2nd Harmonic Tuned to 938.27Hz (MeV)

Spectral Response Analysis of a Classical Brass Church Bell with Second Harmonic at 938.27 Hz

This analysis reconfigures the spectral response of a classical brass church bell so that its second harmonic is precisely 938.27 Hz. Below, we recalculate the fundamental frequency, update the harmonic series, define the spectral response equation, provide a spectral plot, and discuss the role of the quality factor (Q). An HTML implementation is also included for visualization.

Fundamental Frequency Calculation

In bell acoustics, the second harmonic is typically the first overtone, which is twice the fundamental frequency. Given the second harmonic as 938.27 Hz, we calculate the fundamental frequency (( f )) as follows:

[ f = \frac{938.27}{2} = 469.135 , \text{Hz} ]

Thus, the fundamental frequency of the bell is 469.135 Hz. We assume, for simplicity, that the bell’s partials are harmonic multiples of this fundamental frequency (i.e., ( f, 2f, 3f, \ldots )), although real bells may exhibit slight inharmonicity.

Harmonic Series and Amplitudes

The first ten partials are integer multiples of the fundamental frequency, ( f_n = n \times 469.135 ) Hz, where ( n ) is the harmonic number. Their frequencies are:

1. ( 1 \times 469.135 = 469.135 , \text{Hz} ) (fundamental)
2. ( 2 \times 469.135 = 938.27 , \text{Hz} ) (second harmonic, as specified)
3. ( 3 \times 469.135 = 1407.405 , \text{Hz} )
4. ( 4 \times 469.135 = 1876.54 , \text{Hz} )
5. ( 5 \times 469.135 = 2345.675 , \text{Hz} )
6. ( 6 \times 469.135 = 2814.81 , \text{Hz} )
7. ( 7 \times 469.135 = 3283.945 , \text{Hz} )
8. ( 8 \times 469.135 = 3753.08 , \text{Hz} )
9. ( 9 \times 469.135 = 4222.215 , \text{Hz} )
10. ( 10 \times 469.135 = 4691.35 , \text{Hz} )

The amplitude of each partial decreases inversely with the harmonic number, defined as ( A_n = \frac{1}{n} ). Thus, the amplitudes are:

• ( A_1 = 1.000 )
• ( A_2 = 0.500 )
• ( A_3 = 0.333 )
• ( A_4 = 0.250 )
• ( A_5 = 0.200 )
• ( A_6 = 0.167 )
• ( A_7 = 0.143 )
• ( A_8 = 0.125 )
• ( A_9 = 0.111 )
• ( A_{10} = 0.100 )

These amplitudes reflect the natural decay of higher harmonics, contributing to the bell’s timbre.

Spectral Response Equation

The spectral response is modeled as a sum of Lorentzian functions, each representing a resonance peak at the partial frequencies. The Lorentzian for the ( n )-th partial is:

[ L_n(f) = \frac{A_n}{(f - f_n)^2 + \left( \frac{f_n}{2 Q} \right)^2} ]

where:

• ( f ) is the frequency,
• ( f_n = n \times 469.135 , \text{Hz} ) is the resonant frequency,
• ( A_n = \frac{1}{n} ) is the amplitude,
• ( Q = 1000 ) is the quality factor, assumed constant across all partials.

The total spectral response is:

[ S(f) = \sum_{n=1}^{10} \frac{A_n}{(f - f_n)^2 + \left( \frac{f_n}{2 Q} \right)^2} ]

A ( Q ) of 1000 indicates narrow resonance peaks. For example, the full width at half maximum (FWHM) for the fundamental is:

[ \text{FWHM} = \frac{f_1}{Q} = \frac{469.135}{1000} \approx 0.469 , \text{Hz} ]

This narrow bandwidth ensures a sustained and clear tone.

Spectral Plot

To visualize the harmonic structure, we use a bar chart showing the frequencies and amplitudes of the first ten partials:

• Frequencies (Hz): 469.135, 938.27, 1407.405, 1876.54, 2345.675, 2814.81, 3283.945, 3753.08, 4222.215, 4691.35
• Amplitudes: 1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125, 0.111, 0.100

The Chart.js configuration for this plot is:

{

  "type": "bar",

  "data": {

    "labels": [469.135, 938.27, 1407.405, 1876.54, 2345.675, 2814.81, 3283.945, 3753.08, 4222.215, 4691.35],

    "datasets": [{

      "label": "Partials",

      "data": [1.000, 0.500, 0.333, 0.250, 0.200, 0.167, 0.143, 0.125, 0.111, 0.100],

      "backgroundColor": "blue"

    }]

  },

  "options": {

    "scales": {

      "x": {

        "title": {

          "display": true,

          "text": "Frequency (Hz)"

        }

      },

      "y": {

        "title": {

          "display": true,

          "text": "Amplitude"

        }

      }

    }

  }

}

This chart approximates the continuous spectral response with discrete bars, effectively illustrating the harmonic peaks.

Role of the Quality Factor (Q)

The quality factor ( Q = 1000 ) significantly influences the bell’s sound:

1. Long Decay Time: A high ( Q ) results in a prolonged resonance, allowing the bell to ring for an extended period.
2. Sharp Resonances: Narrow peaks enhance tonal clarity by reducing overlap between partials.
3. Rich Timbre: Well-defined partials reinforce the perceived pitch, often a virtual pitch derived from higher harmonics.

A high ( Q ) is thus critical for the bell’s characteristic sound.

HTML Visualization

The following HTML code uses Chart.js to render the spectral plot interactively:

  

  

  

  

This code produces a bar chart visualizing the updated spectral response.

Conclusion

To achieve a second harmonic of 938.27 Hz, the bell’s fundamental frequency is set to 469.135 Hz. The harmonic series, spectral response, and visualization have been updated accordingly, maintaining a quality factor of 1000 and amplitudes of ( 1/n ). This configuration ensures a sustained, clear, and musically rich tone, characteristic of a classical brass church bell, with the spectral plot providing a clear representation of its harmonic structure.