40edo - Revision history

Kili at 18:36, 26 May 2026

2026-05-26T18:36:25Z

← Older revision		Revision as of 18:36, 26 May 2026
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	[[File:40edo_esstemp.png\|thumb\|The 2.5.7/3.11/3 Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 13 and 19 such as 10:13:16:19 can be reached in several ways.]]		[[File:40edo_esstemp.png\|thumb\|The 2.5.7/3.11/3 Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 13 and 19 such as 10:13:16:19 can be reached in several ways.]]
	'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].		'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[Equal temperament\|equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

	40edo can be considered a [[straddle primes\|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s\|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80\|syntonic comma]] is mapped ''negatively'' in 40edo.		40edo can be considered a [[straddle primes\|straddle]]-3, or dual-3, system, as it has both the [[5edo]] fifth of 720{{c}}, and a very flat [[diatonic]] fifth at 690{{c}}, being the smallest 5n EDO to have a diatonic [[perfect fifth]]. 40edo's [[5L 2s\|native diatonic]] scale is nearly [[equiheptatonic]], with a [[hardness]] of 6:5; major and minor intervals of the scale differ by only 30{{c}}. In particular, the major third of the diatonic scale is 360{{c}} (essentially [[16/13]]), generally considered a high neutral or submajor third, and [[5/4]] is mapped not to the major third, but the ''augmented'' third, which implies that the [[81/80\|syntonic comma]] is mapped ''negatively'' in 40edo.

Lériendil: /* Tempered commas */

2026-05-21T03:08:25Z

Tempered commas

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	* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]		* [[456/455]] (abnobismic), equating (5/4)*(7/6) to [[19/13]]
	* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]		* [[540/539]], S12/S14 (swetismic), equating two 7/6s to [[15/11]]
	* [[1573/1568]] (lambeth), equating a stack of two 14/11s to 13/8		* [[1573/1568]], S11/S14 (lambeth), equating a stack of two 14/11s to 13/8
	* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5		* [[1728/1715]], S6/S7 (orwellismic), equating three 7/6s to 8/5
	* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant		* [[3584/3575]], S12/S15, setting the intervals 16/13, 5/4, and 14/11 equidistant

Lériendil: /* Intervals and notation */

2026-05-15T00:41:34Z

Intervals and notation

← Older revision		Revision as of 00:41, 15 May 2026
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	As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.		As 40edo's diatonic fifth is so flat, its native diatonic scale has a chroma of 1 step. Therefore, sharps and flats are one step, and extensions such as [[ups and downs]] therefore do no advantage to the notation; up to triple-sharps must therefore be used to notate all notes of 40edo.

	In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s on J.		In addition to the diatonic, another important notational scale is Orwell[9], generated by the subminor third 9\40. Orwell, being generated by 7/6, and reaching 8/5 in three steps and 12/11 in five, serves as the foundational scale of 40edo's harmony in the 2.5.7/3.11/3 subgroup, comprising its most accurate approximations to simple [[JI]]. By coincidence, the 9-note Orwell scale is also close to equalized with a chroma of 1\40, and therefore sharps and flats will be used to represent a 1-step inflection in Orwell as well as diatonic. Note however, that only double-sharps and flats are needed to represent 40edo's notes using Orwell[9] as a basis. Orwell will be notated with the nominals J through R forming the symmetric mode of 4L 5s (sLsLsLsLs) on J.

	40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.		40edo's approximations to JI will be provided in three separate subgroups, which are 2.5.7/3.11/3.13.19.23; a superset including intervals of 9, 21, 33, and 51 using the dual-3 interpretation; and the [[13-limit]] according to the patent val. [[Inconsistent]] intervals will be italicized, odd harmonics will be bolded, and approximations within 2 cents will be marked in brackets.

Vector at 21:07, 14 May 2026

2026-05-14T21:07:08Z

← Older revision		Revision as of 21:07, 14 May 2026
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	While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.		While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

	Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group, though for structural reasons it may be better to include the 3 regardless.		Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group, though for structural reasons it may be better to include the 3 regardless (and thus to use the patent val).

	As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).		As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).

Vector at 00:29, 11 May 2026

2026-05-11T00:29:03Z

← Older revision		Revision as of 00:29, 11 May 2026
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	While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.		While 40edo has two intervals that can be considered a perfect fifth, its [[patent]] 3/2 is the flat, diatonic one. The 7th harmonic is similar, with the [[7/4]] inherited from 5edo (960{{c}}) being a closer approximation compared to a very sharp mapping at 990{{c}}; as is the 11th. However, 40edo approximates 5/4 rather well, with its 390{{c}} interval, and due to being a multiple of [[10edo]] and [[4edo]], it represents the 13th and 19th harmonics through those EDOs' respective approximations.

	Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group.		Therefore, the case is not dissimilar to [[29edo]]'s treatment of harmonics 5, 7, 11, and 13, as 40edo's patent mappings of 3, 7, and 11 are relatively unambiguous, though damaged, and approximately equally flat. Combining this with primes 5, 13, 19, and 23, we find that 40edo approximates a rather broad [[subgroup]] of 2.5.7/3.11/3.13.19.23, and has a consistent slight sharp tendency for most of the basis elements in this group, though for structural reasons it may be better to include the 3 regardless.

	As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).		As 40edo approximates 9 better than it does 3, a slight extension of this group would be to treat 40edo as a dual-{3 7 11 17} tuning system, implying 9, 21, 33, and 51 as basis elements; this is the interpretation as a subset of [[80edo]]. Of course, the patent approximations can still be used, an interesting consequence of which is that [[6/5]] is mapped to the quarter-octave (300{{c}}), like it is in [[12edo]] (though note that this is not the best 6/5, the 330{{c}} interval being slightly closer).

Lériendil: /* 13edo-derived muddles */

2026-05-10T23:49:47Z

13edo-derived muddles

← Older revision		Revision as of 23:49, 10 May 2026
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	A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.		A notable scale of 40edo is generated by 3\40, stacking thrice to the Orwell generator of 7/6 and nine times to 8/5. The principal generated [[MOS]] is of 13 steps - twelve being length 3\40 and one being length 4\40, and serving as a [[well-temperament]] of [[13edo]] - and contains multiple instances of the 10:13:16:19 tetrad. As this MOS scale is quite nearly an EDO, one is incentivized to take further subsets of it that have useful melodic and harmonic properties - that is, [[MOS muddle]]s.

	As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. [https://scaleworkshop.plainsound.org/scale/~~JxxQLHZ5Q~~ The former sequence] is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.		As taking every 3 generators essentially gives Orwell, the most notable generator chains of 13edo reflected in 40edo in this manner include the stacks of 4\13 and 5\13, representing [[3L 4s]]/[[3L 7s]], and [[oneirotonic]] respectively. [https://scaleworkshop.plainsound.org/scale/_dyrsuMpS The former sequence] is an easy approach to incorporating 10:13:16:19 into a musically coherent scale, as it occurs even in the 7-note muddle if the 4\40 step is placed correctly. The latter provides a simulation of the oneirotonic scale in the largest EDO to formally lack one.

	== Multiples ==		== Multiples ==

Lériendil at 19:18, 4 May 2026

2026-05-04T19:18:42Z

← Older revision		Revision as of 19:18, 4 May 2026
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	[[File:40edo_esstemp.png\|thumb\|The Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 19 such as 10:13:16:19 can be reached in several ways.]]		[[File:40edo_esstemp.png\|thumb\|The 2.5.7/3.11/3 Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 13 and 19 such as 10:13:16:19 can be reached in several ways.]]
	'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].		'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

Lériendil at 17:11, 4 May 2026

2026-05-04T17:11:59Z

← Older revision		Revision as of 17:11, 4 May 2026
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	[[File:40edo_esstemp.png\|thumb\|The Guanyintet chain is central to the ~~harmony~~ of 40edo; additionally, harmonies of 19 such as 10:13:16:19 can be reached in several ways.]]		[[File:40edo_esstemp.png\|thumb\|The Guanyintet chain is central to the structure of 40edo; additionally, harmonies of 19 such as 10:13:16:19 can be reached in several ways.]]
	'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].		'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

Lériendil: added thumb image

2026-05-04T17:09:17Z

added thumb image

← Older revision		Revision as of 17:09, 4 May 2026
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			[[File:40edo_esstemp.png\|thumb\|The Guanyintet chain is central to the harmony of 40edo; additionally, harmonies of 19 such as 10:13:16:19 can be reached in several ways.]]
	'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].		'''40edo''', or 40 equal divisions of the octave (sometimes called '''40-TET''' or '''40-tone equal temperament'''), is the [[equal tuning]] featuring steps of (1200/40) = 30 [[cent]]s exactly, 40 of which stack to the perfect octave [[2/1]].

Lériendil: /* Deeptone */

2026-05-04T12:44:34Z

Deeptone

← Older revision		Revision as of 12:44, 4 May 2026
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	40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.		40edo's native diatonic scale, generated by its flat 690{{c}} fifth, has a step ratio of 6:5, making diatonic melody feel quite washed-out and indistinct compared to Pythagorean or even [[meantone]] diatonics, especially as interval qualities converge significantly towards [[7edo]], with 16/13 submajor thirds in place of 5/4 or 81/64.

	The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of ~~deeptone~~ makes many standard contrasts hard to display in comparison to other diatonic temperaments.		The 12-note chromatic generated by the fifth, [[7L 5s]], is somewhat perversely a very [[hard]] scale with steps of 5\40 and 1\40, which sound more like neutral seconds and commas than conventional semitones. This scale contains only a single ~5/4 interval, though it does contain a few 480{{c}} and 720{{c}} blackwood intervals to contrast the deeptone fifth, as well as two 13:16:19 chords. Still, the [[cluster]]ing nature of Deeptone makes many standard contrasts hard to display in comparison to other diatonic temperaments.

	==== Omnidiatonic/Diasem ====		==== Omnidiatonic/Diasem ====