The beauty of comparing harmonic intervals

Base Frequency in Hz: ()
Duration in seconds:   Wave:

Low Harmonic: High Harmonic :   Low Octave: High Octave:
 1st spiral:   /    2nd spiral:   /    3rd spiral:   /
 4th spiral:   /    5th spiral:   /    6th spiral:   /


Canvas size:   Spiral's line width (negative makes a non-waving line):

THE INTERFACE
The tone spiral is a way to visualize the relationships of musical intervals, specifically, intervals related to harmonics of up to 6 tones. It's a lot like a circular slide rule, in that equal angles correspond to equal interval distances.
Each whorl outward of the tone spiral represents one octave higher. All other intervals spiral out between the octaves.
You can specify up to 6 spirals, starting at any interval. The interval you specify will be transposed into the octave from 1/1 to 2/1. Specifying "0/1" will suppress the spiral.
This way, you can compare the harmonics of tones specified by their individual intervals.
LISTENING TO THE INTERVALS
You can click on an interval's "dot" and hear it as it relates to the Base Frequency, for a duration as set by the duration amount.
This feature is using HTML5's Audio API - you won't hear it if your browser doesn't support it!
If you want to compare two intervals, set the duration to something long enough to hear both. Remember that high harmonics translate to high frequencies, so you may need to drop the base frequency a few octaves. You can either type in that base frequency, or adjust it by octaves or semitones, or set it to A440.
You can also set up the duration of that synthesized tone in seconds, and which waveform to play in the selected drop-down.
This chart has intervals marked out (roughly) as they are in 12 tone equal temperament as "I", "IIb","II", etc. so you don't get too lost.
Preset Links
Normalized Ratios:
Play checks as a chord or Play checks as a strum, for: seconds
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A little theory in a box

A great amount of musical thought examines consonance and dissonance.
Musical compositions play these two basic concepts against each other using timbre, melody, harmony, loudness, phasing, and articulation effects.
Patterns of consonance and dissonance playing out through time, and reacting with memory and anticipation, create musical experience.
There's plenty of psychoacoustic data relating the measurable aspects of sound to what human ears can detect and understand. It generally states that humans ears can hear sounds in a certain frequency range and only if they are "loud enough" at each frequency.
They can show how close two tones' frequencies can be before they are perceived as the same. As the the tones' interval widens, it passes through a period of "roughness" and then dissonant distinction, and occasionally, a resolution to harmony. The ability to recognize consonance is rather dependent of the timbre and loudness of the tones in question. But all things being equal, two tones are perceived as being consonant if their frequencies are related to each other as a simple whole number ratio. This simple ratio of frequencies means that, after a relatively short period of time, the two tones' pressure waveforms will maintain a constant phase relationship. In effect, the two tones merge into one tone.

An explanation for the origin of diatonic and diatonic-like scales

Consider a raw sound source, with a series of pulses at a frequency "F0". Almost any repeating process can generate a series of pulses like this. It doesn't even have to be a pure pulse: often the high harmonic magnitudes are lower or missing.

If this pulse is run trough a tube or transmitted to a string of a certain length, a phenomenon called "comb filtering" occurs, where the frequencies of harmonics of the natural frequency of the resonator ("R0") are emphasized, while other frequencies are attenuated or suppressed. Thus, when the "excitation" frequency F0 is not related to the resonant frequency R0, the resulting wave, played through the resonator, is quieter. The resulting comb filtered sound will appear as a timbre of its own, taken from the energy in the excitation signal.

The harmonics of the resonator are determined by its length (and the speed of sound in the resonating medium). The shorter the resonator, the higher the resonant frequencies. An equivalent phenomenon happens if you keep the resonator at a standard length and change the excitation frequency. As the two frequencies change, different harmonics will be emphasized.
If you make that excitation frequency low, the higher harmonics will be more audible.

And here is the magic part: if you feed an increasing excitation signal into three resonators, one at 4/3 of R0, R0, and 3/2 of R0, the harmonics of each resonator will be excited, and, if most of the harmonic content of the excitation signal is below the 7th harmonic, it creates as part of the series of excited, resonated frequencies, a familiar diatonic scale, each tube resonating in this order:

Harmonics excited in each tube:
 1/19/85/44/33/25/315/82/1
3/2.3..1,2,4.5.
1/11,2,4.5.3,6.(7)(8)
4/33,6..1,2,4.5.3,6

You can see this visualized here

Cents	Ratio	12TET interval +/- cents
0		1/1		I +0	
203.91	9/8		II +3.91	
386.31	5/4		III -13.69	
498.04	4/3		IV -1.96	
701.95	3/2		V +1.95	
884.35	5/3		VI -15.65	
1088.26	15/8	VII -11.74	
You can see that the unison and fifth are highly reenforced in this system. If you expand the audible harmonics up to the 10th harmonic, you get this expanded scale:
4/3,1/1,3/2 ->10th harmonic
Cents	Ratio	12TET interval +/- cents
0		1/1		I +0	
203.91	9/8		II +3.91	
266.87	7/6		IIIb -33.13	
386.31	5/4		III -13.69	
470.78	21/16	IV -29.22	
498.04	4/3		IV -1.96	
701.95	3/2		V +1.95	
884.35	5/3		VI -15.65	
905.86	27/16	VI +5.86	
968.82	7/4		VIIb -31.18	
1088.26	15/8	VII -11.74	
or, if you add resonators for the 5/4 and 4/5 to those of the 3/2 and 4/3, you get this dense scale:
8/5,4/3,1/1,3/2,5/4
Cents	Ratio	12TET interval +/- cents
0		1/1		I +0
203.91	9/8		II +3.91	
315.64	6/5		IIIb +15.64	
386.31	5/4		III -13.69	
498.04	4/3		IV -1.96	
701.95	3/2		V +1.95	
772.62	25/16	VIb -27.38	
813.68	8/5		VIb +13.68	
884.35	5/3		VI -15.65	
1088.26	15/8	VII -11.74	

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