© The Author(s), Published by EDP Sciences, 2024

1 Introduction

Some musical instruments contain a series of tone generators whose pitch is deliberately slightly out of tune with unison. It can be postulated that the knowledge and art of instrument making has evolved empirically over centuries, with builders aiming for a pleasant tone; therefore, the purpose of including acoustic beats is to enhance sensory consonance [1]. Indeed, a recent evaluation of listening tests with a large number of participants demonstrated that slowly beating dyads were perceived as more pleasant than pure intervals without beats [2]. This article examines acoustic beats from a variety of perspectives, including their beat frequency, the frequency resolution of our hearing, the aural perception of detuned dyads regarding pleasantness, and the tuning progression curves of Accordions and Pipe Organs.

The phenomenon of acoustic beats occurs when two tones containing similar frequencies that differ by a few Hertz are simultaneously present. Upon entering the ear, the percept is that of a fluctuation in loudness of one tone at the average frequency, because the auditory system cannot resolve different frequencies within the range of one frequency difference limen. Some musical instruments are designed to generate intentional beats by producing two tones simultaneously, whose frequencies differ just slightly to create beats, sometimes referred to as the “celeste” effect. In the Pipe Organ, the term Celeste rank refers to a set of pipes that has been tuned in a sharp or a flat manner with respect to the true pitch. Such a rank is “purposely tuned slightly sharp or flat to the pitch of the organ, so that, when drawn with another stop, a pleasant undulation of tone is induced” [3]. Several examples occur within the families of organ tone: a flat-tuned Flute (often named Unda Maris), the Diapason Celeste (Voce Umana in the Italian organ tradition) and, most common, String Celestes (such as the Voix Celéste or the Violin Celeste). An audio example of these three families of organ tone is provided in the description of Figure 11.

Pipe Organs do not make use of Celestes in reed ranks, because this would necessitate significant effort to maintain the tuning, which is less stable in reed pipes compared to flue pipes [4]. In contrast, Reed Celestes are frequently observed in Reed Organs and Accordions. To illustrate, selecting the Violin stop of the Accordion draws two reed ranks (one tuned to true pitch, the other slightly sharp), while the Musette stop activates three ranks (tuned flat, true and sharp or, alternatively, true, sharp and very sharp). The beat frequency of Celestes usually typically ranges from 0.5 Hz to 15 Hz, with many instrument builders aiming for a gradual rise in pitch.

The audio recording (1) demonstrates the effect of a Reed Celeste on the song “Sous le Ciel de Paris”. The first phrase is played with one rank, while the second is played with the Violin stop. The tuning of the sharp rank is depicted in Figure 7.

Although the beating tone is a consequence of a detuned unison interval, it remains sounding consonant (for slight detuning) and an additional chorus effect emerges (as is known from a group of strings instruments playing unison), thus allowing the sound to appear not only undulating or shimmering but also fuller.

The following Sections present a review and comparison of the literature with the author’s own observations, addressing four questions as follows:

Section 2 reviews the origin of beats in pure and complex tones in order to clarify the term “beat frequency of a complex tone”. Furthermore, it establishes a relationship between beats, fluctuation strength and undulating tonal timbre. The final part draws attention to other studies in which complex tone dyads that deviate from just intonation have been found to sound pleasant despite their beats.

Section 3 presents measured data and “recipes” for pitch-dependent tuning of Celeste ranks in Pipe Organs and Accordions at which instrument builders have arrived to achieve sensory pleasantness.

Section 4 integrates the rules for Celeste tuning with the theoretical considerations on beats and consonance. This enables the derivation of tuning progression rules for Celestes in musical instruments.

Finally, the conclusions are presented in Section 5.

2 Theoretical considerations

2.1 The beat frequency of complex tones

For the sake of simplicity, we will initially focus on the unison interval, which is the simplest dyad. Two pure tones of similar frequencies, f − Δf/2 and f + Δf/2 (with Δf << f), generate the beat frequency Δf, which typically lies in the order of a few Hertz in this study. In the case of two harmonic complex tones, several frequencies of the form n∙(f − Δf/2) and m∙(f + Δf/2) occur, resulting in multiple beats with the frequencies nΔf (for n = m and integers n and m). The lowest beat frequency is equal to Δf and corresponds to that of the primary beats, which are most clearly perceived [5] in holistic listening. By focusing on one of the lower partials (spectral listening conditions), it is possible to to recognize the beat frequency nΔf of the n-th harmonic (i.e. the n-th partial). This is generally easier for low harmonics and may extend up to the tenth partial. In listening tests with diotically presented harmonic complex tones, in which the amplitude of one of their partials was modulated, the undulating partial was correctly identified with at least 75% probability approximately up to the tenth partial [6]. Figure 1 depicts the situation for two harmonic complex tones with three odd partials.

Figure 1 The beats between the partials of two complex tones with odd harmonic partials, whose fundamental frequency differs by 1 Hz are heard separately in spectral listening (1 Hz, 3 Hz and 5 Hz). In contrast, the beat frequency perceived in holistic listening equals 1 Hz. In the corresponding audio recording (2) the frequencies are added in the order f1, f3, f5 (first harmonic complex tone) followed by f1’, f3’, f5’.

Let us now move on to sounds of musical instruments. Figure 2 illustrates the beats of a harmonic complex tone produced by a typical Celeste rank in a Pipe Organ. The chart depicts the harmonic partials of the sound spectra of two organ pipes, both at pitch C₅. One pipe belongs to the rank Viol di Gamba 8′, while the other is from the Viol Celeste 8. Both ranks are part of the “St. Anne’s” digital pipe organ sample set, from which their long-term average sound spectra have been extracted (see [7] for details). The Pipe Organ in question originally tuned to f(A₄) = 436.5 Hz, which equates to f(C₅) ≈ 519 Hz. To illustrate the resulting beats over time with round numbers, Figure 3 plots both waveforms as if the frequencies of the two pipes were 499.0 Hz and 501.0 Hz, resulting in an exact 2.0 Hz beat frequency. The sound pressure levels (SPL) are plotted on a relative scale, with the magnitude of the first harmonic of the Viol di Gamba 8′ rank set at 60 dB, a value comparable to a listening position situated in close proximity to the pipe.

Figure 2 The sound spectra of the Viol di Gamba 8′ and the sharp tuned Viol Celeste 8′. The frequency shift is exaggerated in the diagram to facilitate the clear separation of the peaks of both ranks. Only the steady part of the sound spectra has been utilised to extract the frequencies with an averaging time of 10 s. In the corresponding recording (3), the note C5 is played first on the Viol di Gamba 8′, then on the Celeste rank alone and finally on both ranks yielding 1.94 Hz beat frequency. The beats with a frequency of 3.88 Hz are the result of interference between the second partials.

Figure 3 The magnitudes of the harmonic partials and the total magnitude exhibit periodic fluctuations over time (top chart). The two selected timbre parameters, plotted in the bottom chart, are discussed in detail in Section 2.2.

In order to address the initial question of the beat frequency of a complex tone is, the top chart of Figure 3 provides two potential answers. The total magnitude (dashed curve) exhibits a periodic variation with a beat frequency of Δf = 2.0 Hz, which is characteristic of holistic listening. However, in the condition of spectral listening the perceived beats occur in integer multiples of 2.0 Hz depending on which partial the listener concentrates. It should be noted that the magnitude of the fundamental (n = 1) oscillates with the same frequency as the beat frequency Δf of the complex tone (black curves). To avoid confusion, it should noted that the term “beat frequency” will be used throughout this article to refer to the quantity written as Δf.

In the case of dyads other than the unison interval, it is necessary to review a second origin of beats, which are termed “secondary beats” or “beats of mistuned consonances” [8]. Two tones whose fundamental frequencies f⁽¹⁾ and f⁽²⁾ are closely related to an integer ratio m/n, as defined by equation (1), generate the beat frequency Δf [8].

(1)

To illustrate this relationship, consider the major third (m/n = 5/4) and select 400 Hz and 500 Hz as the frequencies of the pure interval (corresponding approximately to the dyad G₄−B₄). This yields Δf = 0, which is the desired outcome in just intonation. However, a slightly stretched major third with fundamental frequencies at 280.0 Hz and 351.4 Hz beats with Δf = 5.6 Hz. It is also worth noting that secondary beating occurs even with pure tones, i.e. in the absence of the m-th and n-th partials, which would otherwise interfere with each other. For harmonic complex tones, the resulting beat frequency Δf is equivalent to that observed in spectral listening conditions, with a concentration on the lowest nearly coinciding partials (n = 4 and m = 5 in this case).

Beats are undoubtedly helpful for the tuning of instruments by ear. For instance, unison and octave intervals in Pipe Organs are tuned in a pure manner, that is, without beats. Conversely, all other intervals that deviate from integer ratios in 12-tone equal temperament (12-TET) exhibit the presence of secondary beats. Tabulated values of the beat frequencies (e.g. as number of beats within ten seconds) serve as a template to lay the desired temperament, which is then applied to the entire instrument for tuning. Precise tuning involves analytic listening to identify the secondary beat frequency Δf directly from the interference at the almost equal frequencies of the m-th and the n-th partial.

2.2 Beats, fluctuation strength and periodic change in tonal timbre

The psychoacoustic model of “fluctuation strength” correlates the beat frequency Δf with the temporal difference of the sound pressure level (or the change in magnitude) caused by the undulation [9]. It should be noted that fluctuation strength F and magnitude ΔL are proportional to each other and that a beat frequency of Δf = 4 Hz generates maximum fluctuation strength as expressed by an empirical function with band-pass characteristics [9], as given in equation (2):

(2)

With complex harmonic tones instead of pure tones, the fluctuation strength can decrease due to the contribution of even partials (n = 2, 4, 6, …). The reason is that the nodes of the latter do not coincide with those of the fundamental, which consequently reduces the change in magnitude ΔL. This phenomenon can be observed by comparing the black curves in Figure 3. It should be noted that the fluctuation strength F is proportional to ΔL. Therefore, fluctuation strength of undulating complex tones without odd partials (as with Stopped Flute ranks [7]) or of tones with comparably weak harmonics (as in an Open Flute of wide pipe-scale [7]) is larger than for Diapason ranks [7], String ranks [7], or Accordion reeds, all of which are rich in harmonic content. Given that String ranks are preferred over Flute ranks in order to achieve the Celeste effect in Pipe Organs, it can be surmised that neither Pipe Organ nor Accordion builders are concerned with attaining maximum fluctuation strength, but rather with the periodic alteration of tonal timbre.

If the distinctive auditory quality of Celeste ranks is attributable to their sound spectrum, it is worth to consider two harmonic complex tones with similar fundamental frequencies (as illustrated in Figure 2). In such a dyad the frequency of the n-th harmonic is an integer multiple of the fundamental frequency, that is, f = f₁. Furthermore, each partial (whose frequency is f_n = nf) undulates with its individual beat frequency, nΔf. The SPL of all odd harmonics reach a common minimum twice per cycle; at these times, only the even partials contribute to the spectrum. This results in a periodic change of the spectral centroid c (which is noticeable as “brightness”) calculated by equation (3).

(3)

Furthermore, the slope parameter s is a valuable quantity for organ pipes [7] and Flutes [10], as it correlates with the string quality of the sound, which can be described on a scale ranging from “more fundamental” to “less fundamental” [7]. The calculation is based on the concept of deriving a mathematical measure of the average slope of the sound spectrum. In order to achieve this, it is necessary to sum up all magnitude differences between two consecutive harmonic partials, using a weight factor to reduce the contribution of higher partials, as defined by equation (4).

(4)

In the case of acoustic beats, this quantity varies periodically over time. The origin of the ripples visible in the bottom chart of Figure 3 is a consequence of equation (4) taking no other magnitudes than those of the harmonic partials L_n into account. This makes the slope parameter sensitive to extinguished partials (where L_n → −∞).

The spectral centroid and the slope undergo periodic changes over time, resulting in corresponding variations in the related sound qualities and, consequently, the tonal timbre. For the selected combination of the Viol di Gamba 8′ and the Viol Celeste 8′ (as depicted in Figure 2), the normalised spectral centroid is equal to c/f = 2.67, without beats, and varies within the range 2.12…3.55, with beats. In this example, it is evident that in addition to the variation of tonal timbre, there is a fluctuation in loudness. A closer inspection of the lower chart in Figure 2 reveals that two principal maxima occur in the course of c/f within one beat cycle, while the main maximum of the slope parameter appears only once per period – every 0.5 s in this example. Since this is equivalent to the beat frequency perceived in holistic listening, the slope parameter is a potential candidate to describe the undulating tonal timbre.

It appears that the objective of Accordion and Pipe Organ builders is to achieve a timbre that exhibits a certain degree of fluctuation, or “shimmering”, when incorporating a Celeste rank that is rich in harmonic content. In contrast, a Flute Celeste can only introduce the effect of loudness fluctuation, but it lacks the harmonic content (i.e. the SPL at the harmonic partials) to vary the timbre significantly.

The playing of a single tone on a Flute rank (for which the spectral centroid is approximately equal to the fundamental frequency, i.e. c/f ≈ 1.0 above C₂ [11], Figure 6) with the tremulant (a device to periodically vary the wind pressure of the pipe organ) can result in a timbre that sounds similar to that of a Flute Celeste. In both cases the loudness varies periodically, whereas the changes in the timbre parameters c/f(t) and s(t) are too small to be noticeable by ear. In contrast to the undulation of loudness (and to a small extent that of frequency) caused by a tremulant, the Celeste ranks facilitate the adjustment of the beat frequency freely upon pitch across the tonal compass.

2.3 The relation of beats to sensory pleasantness

Various models for consonance of dyads have focused on the frequency ratios of the two fundamental frequencies involved and group into the following categories: periodicity (harmonicity [2] or compactness [12–14]), interference (acoustic roughness) and cultural familiarity (see [15] for an overview). According to approaches based on periodicity, intervals with frequency ratios consisting of small natural numbers (such as the integer fractions 1:1, 2:1, 3:2, etc.) are traditionally regarded consonant (see the historical review in [16]). Even a slight frequency deviation is sufficient to generate dissonance. A good example is the ongoing debate about how to divide the octave most effectively into twelve semitone steps in order to obtain the most pleasant temperament. Another group of theories assumes that auditory roughness caused by secondary beats generates dissonance (as in [16]). For harmonic complex tones, these “interference models” [2] or “roughness approaches” [12, 14] conclude that the consonant intervals mentioned above exhibit the least dissonance while dyads in just intonation are the most consonant [16]. However, dyads of inharmonic complex tones are perceived as least dissonant in intervals of non-integer ratios, and thus yield to scales that differ from just intonation [17].

The psychoacoustic quantity “sensory pleasantness” describes the extent to which a sound is perceived as pleasant. This sensation is contingent upon the properties inherent in the sound spectrum, including acoustic roughness, sharpness, loudness and tonality [9]. According to this model, the first three quantities have a negative effect on sensory pleasantness, whereas tonality has a positive effect. Consequently, the maximum value of sensory pleasantness is generated by a pure tone. The cited model [9] does not consider further influences, such as cultural familiarity with sounds or the possible influence of fluctuation strength or acoustic beats. However, in order to differentiate between the terms “consonance” and the perceived sound quality, the latter will be referred to as “sensory pleasantness” or “pleasant” in short.

Recent studies have demonstrated that models for perceptual tests of the consonance of dyads and triads exhibit significant improvement when they combine both the harmonicity and the interference approach, rather than just adhering to a single approach [2, 12, 13, 15]. Furthermore, it is possible to extend such a model by incorporating a frequency tolerance, whereby the consonance of pure intervals gradually changes into dissonance within the range of one frequency difference limen (cf. [12] and Fig. 3 in [14]).

Two methods yield similar values for the frequency difference limen. One approach is to estimate is its value as 1/30 of the critical band (as in [12]); an alternative is to equate it with the just noticeable variation in frequency (JNVF). In this study, the JNVF measured using continuous tones [9] has been taken to represent the frequency difference limen. To obtain a frequency-dependent expression for the JNVF an empirical function was fitted to the depicted data [9] resulting in equation (5), which is valid in the range 50 Hz < f < 20 kHz and is plotted in Figure 3 as a dotted curve.

(5)

where f is the frequency and a_i are the following coefficients:

a_–1 = 7.76, a₀ = –19.6, a₁ = 20.2, a₂ = –9.57, a₃ = 2.12, and a₄ = –0.17 yielding R² = 0.99999.

The inclusion of a frequency tolerance in the order of one JNVF is of paramount importance in order to comprehend why musical intervals consisting of harmonic complex tones from two or more sound sources, which are rarely tuned perfectly pure in practice (e.g. in musical performance), can still be perceived as pleasant. For illustrative purposes, consider two harmonic complex tones with different fundamental frequencies, but otherwise identical sound spectra. These two tones elicit a certain degree of sensory pleasantness, when heard in isolation, as well as when combined. Let us consider three special cases:

A slightly stretched unison interval, comprising two of the aforementioned tones with frequencies of 499 Hz and 501 Hz, continues to produce a pleasant auditory sensation. The beat frequency Δf = 2 Hz is too low to cause auditory roughness, which would reduce sensory pleasantness [9], and it is far below the JNVF at 500 Hz (4.2 Hz). The deviation from the unison interval becomes apparent to listeners through the temporal undulation that is associated with a specific fluctuation strength. In the case of analytic listening, beats of the harmonic partials with multiples of 2 Hz also appear.

The secondary beats of a marginally compressed octave (f⁽¹⁾ = 280 Hz, f⁽²⁾ = 558 Hz) occur with Δf = 2 Hz (setting m = 2 and n = 1 in equation (1)). The beat frequency between the second partial of the lower tone and the fundamental of the higher tone is less than the JNVF at 560 Hz. Consequently, this interval remains to be perceived as pleasant.

The stretched major third (“stretched” with respect to just intonation with f⁽¹⁾ = 280.0 Hz and f⁽²⁾ = 351.4 Hz), which has been previously discussed in Section 2.1 generates secondary beats with a frequency of 5.6 Hz. Given that the JNVF at the interfering partials (1.4 kHz) is 8.0 Hz, the sensory pleasantness is retained.

In 12-TET, the only dyads between unison and octave without beats are the perfect unison (P1) and the perfect octave (P8), which refer to 100 ct and 1200 ct, respectively. The major second (M2), the perfect fourth (P4) and the perfect fifth (P5) exhibit secondary beats with frequencies smaller than the JNVF. This is consistent with the straight lines not intersecting the JNVF curve in Figure 4. In contrast, all other dyads generate secondary beats with beat frequencies greater than the JNVF, provided that the pitch of their lower note lies above the value listed in Table 1. The secondary beats are more pronounced in dyads comprising complex tones, where the lower one contains the m-th harmonic partial and the higher note includes the n-th harmonic partial in its sound spectrum, in comparison to pure tone dyads.

Figure 4 This graphical representation illustrates the relationship between the beat frequencies and the pitch of the lower tone of the dyads listed in Table 1. Chart (a) depicts all dyads tuned flat with respect to just intonation, while chart (b) contains the sharp intervals. The lower dotted curve represents the JNVF; beyond 2∙JNVF detuning becomes obvious. Most dyads in 12-TET are tuned so far from just intonation that they exceed the range of one JNVF. The dash-dotted straight lines represent the results of experiments with harmonic complex tones (3 dB/8ve spectral roll-off). In these experiments, listeners adjusted the upper tone of three dyads (M3, M6, P8) until the interval achieved the highest “pleasantness” while the lower tone fixed in the pitch range G3…F4 [2]. The resulting dyads are a flat m6 (878 ct), a flat octave (1194 ct), a sharp M3 (395 ct), a sharp M6 (893 ct) and a sharp octave (1208 ct). The dash-dotted curve represents the preferred average of pitch-dependent octave stretching for orchestra instruments with harmonic partials (Fig. 2 in [19]). It should be noted that for octaves, whose lower pitch is higher than B6, the octaves are enlarged by more than one JNVF in this tuning.

One example may be used to illustrate how to read the diagrams of Figure 4. The major third G₃-B₃ in 12-TET which is a sharp-tuned interval (400 ct) with respect to just intonation and thus included in chart a). In this dyad, the fifth partial of the lower tone interferes with the fourth partial of the higher tone (at pitch B₅). This is represented by the line labelled “M3” and the pitch B₅ on the abscissa. The beat frequency (7.8 Hz) is already greater than the limit given by the JNVF, which consequently results in a reduction in the sensory pleasantness of the M3 interval. A more expedient method for ascertaining whether the beat frequency exceeds the JNVF is to check if the pitch of the lower tone exceeds that listed in Table 1 (A₂ is case of the M3 interval).

The value of the JNVF value can be assigned to the onset of the decrease in sensory pleasantness. Beyond 2∙JNVF dyads are estimated to sound unpleasantly out of tune (cf. Section 4.1).

The results of several online listening tests [2] indicate that pure tone dyads (M3, M6, P8) are perceived as most pleasant when their frequency ratios are exactly 5:4, 5:3 and 2:1, respectively. It is interesting to note that the sensory pleasantness of the same dyads formed by complex tones (–3 dB/oct slope of the SPL) increased when the test subjects adjusted the interval sizes of the M3, M6 and P8 dyads not to integer frequency ratios (pure intervals), but to a small deviation from the values shown in Figure 4 [2]. The aforementioned dyads are represented by straight lines in Figure 4 and are constrained to the investigated pitch range G₃-F₄ of the lower tone [2]. This includes 280 Hz, which has been selected for the compressed P8 and the stretched M3 in the aforementioned examples. Moreover, the beat frequencies used in the previous examples are in close proximity to the two most pleasant-sounding intervals identified in the corresponding listening tests [2]. From the position of these maxima, the authors conclude that a theory to describe pleasant-sounding dyads must include both the “interference” and the “compactness” approach as well as a “liking of slow beats” as a positive influence [2].

The previous demonstration of the M3 and P5 [18] has shown that slow beats are not the dominant cause of the perception of “out-of-tuneness”. Furthermore, orchestra players of wind instruments (woodwinds and brass) and string instrument players prefer stretched tuning (i.e. >1200 ct for the octaves) [19], despite the fact that their instruments generate harmonic complex tones rather than tones with inharmonic sound spectra like the piano does. While the phenomenon of stretched tuning in the piano [20, 21] can be attributed to the inharmonic partials resulting from the mechanical stiffness of the strings, neither the preference for detuned octaves observed in listening experiments with harmonic complex tones [2] nor the use of stretched octave tuning in orchestral instruments [19] can be explained in this manner. The degree of stretched tuning in the symphony orchestra is comparable to that observed in piano tuning [20], with the stretched octaves becoming larger by more than one JNVF in the treble, at pitch B₅ (≈1.0 kHz) and above. This may be related to the psychoacoustic phenomenon whereby timbre affects perceived pitch (as demonstrated for the Viola, Trumpet and Tenor Voice at the pitch A₃ [22]). The rationale for this is likely to be the relationship between pitch and the pitch-dependent sound spectra of (classical) musical instruments. In general, the harmonic content and thus the normalised spectral centroid (c/f) decrease with increasing pitch. This effect is also evident in the Pipe Organ across all its tone families [7]. Consequently, the pitch of complex tones with unfamiliar high harmonic content tends to result in a perception of a higher pitch than the fundamental pitch.

In conclusion, the evidence presented in this section indicates that slow beats enhance the sensory pleasantness of dyads relative to just intervals. This statement holds for harmonic complex tones, but is possibly also true for inharmonic complex tones. This explains why listeners prefer complex tone dyads that deviate from just intonation including the practice of octave-stretched tuning in orchestras. In addition, the practice of instrument builders of incorporating deliberately detuned Celeste ranks into musical instruments such as the Accordion and the Pipe Organ is related to a psychoacoustic phenomenon.

3 Data on pitch-dependent tuning of Celestes

The selection of an appropriate beat frequency is largely dependent on personal taste and influenced by the style of the music. Furthermore, the acoustic properties of the room also play a role, since a long reverb time will even out the time-varying fluctuation strength and the undulating tonal timbre. Nevertheless, an analysis of the beat frequency progressions upon pitch as adjusted by Pipe Organ and Accordion builders reveals some rules for pleasant beating tones reflecting certain properties of the auditory system with respect to sensory pleasantness.

3.1 Celestes in the Pipe Organ

There are several possible approaches to tuning a Celeste rank for the Pipe Organ. One option is to maintain a constant beat frequency (a), which contrasts with the alternative of raising the beat frequency upon pitch to ensure pure octaves within the Celeste rank (b, c). An additional possibility (d) can be considered to lie somewhere between the aforementioned extremes.

Type a)

The objective of this method is to achieve a constant beat frequency (e.g. Δf = 2 Hz) between the Celeste and the unison rank, which is maintained throughout the entire tonal compass. Consequently, the octaves of a sharp Celeste rank are slightly compressed (i.e. flat), although this is still tolerable. As the beat frequency is independent of pitch, the effect is comparable to that of a tremolo with the frequency Δf. For ranks with little harmonic content such as Flute ranks, this results in a certain degree of fluctuation strength, but not a discernible timbre undulation, as the higher partials are insufficiently pronounced [7].

Type b)

The recipe for this variant is as follows: Adjust the beat frequency at a reference tone, say A_4, to the desired value. Subsequently, keep the Celeste rank “dry”, that is, in tune with itself, in order to ensure that its octaves remain pure intervals. The organ builder E.M. Skinner proposed this method for Celeste ranks, tuning 1.7 Hz sharp at pitch A₄ [23], which yields 1.0 Hz at C₄. This method adheres to the principle of tuning all octave intervals in a Pipe Organ to be pure, thereby resulting in a doubling of the beat frequency with each octave. For a key compass of 61 semitones, five doublings occur. Consequently, the beats in the bass may be perceived as too slow while that in the treble range may be considered as too fast.

Apart from these “extreme” recipes, other variants are in use, such as:

Type c)

This option creates a tuning known in German organ building as “Terzschwebung”. It requires the pitch of the pure major third of a Tierce 1³/₅′ rank, which is frequently available in Pipe Organs. The result is a Celeste rank with pure octaves, but tuned in a flat manner with respect to true pitch. This variant is applicable in unified pipe organs [24], in which the same rank of pipes is required to fulfil multiples purposes in order to restrict the number of pipes and thus save building space. For example, the F₂ pipe of the Tierce (with pitch A₄) borrows its pitch to the Celeste rank, which does not have an A₄ pipe of its own. Apart from the difficulty of voicing such a rank to satisfy the duties of a Tierce and a Celeste, the beats are often considered as being too fast in the treble, with 3.5 Hz at A₄. If the Tierce and the Celeste are available as independent ranks, it is possible to tune the Celeste rank either flat or sharp by playing major third intervals on the keyboard while muting the upper pipe in the Tierce and the lower pipe in the Celeste rank, or vice versa.

Type d)

A reasonable compromise between methods a) and b) would be to double the beat frequency at a lower rate, i.e. slower than every octave, while maintaining Δf = 1.7 Hz at A₄ as proposed by E.M. Skinner [23]. In the event that chords are to contain multiple distinct beat frequencies (one for each tone), it is recommended that the interval size after which the beat frequency doubles should be set to a rarely used interval such as 1.5 octaves (the diminished twelfth). With the exception of this infrequently occurring rare interval, the majority of chords will generate beat frequencies that are not integer multiples of each other.

The four variants depicted in Figure 5 demonstrate that the beat frequency of Celeste ranks in Pipe Organs is either constant (type a), or it increases exponentially with pitch. It is of paramount importance for the artistic result to select an appropriate tuning progression curve is critical, as “[t]he effect of a Celeste depends greatly on the “tempering” or “grading” of the tuning through the compass” [3].

Figure 5 Four variants to tune Celeste ranks in the Pipe Organ described as types (a–d) in the text.

3.2 Celestes of the Accordion

It is common for Accordions to contain one or even two Celeste ranks, which serve to enrich the sound. Several traditions for Celeste tuning exist, which can be described as ranging from “dry” to “wet” (referring to Δf ≈ 0.5…7 Hz at A₄₎. It should be noted that knowledge of this value alone is insufficient to describe the course of the tuning progression curve, i.e. how the beat frequency increases upon pitch.

Figure 6 illustrated the tuning of Celeste ranks in two different Accordions and one Harmona (a keyboard instrument with 49 Accordion reeds). The smaller Accordion (manufactured by Hohner, model: Concerto III T) is equipped with three sets of reeds at 16’, 8’ and 8’ pitch; the latter tuned 4.1 Hz sharp at A₄ to generate the Violin stop. The larger Accordion, manufactured by Borsini (model: Borsini Super Star 2000 VE), contains five sets of reeds, each tuned at 16’, 8’, 8’, 8’ and 4’ pitch. The two 8’ Celeste ranks are tuned 3.7 Hz flat and 4.1 Hz sharp, respectively; the remaining 8’ ranks represents true pitch. The combination of the three 8’ ranks yields the Musette stop. In both Accordions, the measured beat frequencies correspond to the average of the two reeds per pitch. One reed acts when the bellow is under pressure, while the other acts when the bellow is being pulled. Figure 6 clearly indicates that both Accordion builders aimed for a linear increase in the beating frequency upon pitch resulting in straight lines.

Figure 6 Tuning progression curves of the Celeste ranks in two different Accordions and a Harmona (an instrument with Accordion reeds, but with an electric blower rather than a hand-pumped bellow.).

The Harmona (Musikus Studio, 16’, 8’, 8’) contains the same ranks as the smaller Accordion. However, the beat frequency of the Violin stop increases exponentially with pitch, before reaching a maximum. The measured values for the highest octave exhibit considerable variability. Following a period of more than 50 years without tuning. However, it seems reasonable to posit that the maximum beat frequency remains constant at approximately 10 Hz.

In summary, the experimental data reveal another important case:

Type e)

In this version, the beat frequency increases linearly in relation to the pitch. This linear dependence appears to be at odds with the intuitive understanding of frequency, which rises exponentially with pitch. Consequently, it would be expected that the beat frequency would also increase exponentially. Nevertheless, the use of such linear progressions may be justified in practice for two reasons: Firstly, their calculation does not require the use of logarithmic calculations; secondly, the difference between a straight line and an exponential curve is almost negligible for a key compass of two to three octaves.

In the other case where the phenomenon was observed, the beat frequency exhibited exponential growth (as previously described in variant b). The measurement on the Harmona demonstrated a doubling of the beat frequency with each octave, with a maximum of 10 Hz (Figure 6). Another method for tuning the Celeste in Accordions involves doubling the beat frequency approximately every 1.5 octaves corresponding to type d).

Listening tests, conducted with the use of pairwise comparisons between Accordions with differently tuned Celestes, indicated that the beat frequency should increase upon pitch, although at a slower rate in the bass range than in the treble range [25]. The findings of this experiment indicated that the tuning progression curve should not exceed the JNVF curve and should roughly follow its course while always increasing, even in the bass range [25]. Based on these assumptions a curve approximately linear with pitch below 500 Hz and increasing exponentially at higher pitch has been proposed [25] (named “lin-exp” in Figure 7) and filed as patent [26]. Nevertheless, in practice, Accordion builders may employ S-shaped tuning progression curves in order to circumvent the steep increase of the beat frequency in the treble. Two examples of such curves fitted to measured data using the hyperbolic tangent function for dry and wet tuning [27] are depicted in Figure 7 alongside with the patented tuning for comparison. It should be noted that some Celeste ranks in Accordions are tuned to a much wetter degree (up to approximately +7 Hz@A4) than the curve for wet tuning might suggest.

Figure 7 Collection of non-linear relations of beat frequency (BF) with pitch including literature data for a dry and a wet Celeste [27] and a proposed tuning progression curve (solid line, “lin-exp type”) for Accordions [25]. The dotted curve represents the exponential curve measured in the Harmona (Figure 4), while the JNVF serves as a comparison.

The solid curve in Figure 7 is approximated by equation (6), which represents a numerical fit to the proposed Celeste tuning for Accordions based on listening tests conducted in the pitch range E₂…A^#₇ [25].

(6)

with b₀ = 1.15, b₁ = 4.42∙10⁻³, b₂ = –3.34∙10⁻⁶, b₃ = 1.63∙10⁻⁹, b₄ = –2.21∙10⁻¹³, R² = 0.99981.

This proposed tuning progression curve for Accordions is based on the idea that the beat frequency should somehow follow the course of the JNVF [25]. However, the limit set by the JNVF is excessively stringent. It is evident that Accordions with a Celeste rank that has been tuned very wetly exceed the aforementioned limit, yet they are still regarded as sounding pleasant. Consequently, it is possible that the aforementioned tuning progression curves may in fact lie above the JNVF curve, and are not necessarily restricted to lie below, as in the case with the proposed optimum [25].

4 Discussion of the data on Celeste tuning

The five variants (a–e) of Section 3 illustrate just a few possibilities for Celeste tuning in Accordions, Pipe Organs and related musical instruments. Nevertheless, if we assume that the objective of instrument builders is to achieve a pleasant sound, the tuning progression curves should be aligned with models of sensory consonance.

4.1 Proposals for Celeste tuning

In order to arrive at a general rule for Celeste tuning it is necessary to formulate four assumptions:

1. The maximum of the beat frequency shall not exceed 15 Hz [27] to prevent auditory roughness [9]. Since such fast beats are uncommon; 10 Hz (as in Figure 4) appears to be a realistic upper limit.

2. The JNVF serves as a “soft” upper bound, that is frequency-dependent. It is crucial to recognise that the JNVF is not a rigid limit. Otherwise the majority of the dyads tuned in 12-TET were dissonant (Figure 4), and any Accordion Celeste ranks tuned much wetter than the JNVF were musically unusable. In practice, a very wet Celeste rank in the Accordion is tuned +7 Hz sharp at A₄ (rarely +8 Hz [28]), where the JNVF amounts to +4.0 Hz. This indicates that the beat frequencies, which are useful in music, should lie approximately below 2∙JNVF, as indicated in Figure 4, Figure 7 and Figure 8.

   Figure 8 Proposed tuning progression curves for Celeste ranks in the range C2…C8 . The dotted curves represent one JNVF and 2∙JNVF. The dashed curve depicts the beat frequency (BF) proposed for Accordion Celestes [25], which rises linearly with pitch for low frequencies (indicated by the dashed straight line) and increases exponentially with pitch (Δf = 0.003∙f) in the treble where it approximates variant b). To facilitate a comparison between the effects of doubling the beat frequency slower than each octave (variants (b) and (c)), namely every 1.5 octaves, the dash-dotted curves have been included.

3. The beat frequency increases upon pitch. An exponential relation appears to be the “natural” choice. To avoid integer ratios of beat frequencies in commonly played chords, doubling the beat frequency every 1.5 octaves is a viable option.

4. The minimum beat frequency in the pitch range, in which melody lines occur is determined by the duration of stressed melody notes. This is because on such tones, at least two beats should occur in order to recognise the periodically occurring beats, rather than just noticing a slight mistuning. The length of these melody notes and thus the tempo of the music is dependent on a number of factors, including the style of music and the reverb time of the room. Pipe Organs are frequently situated in expansive and reverberant spaces, where the echo would obscure the perception of slow beats and render melodic and harmonic progressions indistinct from a certain tempo onwards. Consequently, the selection of an appropriate beat frequency is contingent upon the reverberation time of the room, of individual preference (i.e. personal taste) and it depends on the style of music.

These assumptions permit a brief evaluation of the five tuning variants mentioned above:

Type a)

It is, of course, possible to set the beat frequency to a constant value, independent of pitch. However, this approach misses the chance to give each tone its own, designated beat frequency. The effect is thus similar to that of a tremulant, particularly in the case of Flute ranks.

Type b)

The doubling of the beat frequency each octave, as proposed for Pipe Organs [23], results in fast beats in the treble and very slow beats at low pitch over a key compass of five octaves. In Pipe Organs, there is a risk that the low pipes “draw” each other into tune thereby extinguishing the Celeste effect due to nonlinear coupling [29]. One potential solution to this issue is to place the Celeste ranks at a sufficient distance from each other. In contrast to type a), the tonal timbre of each tone in a chord undulates with an individual beat frequency, which should enrich the sound quality. However, this variant of tuning remains unfortunate insofar, as the beat frequencies of commonly occurring octave intervals are integer multiples of each other for the same reason as stated in a). It should be noted that plotting this variant of a tuning progression curve into Figure 4 would result in a straight line parallel to those of the dyads.

Type c)

The tuning progression curve, commencing with a 3.5 Hz beat frequency at A₄ and doubling each octave, will exceed the JNVF in the pitch range C₅…A₈ (with its maximum at 1.4∙JNVF). This may be tolerable with regard to tuning, but the beats become quite fast in the treble, exceeding 10 Hz at E₆. In the context of fast music and brief reverberation, this may be considered an acceptable solution for Accordions. However, this approach is not optimal for tuning a Pipe Organ Celeste. The plot of this variant would result in a straight line coinciding with that of the major third (M3) in Figure 4b. As this tuning variant exceeds the limit of one JNVF, it intersects the JNVF curve.

Type d)

The doubling of the beat frequency every 1.5 octaves represents an sophisticated choice, given that complex tones played in octaves will not contain beat frequencies that are integer multiples of each other. Furthermore, the range of beat frequencies over a larger pitch range is reduced. For five octaves, this corresponds to the factor 2^10/3 ≈ 10, which can be distributed across the bass range (to increase the beat frequency there) and the treble (to achieve slower beats).

Type e)

A linear course for the beat frequency is incompatible with one of the aforementioned assumptions. Nevertheless, this approach will prove effective in a limited pitch range, specifically below C₅, where the deviation from a non-linear relationship is not readily apparent.

Figure 8 provides examples of tuning progression curves; for further ones see Figure 5 in [1]. It is recommended that the amount of detuning does not exceed the limit of one JNVF, or, in the extreme case, 2∙JNVF. As previously demonstrated in Figure 4, the beat frequencies of numerous dyads tuned in 12-TET exceed the JNVF. Figure 7 depicts an illustrative example of a wet tuned Celeste rank that also lies partly above this limit.

Two personal observations on Celeste ranks are worth mentioning:

Firstly, it can be observed that Celeste ranks with high harmonic content are typically tuned wetter than those of Flute ranks; an impression which is created when comparing the tuning of Pipe Organ Celestes with those of the Accordion. The assertion is corroborated by the findings of recent listening tests [2], which indicate that pure-tone dyads are most pleasing when tuned in just intonation (i.e. without beats), whereas dyads of harmonic complex tones are preferred slightly detuned, as shown in Figure 4, and consequently produce beats. Obviously, the degree of detuning required to achieve maximum sensory pleasantness depends on the harmonic content (at least for harmonic complex tones).

Secondly, single tones appear to be more tolerant of detuning than chords. This phenomenon may be attributed to the fact that a single tone played in conjunction with a Celeste rank corresponds to a slightly detuned unison interval. However, it is notable that the majority of dyads and chords within the 12-TET scale contain detuned unison intervals in addition to detuned dyads (marked as straight lines in Figure 4). The combination of both detuning sources may result in the perception of out-of-tuneness.

These two aspects are of significant importance when determining for the most pleasant beat frequencies at couple of tones within the pitch range of the instrument before connecting them with a smooth tuning progression curve, as exemplified in Figure 8.

The proposed tuning curves depicted in Figure 8 might be more readily utilised together with some tabulated values. Thus, Table 2 provides the values for the lower curve of “type d”, in which the beat frequency doubles every 1.5 octaves and the beat frequency is 1.7 Hz at A₄. The relationship between frequency and pitch is expressed in terms of the pitch category p, which represents the number of semitone steps. For a musical scale in 12-TET, this results in equation (7), where the pre-factor is the frequency of A₄ with the pitch category p(A₄) = 57. In order to facilitate a comparison between the beat frequencies (expressed in Hz) and the pitch (expressed as cent values, which are commonly used in electronic tuners), it is necessary to convert these quantities in accordance with equations (7) and (8).

(7)

(8)

The values are tabulated at pitch distances of 0.25 octaves, which means that the beat frequencies double every sixth row. Tables for alternative tuning progressions are included in the preceding article [1].

4.2 Flat and sharp tuning of pipe organ Celestes

In contemporary Pipe Organs, the majority of Celeste ranks are tuned in a sharp manner, a preference that has been observed since at least 1907 [3]. In contrast, flat tuning, as in the Unda Maris, is less prevalent. In the case of a sharp Celeste, the audible frequency of a rank sounding together with its sharp Celeste is equal to f + ∆f/2. In a sharp Celeste rank that doubles its beat frequency at a rate slower than once per octave (as in “type d”), all intervals are slightly compressed. Nevertheless, this effect is insufficient to shift the positions of the straight lines representing the dyads TT, M3, M6 and M7 in Figure 4 significantly until they no longer cross the JNVF curve and sound fully consonant. In fact the pitch decrease for a Celeste rank tuned in accordance with curve d) in Figure 4 is only a fraction of a semitone, for example 0.07 ct at A₄. Even if the pitch shift were sufficiently large to shift the mentioned dyads into consonace, the intervals in Figure 4a would deteriorate. Therefore, this does not account for the preference of Pipe Organ builders for sharp against flat tuning.

One potential explanation for this phenomenon lies in the pipe-scale (the ratio of pipe diameter to pipe length), which is usually chosen slightly smaller for the Celeste rank compared to the unison rank. In general, narrowing a pipe while maintaining all other parameters constant increases its normalised spectral centroid (c/f) and thus its brightness, as demonstrated for several ranks differing in pipe-scale [7]. Listening tests of harmonic complex tones (Viola, Trumpet and Tenor voice, each at pitch A₃) have shown that tones with a higher spectral centroid evoke a higher pitch height [22]. In order to illustrate this phenomenon, let us consider the two Pipe Organ ranks, whose sound spectra are sketched in Figure 2. It can be expected that a single pipe from the Voix Celeste rank will produce a sound that is somewhat brighter than the corresponding Viola di Gamba pipe. The brightness of their combination will then lie somewhere in between. In the case of a sharp-tuned Celeste rank (as is the case with the majority of Celestes), two effects contribute to the brightening of the sound: the higher spectral centroid of the Celeste and its higher pitch. Conversely, in the case of a flat-tuned Celeste, the increase in brightness caused by the spectral centroid may be partially negated due to the psychoacoustic relationship between (lower) pitch and (less) brightness [22]. Currently, this explanation must remain speculative until further data become available. For the selected example, the values of the ratio c/f of the Voix Celeste are indeed slightly larger than that of the Viola di Gamba rank, but only within the pitch range C₂…A₄, as shown in Figure 9.

Figure 9 Pitch dependency of the normalised spectral centroid c/f for both String ranks in this work, compared with a standard Open Flute [11] and a standard Open Diapason rank [11].

Nevertheless, if this explanation is correct, then the String Celeste and Diapason Celeste ranks should be tuned sharp, as their sound spectra possess significant brightness (i.e. c/f > 1), which is generally true for String and Diapason ranks [7]. It must be noted that these conclusions are contingent upon the Celeste having a smaller pipe-scale than its unison rank. In contrast, the brightness of Flute ranks is almost at the minimum (c/f ≈ 1.0, Figure 9, [11]), meaning that the tuning of Flute Celestes can be either flat or sharp without affecting the perceived pitch height.

4.3 Beat frequencies of more than one Celeste

Some Pipe Organs contain more than one Celeste rank; similarly, larger Accordions may feature a the 3-rank Musette (cf. Figure 6). In order for the Celeste ranks to match in combination, the ratios of their beat frequencies should be small integers and as simple as 1:1 or 1:2. It should be noted that these recommendations only apply, when different Celeste ranks shall be used together with each other.

The reason for the generation of unpleasantly sounding beats when combining Celeste ranks is due to a phase shift occurring in one of the Celeste ranks. To illustrate this phenomenon, consider the case of two ranks, each tuned to 499 Hz and 501 Hz, as depicted in Figure 2. For the sake of simplicity, the amplitudes of the ranks vary sinusoidally over time. As long as only two waves interfere, a phase difference between the two ranks will result in a shift of the interference pattern on the time axis. However, this will not affect the shape of the peaks corresponding to the beats. With the introduction of a second Celeste rank (also modelled by the sine function), the situation changes insofar, as the phase of the additional rank angle changes the shape of the interference pattern. Figure 10 illustrates this difference depending on the third rank being tuned to either 503 Hz or 502 Hz, with the selected phase angle being 3π/4.

Figure 10 The diagrams depict the power of three added oscillations given by the equation 2sin(2π f(1)∙t) + √2 sin(2π f(2)∙t) + sin(2π f(3)∙t + 3π/4). In case (a), only two beat frequencies (2 Hz and 4 Hz) with the ratio 1:2 can be generated from the three given frequencies and both of them are visible as fractions of one period in the top chart. In case (b) three beat frequencies (1 Hz, 2 Hz, 3 Hz) are observed. The combination of them results in a different power distribution among the smaller peaks. The perception of such beats is uneven, reminiscent of stuttering and generally less pleasant than in the upper case. Setting the phase angle to zero results in a regular pattern. However, for independently oscillating tone generators (such as organ pipes) the phase angle remains a free parameter.

In instruments such as the Pipe Organ and the Accordion, the phase angle cannot be controlled. Its value is arbitrary, and thus the beat frequency values of Celeste ranks, which shall sound pleasant in combination, should be restricted to the ratios mentioned above.

To illustrate the potential consequences of combining three ranks (two of which are Celeste ranks), consider the “asymmetrical” tuning: f, f+∆f, f+2∆f. The beat frequencyies for any two combined ranks are ∆f or 2∆f, while for all three ranks sounding together they are ∆f and 2∆f. The ratio of the beat frequencies ratios is 1:2 corresponding to Figure 10a. Other ratios of the two concurrent modulations of all three ranks with arbitrary phase angles result in irregular-sounding beats, as shown in Figure 10b. An alternative approach that is not phase-sensitive is the combination of a flat, a unison and a sharp rank with “symmetrical” detuning (f−∆f, f, f+∆f), as realised in some Musette stops in Accordions. Figure 11 and the accompanying audio recording demonstrate the use of differently tuned Celestes within one music piece.

Figure 11 The last five bars of the Organ piece “Harmonies du soir” (S. Karg-Elert op. 72, reprinted from [30]) demonstrates the three different tone families of the pipe organ (indicated by colour bars) paired with their Celestes as well as several tuning options in (4). Yellow: Violin + Violin Celeste (tuned sharp), first green range: Horn Diapason + Horn Diapason Celeste (tuned sharp), blue: Quintadena Celeste (tuned flat) + Concert Flute Celeste (tuned sharp), final bar (green): Dulciana + Unda Maris (tuned flat). The tuning of the Violin Celeste is adjusted to type d) with +2.1 Hz at A4, while the two Flute ranks (blue) are tuned to type d) with –1.4 Hz and +1.4 Hz at A4, yielding 2.8 Hz beat frequency when used together without a rank tuned to unison pitch. The “green” Celestes ranks could not be retuned, but their beat frequency is about 2 Hz and more or less constant with pitch. The recorded sounds are part of the virtual pipe organ sample set “Paramount 450” (see [7] for details).

5 Summary and conclusions

The undulating sound of musical instruments with Celeste ranks is perceived as pleasant, even more pleasant than without intentional detuning, provided that the beat frequency is sufficiently small. The tolerance of detuning is contingent upon the frequency-dependent JNVF. The incorporation of this frequency deviation into models for sensory pleasantness can elucidate the phenomenon whereby intervals of harmonic complex tones tuned in 12-TET are perceived as pleasant, despite the majority of them deviating from just intonation and generating acoustic beats. Furthermore, this tolerance of sensory pleasantness for beats is a prerequisite for understanding how a group of musicians can evoke a pleasant impression despite numerous small frequency deviations occurring in practical performance of music.

A Celeste rank introduces a loudness fluctuation to the sound, but this is not the only effect involved. As the SPL of the partials within complex tones become time-dependent, the tonal timbre varies in terms of brightness and a tonal timbre related to string quality. The timbre variation is only of consequence in the case of sounds with high harmonic content, such as those produced by String ranks in the Pipe Organ or Accordion reeds. This may provide an explanation for the sought-after effect provided by Celeste ranks.

By comparing the common tuning rules of Celeste ranks in Pipe Organs and Accordions, as well as taking the JNVF values of the auditory system into account, it is possible to formulate guidelines for the tuning of Celeste ranks with a pleasant sound. The practice of tuning Celeste ranks preferably sharp rather than flat may be attributed to the brightness of the resulting sound, which is perceived as more pleasant than that of each contributing rank sounding separately. In the event that different Celestes shall be used simultaneously, their beat frequencies should be equal to the simple ratios 1:1 or 1:2.

The deliberate detuning of Celeste ranks, particularly in the case of Accordions and Pipe Organs, can contribute positively to the sensory pleasantness of their sound, an influence not yet taken into account in this psychoacoustic quantity.

Acknowledgments

I am truly grateful to Mr. Jack Bethards, Chairman and Tonal Advisor of Schoenstein & Co., for portraying some aspects of his approach to the Pipe Organ tone via email. Moreover, I like to thank two anonymous reviewers for their constructive feedback, which has helped to improve the quality of this article.

Funding

The author received no funding to complete this research.

Conflicts of interest

The author declared no conflict of interests.

Data availability statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request. The audio files are publicly available from ResearchGate under the reference [31].

References

All Tables

All Figures

Figure 1 The beats between the partials of two complex tones with odd harmonic partials, whose fundamental frequency differs by 1 Hz are heard separately in spectral listening (1 Hz, 3 Hz and 5 Hz). In contrast, the beat frequency perceived in holistic listening equals 1 Hz. In the corresponding audio recording (2) the frequencies are added in the order f1, f3, f5 (first harmonic complex tone) followed by f1’, f3’, f5’.

In the text

Figure 2 The sound spectra of the Viol di Gamba 8′ and the sharp tuned Viol Celeste 8′. The frequency shift is exaggerated in the diagram to facilitate the clear separation of the peaks of both ranks. Only the steady part of the sound spectra has been utilised to extract the frequencies with an averaging time of 10 s. In the corresponding recording (3), the note C5 is played first on the Viol di Gamba 8′, then on the Celeste rank alone and finally on both ranks yielding 1.94 Hz beat frequency. The beats with a frequency of 3.88 Hz are the result of interference between the second partials.

In the text

	Figure 3 The magnitudes of the harmonic partials and the total magnitude exhibit periodic fluctuations over time (top chart). The two selected timbre parameters, plotted in the bottom chart, are discussed in detail in Section 2.2.
In the text

Figure 4 This graphical representation illustrates the relationship between the beat frequencies and the pitch of the lower tone of the dyads listed in Table 1. Chart (a) depicts all dyads tuned flat with respect to just intonation, while chart (b) contains the sharp intervals. The lower dotted curve represents the JNVF; beyond 2∙JNVF detuning becomes obvious. Most dyads in 12-TET are tuned so far from just intonation that they exceed the range of one JNVF. The dash-dotted straight lines represent the results of experiments with harmonic complex tones (3 dB/8ve spectral roll-off). In these experiments, listeners adjusted the upper tone of three dyads (M3, M6, P8) until the interval achieved the highest “pleasantness” while the lower tone fixed in the pitch range G3…F4 [2]. The resulting dyads are a flat m6 (878 ct), a flat octave (1194 ct), a sharp M3 (395 ct), a sharp M6 (893 ct) and a sharp octave (1208 ct). The dash-dotted curve represents the preferred average of pitch-dependent octave stretching for orchestra instruments with harmonic partials (Fig. 2 in [19]). It should be noted that for octaves, whose lower pitch is higher than B6, the octaves are enlarged by more than one JNVF in this tuning.

In the text

	Figure 5 Four variants to tune Celeste ranks in the Pipe Organ described as types (a–d) in the text.
In the text

	Figure 6 Tuning progression curves of the Celeste ranks in two different Accordions and a Harmona (an instrument with Accordion reeds, but with an electric blower rather than a hand-pumped bellow.).
In the text

	Figure 7 Collection of non-linear relations of beat frequency (BF) with pitch including literature data for a dry and a wet Celeste [27] and a proposed tuning progression curve (solid line, “lin-exp type”) for Accordions [25]. The dotted curve represents the exponential curve measured in the Harmona (Figure 4), while the JNVF serves as a comparison.
In the text

Figure 8 Proposed tuning progression curves for Celeste ranks in the range C2…C8 . The dotted curves represent one JNVF and 2∙JNVF. The dashed curve depicts the beat frequency (BF) proposed for Accordion Celestes [25], which rises linearly with pitch for low frequencies (indicated by the dashed straight line) and increases exponentially with pitch (Δf = 0.003∙f) in the treble where it approximates variant b). To facilitate a comparison between the effects of doubling the beat frequency slower than each octave (variants (b) and (c)), namely every 1.5 octaves, the dash-dotted curves have been included.

In the text

	Figure 9 Pitch dependency of the normalised spectral centroid c/f for both String ranks in this work, compared with a standard Open Flute [11] and a standard Open Diapason rank [11].
In the text

Figure 10 The diagrams depict the power of three added oscillations given by the equation 2sin(2π f(1)∙t) + √2 sin(2π f(2)∙t) + sin(2π f(3)∙t + 3π/4). In case (a), only two beat frequencies (2 Hz and 4 Hz) with the ratio 1:2 can be generated from the three given frequencies and both of them are visible as fractions of one period in the top chart. In case (b) three beat frequencies (1 Hz, 2 Hz, 3 Hz) are observed. The combination of them results in a different power distribution among the smaller peaks. The perception of such beats is uneven, reminiscent of stuttering and generally less pleasant than in the upper case. Setting the phase angle to zero results in a regular pattern. However, for independently oscillating tone generators (such as organ pipes) the phase angle remains a free parameter.

In the text

Figure 11 The last five bars of the Organ piece “Harmonies du soir” (S. Karg-Elert op. 72, reprinted from [30]) demonstrates the three different tone families of the pipe organ (indicated by colour bars) paired with their Celestes as well as several tuning options in (4). Yellow: Violin + Violin Celeste (tuned sharp), first green range: Horn Diapason + Horn Diapason Celeste (tuned sharp), blue: Quintadena Celeste (tuned flat) + Concert Flute Celeste (tuned sharp), final bar (green): Dulciana + Unda Maris (tuned flat). The tuning of the Violin Celeste is adjusted to type d) with +2.1 Hz at A4, while the two Flute ranks (blue) are tuned to type d) with –1.4 Hz and +1.4 Hz at A4, yielding 2.8 Hz beat frequency when used together without a rank tuned to unison pitch. The “green” Celestes ranks could not be retuned, but their beat frequency is about 2 Hz and more or less constant with pitch. The recorded sounds are part of the virtual pipe organ sample set “Paramount 450” (see [7] for details).

In the text