Optimizing MOS scales for DR - Revision history

Inthar: Undo revision 2107 by Inthar (talk)

2025-12-30T04:50:23Z

Undo revision 2107 by Inthar (talk)

← Older revision		Revision as of 04:50, 30 December 2025
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	=== Example ===		=== Example ===
	We start by choosing the [[MOS scale]] and ~~formal octave~~, and the DR chord.		We start by choosing the [[MOS scale]] and equave, and the DR chord.

	For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.		For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Inthar at 04:35, 30 December 2025

2025-12-30T04:35:14Z

← Older revision		Revision as of 04:35, 30 December 2025
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	=== Example ===		=== Example ===
	We start by choosing the [[MOS scale]] and ~~equave~~, and the DR chord.		We start by choosing the [[MOS scale]] and formal octave, and the DR chord.

	For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.		For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Inthar: /* Optimizing a MOS for one DR chord */

2025-12-24T04:08:23Z

Optimizing a MOS for one DR chord

← Older revision		Revision as of 04:08, 24 December 2025
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	TODO: reword		TODO: reword

	A MOS can exactly tune only one DR chord with two non-free deltas. To find such a tuning, we derive and solve an algebraic ~~equatiom~~ for the generator.		A MOS can exactly tune only one DR chord with two non-free deltas. To find such a tuning, we derive and solve an algebraic equation for the generator.

	=== Example ===		=== Example ===

Inthar: /* Optimizing a MOS for one DR chord */

2025-12-24T04:08:12Z

Optimizing a MOS for one DR chord

← Older revision		Revision as of 04:08, 24 December 2025
Line 3:		Line 3:
	TODO: reword		TODO: reword

	A MOS can exactly tune only one DR chord with two non-free deltas.		A MOS can exactly tune only one DR chord with two non-free deltas. To find such a tuning, we derive and solve an algebraic equatiom for the generator.

	=== Example ===		=== Example ===

Inthar: /* Optimizing a MOS for multiple DR chords */

2025-12-24T03:25:46Z

Optimizing a MOS for multiple DR chords

← Older revision		Revision as of 03:25, 24 December 2025
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	== Optimizing a MOS for multiple DR chords ==		== Optimizing a MOS for multiple DR chords ==
	{{adv\|Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we choose a [[DR error measures\|DR error measure]] and we minimize the sum of squares of DR errors~~. For this article, let's choose logarithmic pairwise error~~.}}		{{adv\|Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we choose a [[DR error measures\|DR error measure]] and we minimize the sum of squares of DR errors.}}

	{{cat\|		{{cat\|

Inthar: /* Optimizing a MOS for one DR chord */

2025-12-24T03:10:26Z

Optimizing a MOS for one DR chord

← Older revision		Revision as of 03:10, 24 December 2025
Line 3:		Line 3:
	TODO: reword		TODO: reword

	A MOS can tune ~~exactly~~ one DR chord with two non-free deltas ~~can be tuned exactly~~.		A MOS can exactly tune only one DR chord with two non-free deltas.

	=== Example ===		=== Example ===

Inthar: /* Optimizing a MOS for multiple DR chords */

2025-12-24T03:09:39Z

Optimizing a MOS for multiple DR chords

← Older revision		Revision as of 03:09, 24 December 2025
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	== Optimizing a MOS for multiple DR chords ==		== Optimizing a MOS for multiple DR chords ==
	{{adv\|Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we minimize the ~~square~~ of DR errors.}}		{{adv\|Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we choose a [[DR error measures\|DR error measure]] and we minimize the sum of squares of DR errors. For this article, let's choose logarithmic pairwise error.}}

	{{cat\|		{{cat\|

Inthar at 03:08, 24 December 2025

2025-12-24T03:08:34Z

← Older revision		Revision as of 03:08, 24 December 2025
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	== Optimizing a MOS for multiple DR chords ==		== Optimizing a MOS for multiple DR chords ==
	{{adv\|~~Solve~~ a least-squares error problem; minimize the square of DR errors}}		{{adv\|Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we minimize the square of DR errors.}}

	{{cat\|		{{cat\|
			Atypical ratios
	Delta-rational harmony		Delta-rational harmony
	}}		}}

Inthar at 03:07, 24 December 2025

2025-12-24T03:07:41Z

← Older revision		Revision as of 03:07, 24 December 2025
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	== Optimizing a MOS for multiple DR chords ==		== Optimizing a MOS for multiple DR chords ==
	{{adv\|Solve a least-squares error problem; minimize the square of DR errors}}		{{adv\|Solve a least-squares error problem; minimize the square of DR errors}}

			{{cat\|
			Delta-rational harmony
			}}

Inthar: Created page with "{{expert}} == Optimizing a MOS for one DR chord == TODO: reword A MOS can tune exactly one DR chord with two non-free deltas can be tuned exactly. === Example === We start by choosing the MOS scale and equave, and the DR chord. For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord. Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meanto..."

2025-12-24T03:07:10Z

Created page with "{{expert}} == Optimizing a MOS for one DR chord == TODO: reword A MOS can tune exactly one DR chord with two non-free deltas can be tuned exactly. === Example === We start by choosing the MOS scale and equave, and the DR chord. For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord. Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meanto..."

New page

{{expert}}
== Optimizing a MOS for one DR chord ==
TODO: reword

A MOS can tune exactly one DR chord with two non-free deltas can be tuned exactly.

=== Example ===
We start by choosing the [[MOS scale]] and equave, and the DR chord.

For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are g4/4 -1, g-g4/4. This is because in meantone, 1/1, 3/2, 5/4 are 1, g, g4/4 respectively, so the deltas are identified by subtracting the each term with the one before it.

In this case, we want the difference between our deltas to become 1, so the delta signature will be +1+1.

To achieve this, we take the difference between the first two deltas and set it to zero, so (g4/4 -1) - (g-g4/4) = 0. Put in integer terms, it's g4 - 2g - 2 = 0. Solving for g, the only root that makes sense is g≈1.49453, which in cents is 695.630c. And thus, with this generator, we will have a DR ~4:5:6 meantone chord!

Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: ''g''4 + 2''g'' − 8 {{=}} 0 The latter equation has solution ''g'' {{=}} 1.4960 {{=}} 697.3¢.

If instead we chose a Schismic mapping, the deltas would be g8/8 -1, 2/g - g8/8, which gives a generator of 498.308c for 4:5:6.

== Optimizing a MOS for multiple DR chords ==
{{adv|Solve a least-squares error problem; minimize the square of DR errors}}