightly by recoloring. D became D sharp or D flat, G sharp or G flat, etc.
Prim* / prime
Solmization syllable Do, originally UT. Both the starting tone itself and the first interval are described as prim or unison with the size zero. The Greeks did not know the prime because they defined an interval as the distance between two unequal tones. The Latin numbering is an early Christian rearrangement with the result that all the following interval designations do not match the real interval sizes. This already applies to the prim itself, whose name stands for 1, but whose interval size is zero. A tone that is compared with itself - hence: unisono - shows the greatest possible agreement with itself. It is a self-generated consonance. The prim was held in the highest esteem by Aurelius Augustinus, who stated that nothing was as alike as the 1 of the 1. He described it as the epitome of total fusion and dearest love. Frequency ratio 1:1
Sekund / second
Solmization syllable: Re. The distance between the second root note and the initial note is described as a second. Origin: attributed to Pythagoras. Two fourths are deducted from an octave, arithmetically: 2/1x3/4x3/4 = 18/16=9/8. Interval size: 1 whole tone. Frequency ratio 9:8
Terz / third
Solmization syllable: MI. Interval size: 2 whole steps. Frequency ratio 9/8 x 9/8 = 81/64. Origin: attributed to Pythagoras. For comparison: pure third* 5/4, complementary interval to the pure sixth*.
Quart*/ fourth
Solmization syllable: Fa. Frequency ratio: 4/3. Origin: Determined directly on the monochord. Interval size: 2 whole steps and 1 half step
Quint* / fifth
Solmization syllable: So. Frequency ratio 3/2. Origin: Determined directly on the monochord. Interval size: 3 whole steps and 1 half step
Sext / sixth
Solmization syllable: La. Frequency ratio: 27/16. Origin: attributed to Pythagoras. Interval size: 4 whole steps and 1 half step. For comparison: pure sixth* 8/5, complementary interval to the pure third*.
Septim / sept
Solmization syllable: Si. Frequency ratio 243/128. Origin: attributed to Pythagoras. Interval size: 5 whole steps and 1 half step.
Octav* /octave
Solmization syllable: Do - subsequent addition of the syllable - since it does not belong to the John hymn - in conscious relation to the prim of the same name - reminiscent of the A and Ω. For the same reason, the assignment to the tone designations was made from C, and not following the alphabet, as it says in Off. 22:13: "I [Christ] am the alpha and the omega, the first and the last, the beginning and the end". The Greeks did not use the term octave. They called the interval diapason, which means "through all tones". From this it follows that the interval can be divided into as many parts as one likes - the correctness of the designation would not change - in addition we are dealing with the realization that all intervals outside the octave are repeated octave offset and this interval therefore actually includes all division variants. Because of the consonance, the interval was also called harmonia. Origin: Determined on the monochord by halving the string length. Interval size: 5 whole steps and 2 half steps. Frequency ratio 2:1
Conclusion
In contrast to the multiplication table, which can be regarded as a generally binding set of rules for dealing with numbers, we are dealing here with an ideological influence that put a theological stamp on Western music - as well as on Western culture as a whole including time calculation from the birth of Christ. Contained in the root tones are both timeless physical facts, ancient Greek discoveries such as frequency ratios as such, which came to light 2500 years ago, and rearrangements for aesthetic reasons, such as the Pythagorean sequence of tones. Also not to be overlooked is the early Christian reinterpretation in adaptation to the worldview of the time from a monotheistic perspective. Through the reference to symbols, the music attained a grandeur that the corresponding society of musical sciences, to which J.S. Bach belonged, still tried to defend against the spirit of time of the Enlightenment. (see blog C3: "The creed of Johann Sebastian Bach in image and sound").
With regard to the sequence of tones, a lot has happened in the meantime. Although the keyboards are cut starting from C according to the Pythagorean model - with half-steps between E-F and B-C., the chormatic black keys were inserted in the rear area. Acoustically, too, a new listening experience is offered insofar as we are dealing with equal semitone steps in tempered tuning. Each semitone is 100 cents in size, so the octave is 1200 cents. Aural demonstration of the original stem notes would require retuning the instrument - and a Chopin Prélude in Pythagorean tuning would clearly show that the two things were not made for each other. The traditional construct has been mathematically revised for piano and organ and the difference to the natural tones is manifested in the beats. Previously pushed to the sidelines, they have now become the main aesthetic thing, and a uniform beat pattern extends across all keys.Only the prim and the octave remained untouched.
All this also resulted in an addition in terms of symbolic interpretation: 13 keys with only 12 note designations stand for the 13-1 and thus for the participants at the Last Supper minus Judas, the 5 black keys for the 5 wounds and the 8 diatonic keys for the resurrection on the 8th day. Betrayal, Death, and Resurrection: A Keyboard condensed numerical summary of the Gospel. A consistent implementation of the innovations would have had to result in an adjustment of the keyboards and one might ask what was more important: the reminder of the Pythagorean origin or the Christian symbol reference.
Against the cultural-historical background presented, it becomes clear why Benedict XVI stated in 2015 that music has "its inner source in the liturgy". The Catholic Church still holds on to music, as the Second Vatican Council says: "The treasure of sacred music must be preserved and promoted with the greatest care". In the lecture reproduced in Blog B2, held in Castel Gandolfo, he compares Western music to that of other cultures and even sees in it a "proof of the truth of Christianity". A more intimate relationship between a Pope and musical art is unimaginable.The perception of other cultures - resp. of science - is of course a different one.
A liberation from the "regime of octave tonal spaces" as formulated by Ivan Wyschnegradsky (*1893 - †1979) could only take place with the advent of microtonal music, which in turn has far-reaching historical roots. The arch harpsichord with 36 keys per octave heralds this. Nicola Vicentino conceived it in 1555 and created two instruments. The overabundance of new intervals worth estimating was bound to lead to the breaking of the traditional model of thought one day.
The word of power: "Roma locuta, causa finita" (Rome has decided, the matter is closed) has no longer been valid in musical matters since the Paris Conservatoire was founded in 1795 (see Blog B27: "The liberation of music from the embrace of the church "), which did not prevent the Pontificio Consiglio della Cultura from rejecting the research results in 2015 as scientific theory and to withdraw the publication approval given by the Libreria Editrice Vaticana without reason. (cf. Blog B22: "On the index of society. On the rejection of theological content in the Western sound system" and also menu: "Church environment").).
The following articles also offer in-depth insights: - Blog C4: "Pythagorean tuning system or myth of origin of Western music?" - Video presentation XII: "The Christianization of the Greek sound system" - Video presentation XXVII: "Didactic tools".
Future perspective
Since the French Revolution there has been an intention to refrain from a symbolic interpretation of the western tonal system. The discussions regarding the acceptance of religious symbols in classrooms have already been reported, as well as the recent efforts of the FDFA to reinterpret the Christian cross as a «plus for peace» (see menu "Political environment").
While it used to be about resistance against the usurpation of power, today the focus is more on intercultural dialogue, for which it does not prove to be helpful to carry one's own culture forward like a victory banner. In this sense, Benedict XVI's remarks turn out to be counterproductive. However, we have a history from which lessons must be learned. Rejecting them would be alarming evidence of societal decline. (s. Blog B12: "Symbols. Mirrors of man") Censoring scientific work would be extremely dangerous for democracy.
If we want to understand our own culture, we have to visualize the educational horizon and the worldview of the past and are called upon to apply our intercultural commitment to these worldviews that have become alien to us.
How else could we deal with it if, for example, we find a chapter in the explanations of Johann Heinrich Buttstett (*1666-†1727) with the heading: "Caput. VI. Proves that the music will remain forever and that one day in heaven people will be able to play music with the very same Sonis that are customary here in the world.” (Buttstett, J.H., Ut, Mi, Sol, Re Fa, La, Tota Musica et Harmonia Aeterna, Erfurt 1717)
Precisely because the theory of harmony is about the acceptance of those who think differently, about working together and pulling together, we can prove that we have understood.
© Aurelius Belz 2023