Theoretical Pipe Variations

A set of characteristics of pipe spectra, useful in categorizing actual pipe sounds as well as synthesizing them.

Frequency Variations

• Typical large organ range is from 32' to 2' stops, which puts us around C0 up through C9, or 9 octaves and a note. This is 109 notes.
• For a given note, there are several possible mutation frequencies that pipes could have that would not fall on perfect equal temperament notes:
  • Harmonic 1: Unison (that is, no mutation) (N+0.00)
  • Harmonic 3: Twelfth (N+19.02)
  • Harmonic 5: Seventeenth (N+27.86)
  • Harmonic 6: Nineteenth (N+31.02)
  • Harmonic 7: Twentieth (N+33.69)
  • Harmonic 9: Twenty-third (N+38.04)
• For each mutation of each note, we could be slightly detuned below, perfect, or slightly detuned above, yielding 3 tunings. This will allow things like Voix Celeste, etc. In reality, the detuned mutations would only be used for certain ranks, against certain standard string-like ranks. Most Mixtures would never be detuned, etc. But, the variations could exist. The detuned pipes will probably just be by a couple of cents from the mutated frequency.

These combinations yield 1962 fundamental frequencies that we have to generate pipes for, possibly a handful more for multiple octave mutations that wouldn't normally fall in the C0-C9 range (meaning we'd need unmutated notes up to C12, which at 67kHz is well outside human hearing...)

Spectrum Characteristics

• A given pipe can be open (all harmonics present), stopped perfectly (no even harmonics), or stopped imperfectly (even harmonics severely diminished, say at 1/4 of what it would be if open) (U, Sp, Si)
• A given pipe can be normally blown (all harmonics present), perfectly overblown (fundamental completely missing), or imperfectly overblown (fundamental diminished, say at 1/4 of what it would be if normally blown) (B, Op, Oi)
• A given pipe can be coherent, or have rotating phase shifts (each harmonics is 90 degrees out of phase with previous and next harmonic in the series, so 0, 90, 180, 270, 0..., like a triangle wave) (C, A)
• A given pipe can have harmonic decay in one of multiple modes, where the relative strength of the kth harmonics is k^(-d), where d is one of (Xd):
  • 00, or 0: uniform all the way to the cutoff frequency... not very realistic...
  • 05, or 1/2: subharmonic, slower than a perfect harmonic series
  • 10, or 1: harmonic, a perfect harmonic series. Sawtooth and square waves fall here.
  • 15, or 3/2: subsuperharmonic, slightly faster than a perfect harmonic series
  • 20, or 2: superharmonic, faster than a perfect harmonic series. Like a triangle wave.

Bandwidth Limitations

The spectrum for a pipe can be bandwidth limited as well. All synthesized waveforms for sound should be limited to 22050Hz at a maximum. Cutoff harmonics worth trying are:

• Pure (n=0, perfect). No real transition zone here. Fundamental only, everything else is nonexistant. L0p is just notational convenience.
• n=3. Normal strength fundamental, then diminishing begins.
• n=5
• n=13
• n=37
• None (n=0, imperfect). No cutoff here, all harmonics are full strength, up to the 22050Hz cutoff that everything has. L0i is just notational conveneience.

A bandwidth limitation can be perfect or imperfect (Lnp, Lni). A perfect limitation has a three harmonic transition zone, where the strength of the harmonic before the cutoff is 85% what it would have been, unlimited, the strength of the harmonic at the cutoff is 50%, and the strength of the harmonic after the cutoff is 15%. All harmonics below these are at 100%, and all harmonics above these are at 0%. An imperfect limitation also has a three harmonic transition zone, where strength well before cutoff is 100%, right before cutoff is 81.25%, at cutoff is 62.5%, right after cutoff is 43.75%, and everything above that is at 25%.

Conclusions

Given all that, we have:

• N+M+D determine fundamental frequency, of which there appear to be just under 2000 reasonable ones.
• Xd + U/Sp/Si + B/Op/Oi + C/A determine pipe family, of which there appear to be about 90.
• Lnp/Lni determine pipe color, of which there are 10.

90*10 yeilds 900 rank classes. If each fundamental frequency is represented in each rank class, we would have about 1.8 million different theoretical pipes.

For convenience sake, these theoretical ranks will be named as follows:

XddxxyyzLnq_Mht

• Xdd is one of the decay modes, X00, X05, X10, X15, or X20.
• xx is U, Sp, or Si to indicate any stoppage
• yy is B, Op, or Oi to indicate any overblowing
• z is C or A to indicate phasing
• Lnq indicates color, where n is the cutoff indicator, or 0, and q is either p or i to indicate perfect or imperfect limitation
• Mht indicates that the rank is of mutation harmonic h (1, 3, 5, 6, 7, 9 (possibly 2, 4, or 8 for high notes)), and t indicates either u for under, p for perfect, or o for over to refer to any detuning.

Each pipe file or dft will have the actual fundamental frequency in cents included in the name after another underscore. This will take into account the mutation or detuning. Its the actual frequency of harmonic gap in the spectrum.

So, an example might be:

X10SiOiAL37i_M1p.rank

This indicates a harmonically decaying, imperfectly stopped, imperfectly overblown rank of pipes with alternating phasing, being imperfectly limited above the 37th harmonic, perfectly tuned at unison.

We will likely have 16,200 basic ranks of pipes. Each pipe of mixtures would count as a different rank here.

New Theory

Ok, before we just generated all the pipes but that was crap. Way too much space and time. Now, dftgen only creates one pipe at a time. The new filename format is

XDxxxODxxxEDxxxFxxxExxxSxxxPx_Lllll_ffff.dft

Here, XD determines the decay fraction, 000 is no decay, 100 is harmonic decay, 200 is superharmonic, etc. OD is a linear decay factor for odd harmonics, ED is a linear decay factor for even harmonics. F is the fundamental strength, E is the even strength factor, S is the sustain strength (after the limitation cutoff harmonic). P determines the phasing. L determines the limitation harmonic, and the _ffff is a float that is the fundamental for this pipe.

Rank Creation

• Except for the the fundamental, the name of the pipe spectra is used to determine the rank. Another name can be given as a memory aid.
• We also can specify a delta, in cents, to detune the pipes from equal temperament.
• We specify a length class of the pipes, in fractional feet.
• The final thing we can specify how many notes to generate, and also if we want to add an octave below, or an octave above to allow for offset coupling.

All that will generate the appropriate set of pipes and the rank file to go with them.