Triangulharmonic series - Revision history

BudjarnLambeth at 20:13, 23 April 2023

2023-04-23T20:13:52Z

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	[[Category:Harmonic series]]		[[Category:Harmonic series]]
			[[Category:Xenharmonic series]]

BudjarnLambeth: Categorised this uncategorised page

2023-04-23T20:13:39Z

Categorised this uncategorised page

← Older revision		Revision as of 20:13, 23 April 2023
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			[[Category:Harmonic series]]

Cmloegcmluin at 00:35, 23 March 2021

2021-03-23T00:35:10Z

← Older revision		Revision as of 00:35, 23 March 2021
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	<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>		<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>

	Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are [~~https://en.xen.wiki/w/~~Generalized_superparticulars] ratios: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>		Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are [[Generalized_superparticulars\|superbiparticular]] ratios: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>

	The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.		The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.

Cmloegcmluin at 00:34, 23 March 2021

2021-03-23T00:34:42Z

← Older revision		Revision as of 00:34, 23 March 2021
Line 3:		Line 3:
	<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>		<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>

	Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are [https://~~forum~~.~~sagittal~~.~~org~~/~~viewtopic.php?p=863#p863 superbiparticular~~] ratios: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>		Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are [https://en.xen.wiki/w/Generalized_superparticulars] ratios: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>

	The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.		The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.

Cmloegcmluin at 18:18, 22 March 2021

2021-03-22T18:18:45Z

← Older revision		Revision as of 18:18, 22 March 2021
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	The next occurrence is at 121278 and 242556, which produces a 204 note scale. This scale and beyond are decreasingly likely to be of much interest.		The next occurrence is at 121278 and 242556, which produces a 204 note scale. This scale and beyond are decreasingly likely to be of much interest.

			== References ==

	<references />		<references />

Cmloegcmluin at 18:18, 22 March 2021

2021-03-22T18:18:14Z

← Older revision		Revision as of 18:18, 22 March 2021
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	<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>		<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>

	Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So ~~the~~ unreduced intervals ~~between pitches~~ are [https://forum.sagittal.org/viewtopic.php?p=863#p863 superbiparticular]: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>		Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So its unreduced intervals are [https://forum.sagittal.org/viewtopic.php?p=863#p863 superbiparticular] ratios: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>

	The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.		The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.

Cmloegcmluin: Created page with "The triangulharmonic series is a subset of the harmonic series: only the harmonics which are [https://en.wikipedia.org/wiki/Triangular_number triangular nu..."

2021-03-22T18:17:39Z

Created page with "The triangulharmonic series is a subset of the harmonic series: only the harmonics which are [https://en.wikipedia.org/wiki/Triangular_number triangular nu..."

New page

The triangulharmonic series is a subset of the [[Overtone_series|harmonic series]]: only the harmonics which are [https://en.wikipedia.org/wiki/Triangular_number triangular numbers]. It begins:

<math>1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ...</math>

Essentially, first you skip 1 harmonic, then 2 harmonics, then 3, then 4, then 5, etc. So the unreduced intervals between pitches are [https://forum.sagittal.org/viewtopic.php?p=863#p863 superbiparticular]: <span><math>\frac{3}{1}, \frac{4}{2}, \frac{5}{3}, \frac{6}{4}...</math></span>

The formula for the nth triangular number is <span><math>\frac{n^2 + n}{2}</math></span>.

== Triangulharmonic Modes ==

Infinite pairs of triangular numbers one of which is 2 times the other occur<ref>https://mathgarage.wordpress.com/2019/01/08/finding-a-triangular-number-that-is-twice-another-triangular-number</ref>. If you take a subsequence of consecutive triangular numbers from one of these to the other, you make an octave-repeating scale, like a [[Overtone_scale#Introduction_-_Modes_of_the_Harmonic_Series|mode of the overtone series]].

The first occurrence of such a pair is near the beginning: 3 and 6. But that doesn't make an interesting scale, because when assuming octave equivalence, it's only a single note.

=== First Triangulharmonic Mode ===

The next occurrence is at 105 and 210. This gives us a [https://en.wikipedia.org/wiki/Hexatonic_scale hexatonic] scale, a subset of the 105th harmonic mode:

<math>\frac{120}{105}, \frac{136}{105}, \frac{153}{105}, \frac{171}{105}, \frac{190}{105}, (\frac{210}{105})</math>

which reduces to:

<math>\frac{8}{7}, \frac{136}{105}, \frac{51}{35}, \frac{57}{35}, \frac{38}{21}, (\frac{2}{1})</math>

You could think of it like a [[Primodality#Neji|NEJI]] for [[6edo|6-EDO]]. This is the first triangulharmonic mode.

=== Second Triangulharmonic Mode ===

The next occurrence of such a pair of triangular numbers isn't until 3570 and 7140, which produces a 35 note scale with step sizes ranging from 40.7¢ at the bottom to 29.3¢ at the top. That would be called the second triangulharmonic mode.

=== Third Triangulharmonic Mode ===

The next occurrence is at 121278 and 242556, which produces a 204 note scale. This scale and beyond are decreasingly likely to be of much interest.

<references />