Conditions foe the summability by triangular methods of the cauchy product of two number series

I define a new set of microtonal scales based on the prime number series, and containing 41 to 91 pitches spread over the whole audible range, rather than subdividing the octave. I designed these scales for metaphorical purposes, and applied one within my performance piece Ubasuteyama (2008), written with Hazel Smith, for speaker, computer sound and digital processing. Simple timbres using partials bearing prime number ratios to their fundamental were also used to embody the scale. The scales and timbres will be amongst the subjects of cognitive studies of pitch combinations, of large and unbroken pitch intervals in melodies, and of the relation between scale and timbre.

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On the reliability of the Wolf number series for estimating long-term periodicities

Journal of Geophysical Research Atmospheres ◽

10.1029/ja082i007p01271 ◽

1977 ◽

Vol 82 (7) ◽

pp. 1271-1272 ◽

Cited By ~ 7

Author(s):

P. N. Mayaud

Keyword(s):

Wolf Number ◽

Number Series

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Sunspot Number Series Envelope and Phase

Australian Journal of Physics ◽

10.1071/ph851009 ◽

1985 ◽

Vol 38 (6) ◽

pp. 1009 ◽

Cited By ~ 19

Author(s):

RN Bracewell

Keyword(s):

Analytic Function ◽

Sunspot Number ◽

Hilbert Transform ◽

Cycle Length ◽

Time Varying ◽

Number Series ◽

Third Harmonic ◽

The Third ◽

Envelope Amplitude ◽

Time Varying Medium

The sunspot number series R( t) from 1700 to date is found to be representable by R( t) = I Jf' {Re( E( t) exp[i {wo t + ( t) I]) + U( t) 1 I, where Wo is the angular frequency corresponding to a period of 22 years, E(t) is the instantaneous envelope amplitude, (t) is the instantaneous phase of a complex time-varying analytic function, U(t) is an undulation of low amplitude and period about 30 (22-year) cycles and jy is a nonlinear operator whose main effect is to introduce a small amount of third harmonic (period about 7 years). The justification for the 22-year period is the known fact that the observable sunspot magnetic fields reverse polarity every 11 years or so at the time of sunspot minimum; the undulation has been demonstrated, and its period determined, in fossil records discovered by Williams; and the third harmonic is an expected consequence of minor nonlinearity in the dependence of the arbitrarily defined R( t) on the physical cause of sunspots. The algebraic representation is established by the Hilbert transform method of forming a complex analytic function as proposed by Gabor. The method reveals three obscuring features that may be alleviated as follows: use the alternating series R� (t) in which alternate II-year cycles take opposite signs, remove the third harmonic, and subtract the undulation. These justifiable steps remove artificial components, such as sum and difference frequencies, that are gratuitously and nonlinearly introduced by conventional Fourier analysis as applied to the rectified, or absolute, value of the 22-year oscillation. Then a complex envelope E( t) exp {i ( t)j can be discerned whose intrinsic behaviour can be studied to reveal statistics that bear on the physical origin of the solar cycle. The results favour a deep monochromatic oscillator whose influence is propagated to the observable surface via a time-varying medium. The r.m.s. value of the component of E(t) is 0�4 of the mean and the characteristic time is a century. Frequency analysis of the envelope does not support a 78-year period in the modulation noticed by Wolf. Both the statistical frequency distribution of the amplitude E( t) and its spectrum are subject to refinement by analysis of fossil solar records. The results do not favour the theory that the 22-year period is set by the natural frequency of a resonator with characteristic damping subject to random turbulent excitation. Also disfavoured is the theory of energy release at intervals determined by a relaxation process. Correlation has been found between the phase departure ~(t) from linear and envelope amplitude and attributed to propagation of the magnetic .cycles through a time-varying, such as a convecting, medium. A correlation not depending on Hilbert transform analysis is predicted between the reciprocal cycle length and envelope amplitude and found to� exist. Variability of the sunspot cycle length can be viewed as a Doppler shift due to propagation in a time-varying medium and the Wolf modulation then represents the concomitant intensity change. Agreement has been found between E(t) and '(t) but not explained. If the explanation is dispersion in the propagation of the assumed magnetic flux waves then there is a mode of oscillation. that has the characteristics required for the undulation U( t). Extra buoyancy possessed by the magnetic field of strong cycles accounts for the fast rise time of strong cycles.

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Several Populations of Sunspot Group Numbers – Resolving a Conundrum

10.5194/egusphere-egu21-282 ◽

2021 ◽

Author(s):

Leif Svalgaard

Keyword(s):

Solar Activity ◽

Sunspot Number ◽

Degrees Of Freedom ◽

18Th Century ◽

Number Series ◽

Solar Observations ◽

Group Sunspot Number ◽

Group Number ◽

Working Groups

<p>The long-standing disparity between the sunspot number record and the Hoyt and Schatten (1998, H&S) Group Sunspot Number series was initially resolved by the Clette et al. (2014) revision of the sunspot number and the group number series. The revisions resulted in a flurry of dissenting group number series while the revised sunspot number series was generally accepted. Thus, the disparity persisted and confusion reigned, with the choice of solar activity dataset continuing to be a free parameter. A number of workshops and follow-up collaborative efforts by the community have not yet brought clarity. We review here several lines of evidence that validate the original revisions put forward by Clette et al. (2014) and suggest that the perceived conundrum no longer need to delay acceptance and general use of the revised series. We argue that the solar observations constitute several distinct populations with different properties which explain the various discontinuities in the series. This is supported by several proxies: diurnal variation of the geomagnetic field, geomagnetic signature of the strength of the heliomagnetic field, and variation of radionuclides. The Waldmeier effect shows that the sunspot number scale has not changed over the last 270 years and a mistaken scale factor between observers Wolf and Wolfer explains the disparity beginning in 1882 between the sunspot number and the H&S reconstruction of the group number. Observations with replica of 18th century telescopes (with similar optical flaws) validate the early sunspot number scale; while a reconstruction of the group number with monthly resolution (with many more degrees of freedom) validate the size of Solar Cycle 11 given by the revised series that the dissenting series fail to meet. Based on the evidence at hand, we urge the working groups tasked with producing community-vetted and agreed upon solar activity series to complete their work expeditiously.</p>

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