A Handbook on the Middle TuningSecond Edition with corrected and improved tuning indications

Bevis Stevens – Maria Renold

The Scale of Twelve Fifths

A Handbook on the Middle TuningSecond Edition with corrected and improved tuning indications

A temperament founded on the tones c = 128 Hz, gelis1 = 362.41 Hz and a1 = 432 Hz

1 Gelis is Maria’s term for her newly discovered tone corresponding to f sharp. See further the Explanation of Terms

Private Press

© First Edition, Copyright 2006by Bevis Stevens, Dornach – Switzerland

Second Edition © 2012Havelock North – New ZealandEmail: stevens@eurythmy.co.nz

Contents

Forward to the second edition! 6

Forward to the first edition! 6

........................................................................................Background! 8

...............................Why a new concert pitch? Why a new temperament?! 10

............................................................................................................Origins! 10

.........................................................................................Aural experiments! 11

The mathematical-cosmic origin of the frequencies C = 128 Hz and A = .............................................................................................................432 Hz! 11

Table 1 ........................................The 10 x 7 factors of the Platonic cosmic year 12Table 2 .......................................................................Intervals of the major scale 13Table 3 ......................................................................Intervals of the minor scale 13

............................................................................Qualitative characteristics! 13

...............................................................................New discoveries ! 16

..............................................................................................‘Open intervals’! 18Table 4 ....................Measurements of the double octave–Interval sizes in cents 18

..................................................................................The secret of the fifths! 18Table 5 ................................................Fifths and Fourths–interval sizes in cents 19Table 6 .......................................Oscilloscope pictures of differently sized fifths 20Table 7 ....................................................................(Renold, 2004, p. 136, fig. 4) 20

..........................................................................Tuning instructions ! 22

..................................................The structure of the scale of twelve fifths! 24

...............................................................Tuning instructions–Maria Renold! 24

.....................................................................Tuning open fourths and fifths! 24Table 8 ....................................................................Difference tone of the fourth 25Table 9 .......................................................................Difference tone of the fifth 25Table 10 ...........................................Tuning to the scale of twelve fifths - Renold 28

......................................................The Ideal Mathematical Representation! 29Table 11 ......................................................................Tuning instructions – Davis 29Table 12 .............................................................Offset for Renold 2 for the Tuner 30

...........................................................................................Appendix! 31

...........................Experiences in doing Eurythmy with the Middle Tuning! 33

....................................................................................Explanation of Terms! 35

..................................................................................................Bibliography! 36

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4

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Forward to the second editionSince writing the first edition I have learnt to tune the piano myself and Paul Davis’

appendix to Maria Renold’s book (2004) became understandable. This made me aware of the fact that following her and Thomma’s indications (Renold, 2004) actually results in a third version of the tuning! Although I like it, as it has even more key colour, it is not what Maria intended. Now Paul has further refined the mathematically ideal tuning to a point where it accurately represents Maria’s intentions. This has made a total revision of the whole tuning section necessary. I have kept Maria’s description of the tuning method, abandoned those given by Henken and Thomma and laid out the tuning in the style developed by Jorgensen (1991) which is generally known and accepted by tuners worldwide.

On a historical note: The research of Michael Kurz at the Goetheanum has shown that Rudolf Steiner could not have given an indication for the pitch of the A = 432 Hz to the recorder maker Ziemann-Molitor. However this does not affect the validity of this concert pitch for Maria’s temperament as it results as a matter of course when beginning from her original starting point of Steiner’s recommendation for C = 128 Hz. As a result the merits of A = 432 Hz will be still covered.

– Bevis Stevens March 2012

Forward to the first editionThis booklet is an attempt to satisfy the need for a short introduction into the twelve

fifth–tones scale. It is directed towards all interested people who wish to have their instrument tuned in this tuning, as well as professional tuners who require a handbook. Although this booklet follows the content of, and in deed adds to Maria Renold’s book (2004), it by no means replaces it. This booklet deals with the twelve fifth–tones tuning alone, whereas Renold’s book also goes into the construction and historical development of the Western scales and the origin of their tones and intervals, researches why the human ear experiences some intervals to sound false, while others sound genuine, and verifies the existence of an ethical effect of the pitch of single tones. Renold also looks into various remarks on music and specifications given by Rudolf Steiner2.

There are two versions of the tuning. The first was discovered in 1962 (Renold, 2004, p. 57).3 Over the years, the tuning was developed further and new discoveries made a second more beautiful version of the tuning possible. It is the second version of the tuning, which will be introduced here.

– Bevis Michael Stevens, June 2006

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2 The Founder of Anthroposophy. See also footnote 4

3 After Renold discovered the 1st tuning method, she found out, more or less by chance, that Henricus Grammateus had already constructed the scale mathematically in 1518 (Jorgensen, 1991, p. 332). Because it has often occurred that Renold has been accused of simply copying this scale, it must be stressed that it was a new discovery, through hearing and not through mathematical construction.

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Background

8

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Why a new concert pitch? Why a new temperament?One may well ask: why use a different concert pitch, haven’t our ears long become

accustomed to a = 440 Hz? Understandably this question is often posed by musicians who have the so-called gift of absolute pitch. But the definition absolute pitch is misleading, as the pitch to which this ‘ability’ is set varies greatly. Therefore absolute pitch cannot be used as a guide to determine the correctness of a concert pitch. The correctness of a pitch is of course a contentious issue but Maria Renold’s research shows that small variations in the pitch of a tone give rise to remarkably different qualities. The Concert pitch A = 432 Hz is a result of the temperament itself which takes its starting point from Rudolf Steiner’s recommendation for the pitch of C.

But what does one want to achieve with yet another tuning method? Is our equal tempered tuning not perfect enough already? Yes indeed, but it’s ‘equalness’ ultimately lessoned and restricted music’s expressiveness and variation in tonal colour. The intention is therefore not one of wanting to replace something or wishing to find something better, but rather one of expanding the expressive possibilities of music and thereby enriching our musical experience. Qualitative differences between pitches are often first noticeable when directly compared. One is then often amazed at the great effect that small differences in the pitch of a tone can have. Furthermore it is striking how quickly one can get used to a new temperament and its concert pitch.

It needs to be added at the outset that this new method of tuning arose through hearing and not through theory. Therefore in order to evaluate the tuning it is necessary to hear it! Herewith a limitation of this booklet is addressed. However, an attempt to describe this tuning and talk about its attributes as well as giving tuning instructions will be made

OriginsThe experiences that Maria Renold had – on the one hand as a violinist and violist in

the internationally renowned Busch Chamber Orchestra and Busch String Quartet and on the other hand with eurythmy4 – were important. Her tonal world was one of perfect fifths and so-called Pythagorean tuning. The discrepancy posed by equal-tempered tuning bothered her. She wondered if it weren’t possible to tune the piano in a way, which didn’t sound so false to her string-player ears.

In 1962, beginning with a diatonic scale based on perfect fifths, Maria Renold found the five chromatic tones between them through inner listening. As these are in the middle of the other tones, they are mathematically geometric mean tones. The result was a scale in which all major and minor keys could be played and which sounded aurally genuine, having more tonal variation and key colour than equal temperament. Renold called the resulting scale “the scale of twelve fifths”. She continued to experiment and develop the temperament and a second version was published in 1991. The two versions are often referred to as Renold1 and Renold2. Renold2 is also known as the “Middle Tuning” by which it is referred to here because of its grounding, centering quality and the way it speaks directly to the middle of the human being, to the heart. It has a sun-like, radiant quality.

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4 The art of movement inaugurated by Rudolf Steiner. For more information on eurythmy go to www.eurythmie.com and for more on Rudolf Steiner and Anthroposophy see www.goetheanum.org or www.anthroposophy.org

Aural experimentsThe first time the piano was tuned in the Middle Tuning, Maria Renold used the

concert pitch a1 = 440 Hz. It sounded nice but there arose an unsocial, hostile mood amongst the members of the family–a rare occurrence! Some time later, she heard of Rudolf Steiner's specification c = 128 Hz, which in this method of tuning results in the concert pitch a1 = 432 Hz. The piano was retuned and harmony reigned amongst those present (Renold, 2004).5

Amazed by this experience of hers, Renold began to make countless aural experiments in order to test the objectivity of her observation and to verify the importance of Rudolf Steiner's specification6. She tested musically trained and untrained people in America and Europe.

She discovered that, although a1 = 440 Hz and c1 = 261.656 Hz were the more familiar pitches, over 90% of those tested preferred a1 = 432 and c = 128 Hz and it’s octave 256 Hz. Some statements for C were:

c1 = 256 Hz belongs to the human being, gives space, sounds peaceful, pleasant and full and sounds as prime within the whole human being.

To c1 = 261.656 Hz over 90 % said it sounded jabbing, irritating, unpleasant, heady, intellectual, makes one nervous…

Further experiments were made with the correspondingly lower C, c1 = 252 Hz. Those tested said that this tone gives rise to a feeling of bodily comfort, made one drowsy, but at the same time had a calculating, merciless quality…

Thus Renold showed that a sense for the ethical quality of a tone described by the Greeks, still exists today, and that it was important to take this fact into account in the development of a new method of tuning. Further results of these experiments can be found in her book (Renold, 2004, pp. 76-79).

The mathematical-cosmic origin of the frequencies C = 128 Hz and A = 432 Hz

Everyone discovers soon enough that numbers have a lot to doing with music. They are connected with all kinds of rhythms and are thereby capable of showing the correlation between macrocosmic rhythms, microcosmic rhythms and musical harmonies. Numbers are not allegations but are pure phenomena, and as such enable us to gain the required objective basis.

Music has a very interesting and important connection to time. If something (e.g. a string) moves 16 times, and the speed at which it does so is fast enough for us to hear it, we can say with certainty: ‘that is a second’. We are unable to hear one cycle per second, but one

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5 In this connection it is interesting to note that Rudolf Steiner originally gave this specification to Kathleen Schlesinger who had noticed that her newly rediscovered modes (1939) had different effects on the listeners, depending on what pitch they were tuned to.

6 For a while the story has been upheld that Rudolf Steiner suggested the concert pitch of a=432 Hz be used on the flutes built by Mr and Mrs Ziemann-Molitor for the second Waldorf School. However recent research made by Michael Kurtz (responsible for the music department at the Section for Eurythmy, Speech, Music, Puppetry and Drama at the Goetheanum, Switzerland) has proven this to be a myth. However, as Maria’s Temperament gives rise to this concert pitch as a matter of course when taken from c=128 Hz, and because tuning usually begins with A the merits of this pitch are also gone into here.

cycle times 24 (2 x 2 x 2 x 2) is 16 Hz, which is the lowest tone we are able to hear. 164 = 128 Hz, i.e. 128 Hz7 is the 4th octave of 16 Hz.

Table 1 The 10 x 7 factors of the Platonic cosmic year

Herewith the number 128 finds its musical substantiation, and at the same time its importance for music and the human being. This tone is also known as the philosophic pitch.The number 432 is found in the harmony of the rhythms contained within the so-called platonic cosmic year, which takes 25920 years. This number represents the progression through the whole zodiac of the point where the sun rises at the spring equinox.The number 25920 has 70 factors, which are connected to each other in a wonderful, harmonic rhythm. E.g. 1 occurs 25920 times, while 2 occurs simultaneously 12960 times, 3 8640 times etc. One of these rhythms is 432 : 60; 432 divides 25920 60 times. The number 60 is to be found in the rhythm of the Saturn and Jupiter conjunctions: the conjunction between Saturn and Jupiter takes place every 20 years. Every 60 years they meet again at the point of departure.Microcosmically, the cosmic number 25920 has a connection to the average breaths a human being takes in a day: 18 breaths a minute multiplied by 1440 (60 minutes x 24 hours) makes 25920.

“This number corresponds also exactly with the number of days, which–in accordance with the old Babylonian year of 360 days–make 72 years. The statement of Rudolf Steiner’s, that esoterically the human life spans 72 years, is based on this. After this time the sun ‘covers’ another point of the heavens and ‘releases’ the star governing ones destiny .8 Thus we breathe an average of 25920 times a day and our life contains 25920 days” (Glöckler & Glöckler, 2007, p. 186).

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7 Also 27 = 128.

8 See Rudolf Steiner, GA 237, lecture of 5th July, 1924

1 x 25 9202 x 12 9603 x 8 6404 x 6 4805 x 5 1846 x 4 3208 x 3 240

9 x 2 88010 x 2 59212 x 2 16015 x 1 72816 x 1 62018 x 1 44020 x 1 296

24 x 1 08027 x 96030 x 86432 x 81036 x 72040 x 64845 x 576

48 x 54054 x 48060 x 43264 x 40572 x 36080 x 32481 x 320

90 x 28896 x 270108 x 240120 x 216135 x 192144 x 180160 x 162

If we look more closely at the 24th to 48th factors we find the intervals of the major scale: 24:27:30:32:36.40:45:48.

“The intervals of the major scale begin with the prime on the factor 24. The proportion 24:24, completely reduced, is the same as the proportion 1:1, i.e. the prime… The proportion 24:27, when completely reduced, corresponds with the proportion 8:9, i.e. the second. And so on till 24:48 (1:2), the octave…

Table 2 Intervals of the major scale24 : 24 = 1 : 1 Prime24 : 27 = 8 : 9 Second24 : 30 = 4 : 5 Major third24 : 32 = 3 : 4 Fourth24 : 36 = 2 : 3 Fifth24 : 40 = 3 : 5 Major Sixth24 : 45 = 8 : 15 Major Seventh24 : 48 = 1 : 2 Octave

…The Classical intervals of the minor scale arise when we depart from the factor 360

Table 3 Intervals of the minor scale360 : 360 = 1 : 1 Prime360 : 405 = 8 : 9 Second360 : 432 = 5 : 6 Minor third360 : 480 = 3 : 4 Fourth360 : 540 = 2 : 3 Fifth360 : 576 = 5 : 8 Minor sixth360 : 648 = 8 : 15 Minor seventh360 : 720 = 1 : 2 Octave

So we find, hidden within the two factor rows, as a continuous proportion, the intervals of the major and minor scales. Therein included is the number 432 [27]–the vibration frequency [Hz] of the archetypal concert pitch” (Glöckler & Glöckler, 2007)

If we include the lower and higher octaves of 432, we discover that this concert pitch appears 7 times within the 70 factors (27, 54, 108, 216, 432, 864 and 1728)

Qualitative characteristicsBecause the size of the various types of intervals in the Middle Tuning varies, each

chord and consequently each key gains its own definite character. A modulation is thereby experienced more strongly. Because of this an organ builder and piano tuner once said that the Middle Tuning in comparison to equal tempered tuning is like a relief in contrast to a flat map. Besides this enrichment in the keys, a greater sonority is achieved through the difference tones, which arise through the size of the tuned fifths.9 Because of this even upright pianos sound a lot better as a result of the tuning.

139 For more about the difference tones, see ‘Tuning open fourths and fifths’ below

In contrast to the bright, even tense-sounding equal temperament on concert pitch 440 Hz, the Middle Tuning is more relaxed and warmer. Many people experience that the music speaks more directly to the heart and that the barrier between audience and music is broken down.10

But enough has been said! Let us proceed to a discussion of the Middle Tuning itself, before proceeding to the tuning instruction, so that you will be able to make your own experiences!

1410 See the Appendix for a report on doing eurythmy with the Middle Tuning.

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New discoveries

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‘Open intervals’The second method of tuning, which is introduced here, was made possible by the

discovery of open fourths and fifths, as well as the discovery that a genuine sounding octave is bigger than the ‘perfect’ proportion 2 : 1.Even though the latter is a surprise, ‘stretching’ the octaves has long become established tuning practice.

As a string player, Maria Renold was aware of the problem posed by the double octave flageolet. In the lower regions it sounds flat because of the inharmonicity of the strings; in higher regions in sounds low because of so-called psycho-acoustic reasons. She made several measurements of the sizes of the double octave:

Table 4 Measurements of the double octave–Interval sizes in cents

Grand piano 2404.6 2406.8 2411.4 2418.6 Bechstein ENCello,flageolet von c 2398.7 2399.2 2399.8

c1 sounded too low

Cello,stopped 2401,7 2404,4 2406,7 c1 sounded right

“Note: If one multiplies the widest stopped double octave on the cello by the factor 3.5 ( = seven octaves), the result is the same interval as that of twelve perfect fifths; the Pythagorean comma does not apply: 2406.7 x 3.5 = 8423.45; 701.955 [= cents of perfect fifth] x 12 = 8423.46” (excerpt from Renold, 2004: Table 27)

The secret of the fifthsMaria Renold also discovered that different perfect fifths exist:“Further observations can be made with fourths and fifths on resonant instruments.

Grand pianos, harpsichords and pipe organs are best for first attempts because of the required precision. Let us begin with the beat-free perfect fifth. It sounds peaceful, almost solid. The lower octave of the fundamental hums along, totally absorbed in the sound, giving a silvery colouring to the whole. If one minimally enlarges the fifth–by either raising the pitch of the higher, or lowering that of the bottom tone–the whole texture is set in motion. First of all the interval vibrates violently, then it opens up and the fifth sounds peaceful and clear once again but also wide. The difference tone emerges sonorously and sounds stable. Some people recognise the openness of the interval by a relaxation in the area of the diaphragm. If the fifth is enlarged further, the difference tone begins to beat, the sound loses its coherence and the beats dominate. If one makes the fifth smaller than perfect, it immediately loses its lustre and strength and grows milder and more inward. When the interval is smaller than a perfect fifth, the difference tone does not stabilise, being weak at first then suddenly breaking off. Where the beating sets in, the interval is roughly the same size as the equal-tempered fifth. If one makes it still smaller, the fifth will be utterly calm again.

Much the same applies for the fourth…”(Renold, 2004, p. 129)

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The fourth and fifths were also measured:

Table 5 Fifths and Fourths–interval sizes in centsInstrument IntervalIntervalInterval Comments

Open fifths, sonorous difference toneOpen fifths, sonorous difference toneOpen fifths, sonorous difference toneGrand piano 703.5 702.9 704.2 Bechstein ENPiano 703.4 702.7 704.2 SabelViolin 703.5 703.7 704.2 open strings, steelViola 703.6 704.0 704.8 open strings, steelCello 702.9 703.8 706.5 open strings, steelOrgan pipes 703.0 704.3 705.5 stopped

Perfect fifths. silvery difference tonePerfect fifths. silvery difference tonePerfect fifths. silvery difference toneCello 701.0 702.2 open strings, steelOrgan pipes 702.0 Stopped

Small fifth. no difference toneSmall fifth. no difference toneSmall fifth. no difference toneCello 698.5 699.0 700.4 open strings, steel

Open fourths. sonorous difference toneOpen fourths. sonorous difference toneOpen fourths. sonorous difference toneGrand piano 499.7 499.9 500.8 502.2 Bechstein ENPiano 499.6 499.9 501.3 502.3 SabelChord zither 499.5 499.7Organ pipes 499.7 499.9 500.9 502.5 Stopped

(Renold, 2004, p. 27)

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Oscilloscope pictures of the different fifths were made:

Table 6 Oscilloscope pictures of differently sized fifths

Open fifth

Transition

Perfect fifth

Transition

Small fifth

Table 7 (Renold, 2004, p. 136, fig. 4)20

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Tuning instructions

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The structure of the scale of twelve fifthsThe octave proportion used here is: 2.003873819 which amounts to 1203.35 cents

Maria Renold specifies three pitches:1. c = 128 Hz2. a1 = 432 Hz3. gelis1 = 362.4 Hz

The first one originates from Rudolf Steiner; the first was given to the musicologist and rediscoverer of the Aulos modes, Kathleen Schlesinger, the second was given to Mr and Mrs Ziemann-Molitor, the builders of the flutes for the second Waldorf School. Both tones originate in the duodecimo row of c.

The scale of twelve fifths consists of two groups of duodecimo rows: the first of seven diatonic open-fifth tones (F, C, G, D, A, E, B) which belong to the Pythagorean-Dorian octachord, and the second from five geometric mean tones (Gelis, Alis, Belis, Delis, Elis). The latter is built, beginning from the geometric mean of the minimally enlarged octave (1203.350 cents) c = 256 Hz –c1 = 512.99 Hz, gelis1 = 362.4 Hz and progressing in open duodecimos.

The audible difference tones are an important part of the scale of twelve fifths. They add resonance and fullness to the sound. Therefore Maria Renold recommends that instead of tuning to octaves one tunes to fourths and fifths. But in the extreme regions this is impractical, where octave tuning is necessary.

Tuning instructions–Maria RenoldThe method of tuning “is made up of three fundamental parts:

1. gelis1 = 362.04 Hz and Rudolf Steiner's pitch indications c = 128 Hz and a1 = 432 Hz (for these pitches it is necessary to have three tuning forks that must be calibrated to within ± 0.5 Hz);

2. the open (minimally enlarged) octave, fifth and fourth intervals which, despite background beating, sound totally calm and open and have sonorous difference tones;

3. tuning in contrary direction of the Apollonian Sun scale: the pre-Christian descending direction of the true Dorian octachord and the Christian ascending direction of the true-tone C major resurrection scale(Renold, 2004, p. chapter 19 and Table 11)...”(Renold, 2004, p. chapter 24)

As can be seen through the measurements of the fourths and fifths above, it is evident how exactly one has to listen in order to attain the required intervals.

“The results of the interval measurements as presented in [Table 4 and Table 5 above] therefore showed a surprising variation in the size of the named intervals. This means that the cents can only be a rough guide and that very exact hearing and the ability to recognize these intervals is vital for successful tuning of the scale of twelve fifths. Every tone must always be fine-tuned by ear.”

Tuning open fourths and fifthsFirst tune a perfect fourth or fifth, making sure that the interval has absolutely no

beats. “If it is perfect it sounds stable and the difference tone, though not strong, is clearly audible and sounds silver.” (Renold, 2004, p. 140)

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Fourths are easier to tune as the difference tone is lower than that of the fifth and is therefore easier to hear. The difference tone is the difference in frequency between the lower and upper tone, e.g. 341.3 Hz (F) minus 256 Hz (C) = 85.3 Hz (F). Written musically this is as follows:

Table 8 Difference tone of the fourth

Fourth:

Differenzton:

An example of the fifth is: 384 Hz (G) minus 256 Hz (C) = 128 Hz (C), in notes:

Table 9 Difference tone of the fifth

Fifth:

Difference tone:

Next, enlarge the interval slowly. “The interval will begin to beat intolerably: this has often shocked piano tuners so much that they hardly dared to continue. Do not give in to ‘fear of beats’, but calmly increase the interval in minute steps. The beats will gradually get slower until they suddenly disappear altogether, the sound of the interval opens up and the differential begins to sound strongly and sonorously… these open fourths and fifths, together with their sonorous difference tones, lend their richness of colour to the scale of twelve fifths…

[Table 9] gives a musically notated tuning guide. The open note head of each interval indicates the tone from which one tunes, and the solid head the one to be tuned. The diamond-shaped heads on the base stave show the differential of the interval directly above… The three tones on the extreme left indicate the tones c1 = 256 Hz, gelis1 = 362.40 Hz and a1 = 432 Hz. First tune these three tones as exactly as possible. They must equal the tone of the tuning fork so exactly that they are beat-free. This is best achieved when one inwardly experiences the tuning fork's tone and then recreates the experience on the instrument.

Tuning proceeds in the two directions of movement belonging to the Apollonian scales: the pre-Christian descending direction of the true Dorian octachord and the ascending direction of the true-tone C major resurrection scale. Starting with c1 = 256 Hz, tune an ascending open fourth to f1 and open fifth to g1. Then tune a descending open fifth from a1 = 432 Hz to d1 and an open fourth to e1. From this e1, tune a descending open fourth to b and an

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ascending open fifth to b1. The seven non–altered diatonic fifth tones of the scale of twelve fifths have now been tuned. The first open octave sounds between b and b1…

As the two directions of movement meet in the fourth d1–g1, the first steps must be carefully tuned. The first formed fifth appears between b and gelis1. It sounds serious, but it must be calm and totally harmonically acceptable. If this is not the case, look first for the cause of the problem in the b which has probably been tuned too sharp, and not in the gelis1 which has been tuned from the tuning fork. Then go back to the descending open fourth a1–e1 and check this thoroughly. Pay attention to tuning the e1 really flat enough. This fourth is wide and open and is correctly tuned when its differential sounds really strong and sonorous like the sound of an organ. The same applies to the descending fourth e1–b. When these two fourths are tuned correctly, then the formed fifth b–gelis1 also sounds correct.”

My experience has been that the descending fourths a-e and e-b need to be particularly wide, otherwise the b is too high, resulting in an intolerable 5th b-gelis. A beginning help is to tune the 5th b-gelis first. (it should have about 2.5 – 3 beats) and then tune the e to fit between the b and the a.

“Now proceed from gelis1 and tune a descending open fourth of the same size to delis1, and from there an ascending open fifth to alis1; from alis1 a descending open fourth to elis1 and from elis1 a descending fourth to belis1. The second formed fifth sounds between belis1 and f1. What applied above is also applicable here: if the formed fifth belis1–f1 sounds wrong, go back to gelis1 and tune the tone sequence delis1, alis1, elis1, belis1 once again until they are correct and the formed fifth belis–f1 sounds right.”

As above, the fifth belis-f may be tuned first, beating about the same as the fifth b-gelis, and then progression back to gelis. The fourths need to be wide and the fifth delis-alis relatively small, i.e. perfect rather than open.

“Finally, tune the two ascending open fifths elis1–belis1 and f1–c2 and check the latter with the open fourth g1 = c2. The 9th belis–c2 has thus been tuned to the scale of twelve fifths and the three minimally enlarged octaves b–b1, belis–belis1 and c1–c2 have been gained.

Success in tuning the lower octaves downwards is again dependent on tuning the open fourths and fifths large enough, i.e., tuning the lower tone flat enough. To make sure that tuning has been successful, the following check is imperative. Having completed the lower octaves between c2 and subcontra 2A, play all 24 major and minor arpeggios in ascending order one after the other. If they sound pleasant to the ear they have been correctly tuned. If each higher octave sounds too flat, then the open fourths and fifths have been tuned too small, with each new tone then tuned too sharp. In this case, return to e1 and tune again until the correct size has been achieved and all chords sound pleasing.

To tune the higher octaves of the instrument, proceed from gelis1 and tune in ascending direction to open fourths and fifths… A similar problem exists here as above, but the opposite. Take care that each new tone is tuned sharp enough, in other words that the open fourths and fifths are really tuned open. The sonorous resonating of the differential can be a helpful guide. When tuning is complete, play all 24 major and minor arpeggios again in ascending direction through all registers. The incorrectly tuned tones will be immediately and uncomfortably noticeable. When they have been corrected and all chords sound pleasing, the

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instrument is ready to be played; you can play in all styles and rejoice in a true and beautiful sound.

When testing the 19ths of the base tones, it will be noticed that some beat more than others. E.g. the 19th f-a beats (this interval is Pythagorean) very fast while the 19th e - g hardly beats (this interval is almost just)

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Table 10 Tuning to the scale of twelve fifths - RenoldTemperament octave

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Open note head: tone from

which tuning proceeds.

Black note head: tone to be tuned.

Diam

ond note head: difference tone.N

otes preceded by a cross are geometric m

ean tones.Tuning forks at the follow

ing frequencies are required for the tuning: c1= 256 H

z, gelis 1 = 362.4 Hz and a

1 = 432 Hz.

The Ideal Mathematical RepresentationThe word ideal is used because such a temperament is neither obtainable nor

desirable. A temperament will always be adjusted by ear to suite a particular instrument. However, for the sake of an accurate representation of this temperament this is vital and is a milestone in the development of this tuning. This is shown first in the style of Jorgenson (1991) for tuning by ear and then a table is given showing the offset for setting a tuner.

Table 11 Tuning instructions – Davis

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& œ œ œ#Tune totuning forks

œ̇–1.32

¿Listen fordifferencetones

?œ̇

+0.50

¿œ̇+0.56

¿ etc.

œœTest–1.49 œ̇–1.67

œ̇–1.25

œœ#Test–2.34

œ̇+0.63

& œœ+0.94Test

œ̇+0.66 œœ

+0.99Test

œ̇b–2.21

œœ#–2.34

Test: beats faster

œ̇##–1.40

œ̇##+0.53

œ̇##–1.58

œ̇bb+0.59 œœbb

+0.89Test

& ˙̇b+8.33

Major thirds

˙̇#+8.83

˙̇+14.33

˙̇b+9.89

˙̇#+10.48

˙̇b+11.12

˙̇#+11.79 ˙̇+19.13 ˙̇##

+20.28 ˙̇+21.51 ˙̇b+14.84

& ˙̇bb–14.31

Minor thirds

˙̇–15.17

˙̇b–10.13

˙̇#–10.74

˙̇–18.02

˙̇bb–19.10

˙̇–20.25 ˙̇b–13.52 ˙̇#

–14.33 ˙̇b–15.20 ˙̇#–16.12 ˙̇–27.04

Renold 2Tuning instructions

© BMS

Tune to Tuning forksc1 = 256 Hza1 = 432 Hzgelis1 = 362.40 Hz

+ before the beats indicates beats faster – beats that are slower than equal temperamentblack note heads: ntoes arleady tunedopen note heads: notes to be tuned

Beats:

Table 12 Offset for Renold 2 for the Tuner

30

-12

03

.35

Dow

nwards

12

03

.35

Upw

ards

octavec octaves unstretched

Renold c-octaves stretched

e.t. c from A

=

432 HZ

Difference

octaves of A 440

e.t. c from A

=

440

difference to Renold c-octaves stretched

C4

-5.87C

0to B

016

15.87716.05

-19.27427.5

16.35-51.032

-45

.16

C0

delis4-4.47

C1

to B1

3231.815

32.11-15.924

5532.70

-47.682-4

1.8

1C

1D

4-3.07

C2

to B2

6463.753

64.22-12.574

11065.41

-44.332-3

8.4

6C

2elis4

-1.68C

3etc

128127.753

128.44-9.224

220130.81

-40.982-3

5.1

1C

3E4

-0.28C

4256

256.000256.87

-5.874440

261.63-37.632

-31

.76

C4

F4-5.59

C5

512512.992

513.74-2.524

880523.25

-34.282-2

8.4

1C

5gelis4

-4.19C

61024

1027.9711027.48

0.8261760

1046.50-30.932

-25

.06

C6

G4

-2.79C

72048

2059.9232054.96

4.1763520

2093.00-27.582

-21

.71

C7

alis4-1.40

C8

40964127.827

4109.927.526

70404186.01

-24.232-1

8.3

6C

8A4

0.00belis4

1.40B4

2.79

Offset for R

enold

2 –

regu

larised accord

ing

to Pau

l Davis

octave stretch in centsU

se the last two colum

ns of the following table (red) to set the stretch individually for each octave if an overall stretch

can not be set.

set the offset for th

e n

otes of the ch

romatic

octave as follows:

offset in relation

to C4

. Set

the corresp

onid

ng

octaves to th

is offset

Appendix

31

32

Experiences in doing Eurythmy with the Middle Tuning In repeated experiments, the concert pitches c = 128 Hz and c = 130.828 Hz and the

equal-tempered and twelve fifth-tones tuning have been eurythmically compared with one another, singly and in groups.

The tone c = 128 Hz as prime streams evenly and harmoniously through the whole stature. Even the feet are self-evidently 'there' and can be reached without exertion.

With the tone c = 130.828 Hz the eurythmist experiences himself to be drawn upwards, away from the feet, because the tone does not sound through but outside of the body.

The same fundamental tendencies appear with the 30° tone movements. The tones relating to c = 128 Hz rest within the area of the muscle and bone structure of the human being. There exists a wonderful correspondence between gesture and tone. With c = 130.828 Hz the eurythmist can imagine these movements to lie within the body but they never become real and direct experience there. If the eurythmist tries to find where they lie, the place, and/ or angle cannot be found; he experiences only that they lie ‘outside’.

By eurythmical improvisation to the C major prelude from J. S. Bach's Well-Tempered Clavier, the following experiences arose. With c = 130.828 and equal-tempered tuning one experienced a world of most beautiful radiance. Nothing more beautiful exists. In striving after this ‘world’ the eurythmist initially feels bigger, like a ‘wonderful soloist’. It is a great feeling! But it deludes one; this world’ of most beautiful radiance is outside of him and his true being.11 It proves to be an unattainable illusion, and the human being loses his humanity, his freedom. If the attempt is made to bring this world of beautiful radiance to expression–and this is only possible in that the eurythmist strives after this world, because it does not come to him–then he tires very quickly. For the observer, the eurythmist looks tense and small and the movement angular. This is due to the strong muscle tension which is induced. A group of eurythmists move very inharmoniously together.

With the twelve fifth-tones tuning on c = 128 Hz the eurythmist feels as if he is being invited to move. He moves his arms and the world streams in. The eurythmist feels himself in balance between centre and periphery. The space between the arms is also filled, bringing about relationship and conversation between the arms. The arms find a connection to one another. The eurythmist does not need to strive after something. Rather he is left free to approach the music with questioning interest. He then receives an answer. He experiences a grace-given peace, peace as joy and quiet at the same time. Seen by the viewer, the movement is peaceful, but big, and the periphery is filled. A group of eurythmists move together harmoniously. In eurythmy terms one would say that the etheric surrounding and streaming is full and evident.

Both pianos were in the same practice room, so comparisons could easily be made. Sometimes a period of adjustment from one piano to the next was necessary-after the brilliance and tension of equal-tempered tuning and c = 130.828 Hz (a = 440 Hz), the twelve

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11 This feeling of outside of one, is not just caused by the pitch but also arises through equal tempered tuning. This feeling is possibly the reason many concert goers experience the music to be distant, disconnected to one. Only accomplished musicians are able to overcome the quality of the pitch and the tuning through their inner activity.

Equal tempered tuning may leave one free because the archetypal music does not immediately sound. But the eurythmist needs tones and intervals which bring to expression what he or she shows in the gesture.

fifth-tones tuning and c = 128 Hz could occasionally feel empty. Here the above-mentioned feeling of questioning interest was the key to finding the new quality. On other occasions the change was experienced as a relief. A change the other way around was mostly experienced as being unpleasant. On one day we would begin with the one piano and then we progressed to the other; the next day we started with the other one. Sometimes we stayed with one piano for the whole session.

On one occasion the pianist told how the one piano required to be played very differently to the other and suggested that she could try to swap the difference around, playing on the one as if for the other. The effect was immediate. Playing on twelve fifth-tones tuning as if for equal-tempered tuning, the falseness was clearly observed. The feeling was like having caught a thief red-handed. The other way round gave the feeling that what was wanting to be musically created was being attacked and destroyed by the equal-tempered tuning, or, seen from the other angle, just did not speak–like an unsalted meal.

A piano tuned to a pitch correspondingly lower than c = 128 Hz as c = 130.828 Hz is higher (c = 126 Hz) was not available. but comparisons were made with tuning forks and tones played on a monochord. With the tone c = 126 Hz as prime, the eurythmist feels as if he is pressed so far into his body that the air is squeezed out of him. The head area feels very small and the inner movement weighs downwards. One is drawn through the feet into weight.

These experiences show that a relatively small difference in pitch and method of tuning have a big effect on our experience. The Middle Tuning is a tremendous discovery and contributes to the expansion of musical expression and is an enrichment to musical sensation.

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Explanation of Terms• What is the duodecimo row? This is an independent row within the overtone row of

C, in which the tones appear in the proportion of 3:1 (a duodecimo), e.g. C, g, d2, a3… or the 1st, 3rd, 9th, 27th… overtones. Tones, which belong to this row, are also called fifth tones. Two of the pitch specifications belong to this row: the tone a1 = 432 Hz is the 4th tone of the duodecimo row beginning on 2C = 16 Hz (c = 128 Hz is the 3rd octave of this C).

• What is Pythagorean tuning? This is the tuning based on perfect fifths. A characteristic of this tuning is the relatively large Major third, which sounds bright, and to unaccustomed ears almost sharp.

• What are geometric mean tones? There are three means that build musical intervals: the harmonic, arithmetic, and geometric mean. In the octave C–c the fifth, G, is arrived at through the arithmetic, the fourth, F, through the harmonic and the geometric mean gives the middle between g and f12, to the equal tempered tritone f sharp or g flat. However this tone is the middle of the octave and is therefore neither the sharpened tone f sharp nor the flattened tone g flat, but the middle of the two. Therefore Maria Renold gave these geometric mean tones new names: Gelis (for the equal tempered tritone g flat/f sharp), Alis, Belis, Delis and Elis.13 The third tone specification gelis1 = 362.4 Hz is therefore a geometric mean tone.

• What are formed intervals? This is the name given by Maria Renold to the intervals, which lie between a diatonic tone and a geometric mean tone. E.g. b–gelis (formed fifth).

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12 Musically, the middle of an interval is arrived at through the geometric mean. I.e. the resulting intervals are equally large. (Purely numerically, the middle is built through the arithmetic mean.)

13 In German a flattened tone is called Ges and a sharpened tone Giss. The ending (e)lis is derived from these two terms.

BibliographyGlöckler, G., & Glöckler, M. (2007). Das Musikalrische geheimnis des Platonischen

Weltenjahres. In A. Husemann (Ed.), Menschenwissenschaft durch Kunst Verlag Freies Geistesleben.

Jorgensen, O. (1991). Tuning: containing the perfection of eighteenth-century temperament, the lost art of nineteenth-century temperament, and the science of equal temperament, complete with instructions for aural and electonic tuning East Lansing, Mich.: Michigan State University Press.

Renold, M. (2004). Intervals, scales, tones and the concert pitch c = 128 Hz (B. M. Stevens, Trans.). Forest Row: Temple Lodge.

Schlesinger, K. (1939). The Greek aulos: a study of its mechanism and of its relation to the modal system of ancient Greek music. London: Methuen.

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